diff --git a/graphiques/CH3-SIMGAMMAGAUSS.csv b/graphiques/CH3-SIMGAMMAGAUSS.csv new file mode 100644 index 0000000..f25a8db --- /dev/null +++ b/graphiques/CH3-SIMGAMMAGAUSS.csv @@ -0,0 +1,1002 @@ +"","Temps","Prix" +"1",0,100 +"2",0.000719988132698157,100 +"3",0.000719988132698157,100.162025133946 +"4",1.14595797191896,100.162025133946 +"5",1.14595797191896,100.824966046001 +"6",1.32784224547605,100.824966046001 +"7",1.32784224547605,100.727669883526 +"8",1.66113207980994,100.727669883526 +"9",1.66113207980994,100.928786894497 +"10",5.34359820315648,100.928786894497 +"11",5.34359820315648,100.930760681521 +"12",5.84363304016895,100.930760681521 +"13",5.84363304016895,100.730181698112 +"14",6.15648892946118,100.730181698112 +"15",6.15648892946118,100.631014880029 +"16",6.32331095646652,100.631014880029 +"17",6.32331095646652,100.636969437846 +"18",6.82272960818223,100.636969437846 +"19",6.82272960818223,100.835346546454 +"20",7.1311000052793,100.835346546454 +"21",7.1311000052793,100.592691099354 +"22",7.13428116885096,100.592691099354 +"23",7.13428116885096,100.436163882211 +"24",7.15041865102238,100.436163882211 +"25",7.15041865102238,100.489739843708 +"26",7.37590196884696,100.489739843708 +"27",7.37590196884696,100.669561608621 +"28",7.67012042614077,100.669561608621 +"29",7.67012042614077,100.613746497844 +"30",7.76821099233034,100.613746497844 +"31",7.76821099233034,100.178730988894 +"32",7.77034543061228,100.178730988894 +"33",7.77034543061228,100.509149284046 +"34",7.82281619922248,100.509149284046 +"35",7.82281619922248,100.683911259243 +"36",8.04312131466111,100.683911259243 +"37",8.04312131466111,101.01759140828 +"38",8.22654658399415,101.01759140828 +"39",8.22654658399415,101.197676819293 +"40",8.97808851109157,101.197676819293 +"41",8.97808851109157,101.453696458073 +"42",8.99844862387488,101.453696458073 +"43",8.99844862387488,101.934975198409 +"44",9.10158323696821,101.934975198409 +"45",9.10158323696821,101.72587195009 +"46",9.10159325487976,101.72587195009 +"47",9.10159325487976,101.91928701266 +"48",11.3702138534419,101.91928701266 +"49",11.3702138534419,101.730100587102 +"50",12.7190026574907,101.730100587102 +"51",12.7190026574907,101.470407434547 +"52",14.7009751737469,101.470407434547 +"53",14.7009751737469,101.347921721727 +"54",15.2352816726132,101.347921721727 +"55",15.2352816726132,101.234212821188 +"56",15.2644954485092,101.234212821188 +"57",15.2644954485092,101.570166975435 +"58",15.8527144569417,101.570166975435 +"59",15.8527144569417,101.587485754386 +"60",15.8914514079836,101.587485754386 +"61",15.8914514079836,102.148749375339 +"62",16.0301926022572,102.148749375339 +"63",16.0301926022572,102.305931450697 +"64",16.0578441244736,102.305931450697 +"65",16.0578441244736,102.398414860262 +"66",16.0766904077548,102.398414860262 +"67",16.0766904077548,102.658165861338 +"68",16.7413180703984,102.658165861338 +"69",16.7413180703984,102.523130320799 +"70",18.271984917225,102.523130320799 +"71",18.271984917225,102.687238702931 +"72",19.6897428652268,102.687238702931 +"73",19.6897428652268,102.596982612121 +"74",20.1735754346584,102.596982612121 +"75",20.1735754346584,102.318557290302 +"76",20.1741188062209,102.318557290302 +"77",20.1741188062209,102.711577019815 +"78",20.1741192902794,102.711577019815 +"79",20.1741192902794,102.526878214741 +"80",20.1985626287109,102.526878214741 +"81",20.1985626287109,102.705297508211 +"82",20.2156223868917,102.705297508211 +"83",20.2156223868917,102.584608614598 +"84",20.458788632231,102.584608614598 +"85",20.458788632231,102.364038199106 +"86",21.7923326055737,102.364038199106 +"87",21.7923326055737,102.022317391175 +"88",21.8379790276113,102.022317391175 +"89",21.8379790276113,101.996362079192 +"90",21.8395357305824,101.996362079192 +"91",21.8395357305824,101.842620315322 +"92",21.9614430379367,101.842620315322 +"93",21.9614430379367,101.898019701615 +"94",23.5081676146297,101.898019701615 +"95",23.5081676146297,101.728369488738 +"96",23.5593423841574,101.728369488738 +"97",23.5593423841574,101.656217095133 +"98",24.9223916874489,101.656217095133 +"99",24.9223916874489,101.088075157866 +"100",24.9523379971733,101.088075157866 +"101",24.9523379971733,101.073780971747 +"102",26.7149409950213,101.073780971747 +"103",26.7149409950213,101.65478648016 +"104",28.6924866961458,101.65478648016 +"105",28.6924866961458,101.46931044942 +"106",29.0133116134858,101.46931044942 +"107",29.0133116134858,101.389932120955 +"108",29.4708992825933,101.389932120955 +"109",29.4708992825933,101.351870182856 +"110",29.5454979873755,101.351870182856 +"111",29.5454979873755,101.025498031558 +"112",29.5903209678279,101.025498031558 +"113",29.5903209678279,100.991189778656 +"114",30.1921688244166,100.991189778656 +"115",30.1921688244166,100.786094701358 +"116",32.0648322389694,100.786094701358 +"117",32.0648322389694,100.341812622297 +"118",32.2955952856784,100.341812622297 +"119",32.2955952856784,100.449008713116 +"120",32.4834501207111,100.449008713116 +"121",32.4834501207111,100.397845022733 +"122",32.4972106473332,100.397845022733 +"123",32.4972106473332,100.330428742282 +"124",33.4858066962176,100.330428742282 +"125",33.4858066962176,100.569436897857 +"126",33.6630777099852,100.569436897857 +"127",33.6630777099852,100.520758293708 +"128",33.6732873726573,100.520758293708 +"129",33.6732873726573,100.363460833222 +"130",33.6782361362639,100.363460833222 +"131",33.6782361362639,100.758877458567 +"132",34.1195134178812,100.758877458567 +"133",34.1195134178812,100.158051658509 +"134",35.4397604813055,100.158051658509 +"135",35.4397604813055,99.9445409143547 +"136",36.0622421404576,99.9445409143547 +"137",36.0622421404576,99.7834018771209 +"138",36.1425092803494,99.7834018771209 +"139",36.1425092803494,99.9209750399512 +"140",36.477138355161,99.9209750399512 +"141",36.477138355161,99.6612981599757 +"142",36.6400398933603,99.6612981599757 +"143",36.6400398933603,99.6416022490182 +"144",36.6500202462516,99.6416022490182 +"145",36.6500202462516,99.0609287896255 +"146",37.0133571920176,99.0609287896255 +"147",37.0133571920176,98.9634260068622 +"148",37.9727285782478,98.9634260068622 +"149",37.9727285782478,98.5195069326089 +"150",38.0637484850072,98.5195069326089 +"151",38.0637484850072,98.4313722197006 +"152",40.3397903590365,98.4313722197006 +"153",40.3397903590365,98.6796735697588 +"154",40.3399975900068,98.6796735697588 +"155",40.3399975900068,98.6573335710121 +"156",40.3580510026491,98.6573335710121 +"157",40.3580510026491,98.5277481167292 +"158",40.8541127622594,98.5277481167292 +"159",40.8541127622594,98.8588656206599 +"160",40.8562539388624,98.8588656206599 +"161",40.8562539388624,99.3204774038521 +"162",40.8798921675571,99.3204774038521 +"163",40.8798921675571,99.4701709461309 +"164",41.5129991617353,99.4701709461309 +"165",41.5129991617353,99.3795980477816 +"166",41.5240693610811,99.3795980477816 +"167",41.5240693610811,99.586558291884 +"168",41.845732870841,99.586558291884 +"169",41.845732870841,99.5932057247737 +"170",42.6167095106289,99.5932057247737 +"171",42.6167095106289,99.6024459390569 +"172",42.6221083313092,99.6024459390569 +"173",42.6221083313092,99.5830947237184 +"174",42.7538811241523,99.5830947237184 +"175",42.7538811241523,99.4416634722 +"176",42.8172582265542,99.4416634722 +"177",42.8172582265542,99.4935661760613 +"178",42.8245945531865,99.4935661760613 +"179",42.8245945531865,99.1705444628276 +"180",42.8596968853159,99.1705444628276 +"181",42.8596968853159,99.1654296753911 +"182",43.8646605321094,99.1654296753911 +"183",43.8646605321094,99.4481810978207 +"184",43.9507325732908,99.4481810978207 +"185",43.9507325732908,99.8237953404081 +"186",44.9348311313217,99.8237953404081 +"187",44.9348311313217,99.5830836553512 +"188",44.9727914107315,99.5830836553512 +"189",44.9727914107315,99.3807356890061 +"190",45.0119383841678,99.3807356890061 +"191",45.0119383841678,99.7757717430826 +"192",45.0132631076167,99.7757717430826 +"193",45.0132631076167,99.7121815255067 +"194",45.2900938316073,99.7121815255067 +"195",45.2900938316073,99.5219937094718 +"196",45.2965785915418,99.5219937094718 +"197",45.2965785915418,99.8615396977596 +"198",45.4739102668492,99.8615396977596 +"199",45.4739102668492,100.025617232359 +"200",46.024060358217,100.025617232359 +"201",46.024060358217,100.05222099957 +"202",46.1121099261557,100.05222099957 +"203",46.1121099261557,100.005245395173 +"204",46.1143366010321,100.005245395173 +"205",46.1143366010321,100.337623287571 +"206",46.3116587452357,100.337623287571 +"207",46.3116587452357,100.211327139594 +"208",46.7639250064278,100.211327139594 +"209",46.7639250064278,100.215024192824 +"210",47.2491650785816,100.215024192824 +"211",47.2491650785816,100.400902925792 +"212",47.9276668600945,100.400902925792 +"213",47.9276668600945,100.408513111695 +"214",47.9283426759815,100.408513111695 +"215",47.9283426759815,100.519121157885 +"216",48.0854129467997,100.519121157885 +"217",48.0854129467997,100.525838384374 +"218",48.8176060710966,100.525838384374 +"219",48.8176060710966,100.181203623532 +"220",49.1562237847937,100.181203623532 +"221",49.1562237847937,99.8113481585399 +"222",49.5821452743037,99.8113481585399 +"223",49.5821452743037,99.475617756969 +"224",50.1501457821589,99.475617756969 +"225",50.1501457821589,99.6057792170918 +"226",50.3850067979033,99.6057792170918 +"227",50.3850067979033,99.7463794202741 +"228",50.4004729218816,99.7463794202741 +"229",50.4004729218816,100.025583936126 +"230",50.413001865463,100.025583936126 +"231",50.413001865463,100.055494869298 +"232",50.4885913344814,100.055494869298 +"233",50.4885913344814,99.831179623552 +"234",50.5349510768367,99.831179623552 +"235",50.5349510768367,99.6016230412913 +"236",50.6893444917358,99.6016230412913 +"237",50.6893444917358,99.4210038511918 +"238",52.4655880980322,99.4210038511918 +"239",52.4655880980322,99.0824632541606 +"240",52.50275937663,99.0824632541606 +"241",52.50275937663,99.2310512780639 +"242",52.6809807350118,99.2310512780639 +"243",52.6809807350118,99.1998040168633 +"244",53.9820251596862,99.1998040168633 +"245",53.9820251596862,98.7167717316166 +"246",56.1783575050804,98.7167717316166 +"247",56.1783575050804,98.81457772695 +"248",56.5407342516267,98.81457772695 +"249",56.5407342516267,98.7561390235546 +"250",57.1339157964347,98.7561390235546 +"251",57.1339157964347,98.6980388194377 +"252",57.2283331411447,98.6980388194377 +"253",57.2283331411447,98.7517170237811 +"254",57.2604403521564,98.7517170237811 +"255",57.2604403521564,98.7158974336602 +"256",57.3229048738097,98.7158974336602 +"257",57.3229048738097,98.3217019981636 +"258",57.4176274575992,98.3217019981636 +"259",57.4176274575992,97.8149616102364 +"260",61.8949222511747,97.8149616102364 +"261",61.8949222511747,97.8922138283196 +"262",62.9396351462508,97.8922138283196 +"263",62.9396351462508,98.1633064284921 +"264",64.4952213151433,98.1633064284921 +"265",64.4952213151433,98.2699226874911 +"266",64.6020643352388,98.2699226874911 +"267",64.6020643352388,98.4809667707387 +"268",65.3024745056854,98.4809667707387 +"269",65.3024745056854,98.3724265156954 +"270",65.4561206017453,98.3724265156954 +"271",65.4561206017453,98.6187034947076 +"272",65.5138986437635,98.6187034947076 +"273",65.5138986437635,98.7280486635978 +"274",65.7863328299992,98.7280486635978 +"275",65.7863328299992,98.4348320508354 +"276",65.9387398889294,98.4348320508354 +"277",65.9387398889294,98.2021854813762 +"278",66.0972971484421,98.2021854813762 +"279",66.0972971484421,97.891966450629 +"280",66.2532818793164,97.891966450629 +"281",66.2532818793164,98.1416189436532 +"282",66.6747563676861,98.1416189436532 +"283",66.6747563676861,98.4384494051809 +"284",68.697773949917,98.4384494051809 +"285",68.697773949917,98.445542013548 +"286",70.5649366968052,98.445542013548 +"287",70.5649366968052,98.7010804368015 +"288",70.5869297687509,98.7010804368015 +"289",70.5869297687509,98.1003948474195 +"290",71.1445967677769,98.1003948474195 +"291",71.1445967677769,97.9328247756465 +"292",71.1452705066867,97.9328247756465 +"293",71.1452705066867,98.1750693236743 +"294",71.159848833635,98.1750693236743 +"295",71.159848833635,98.3073776162841 +"296",71.2354996360603,98.3073776162841 +"297",71.2354996360603,97.9036742296927 +"298",71.2556520881329,97.9036742296927 +"299",71.2556520881329,97.986070059059 +"300",72.1644651037524,97.986070059059 +"301",72.1644651037524,98.2364477952978 +"302",72.3204540440295,98.2364477952978 +"303",72.3204540440295,98.3765601979046 +"304",75.979464824543,98.3765601979046 +"305",75.979464824543,98.3773386919351 +"306",76.0895668231868,98.3773386919351 +"307",76.0895668231868,98.5579767053178 +"308",76.4361718450675,98.5579767053178 +"309",76.4361718450675,98.6002601043247 +"310",78.0635652435199,98.6002601043247 +"311",78.0635652435199,98.8183818235005 +"312",78.1779947988539,98.8183818235005 +"313",78.1779947988539,98.9238767590722 +"314",78.2277655512106,98.9238767590722 +"315",78.2277655512106,99.1235335806155 +"316",79.6568717302812,99.1235335806155 +"317",79.6568717302812,98.6702800614203 +"318",79.7591024927365,98.6702800614203 +"319",79.7591024927365,98.6968295316937 +"320",79.9319118814403,98.6968295316937 +"321",79.9319118814403,98.9950885881525 +"322",80.142283342979,98.9950885881525 +"323",80.142283342979,98.6813994165489 +"324",80.442095050025,98.6813994165489 +"325",80.442095050025,98.6570177714324 +"326",80.6821031699262,98.6570177714324 +"327",80.6821031699262,98.5824336260168 +"328",80.692868544375,98.5824336260168 +"329",80.692868544375,98.6544632252978 +"330",81.6661634637829,98.6544632252978 +"331",81.6661634637829,98.6702930632244 +"332",82.0321225726176,98.6702930632244 +"333",82.0321225726176,98.4509669515179 +"334",82.4162437366812,98.4509669515179 +"335",82.4162437366812,98.5587372351529 +"336",82.9208968895288,98.5587372351529 +"337",82.9208968895288,98.9147735865481 +"338",83.0655626166954,98.9147735865481 +"339",83.0655626166954,99.0491033329015 +"340",83.0804720399956,99.0491033329015 +"341",83.0804720399956,98.9385440111567 +"342",83.721085585585,98.9385440111567 +"343",83.721085585585,99.2586020074176 +"344",83.9154851950506,99.2586020074176 +"345",83.9154851950506,99.0716817962147 +"346",83.9234392259092,99.0716817962147 +"347",83.9234392259092,99.0071812215389 +"348",84.2292954363594,99.0071812215389 +"349",84.2292954363594,98.8262897955131 +"350",84.3537401819849,98.8262897955131 +"351",84.3537401819849,98.802021359083 +"352",84.4320255609143,98.802021359083 +"353",84.4320255609143,98.7957988269972 +"354",85.4952687859085,98.7957988269972 +"355",85.4952687859085,98.3860302035911 +"356",85.4952701561962,98.3860302035911 +"357",85.4952701561962,98.4316414467673 +"358",85.5060406533293,98.4316414467673 +"359",85.5060406533293,98.8139645791018 +"360",85.648761264737,98.8139645791018 +"361",85.648761264737,98.6550642795492 +"362",86.9707877284635,98.6550642795492 +"363",86.9707877284635,98.8663389982632 +"364",87.5405423460478,98.8663389982632 +"365",87.5405423460478,99.1213235250201 +"366",88.2744842637848,99.1213235250201 +"367",88.2744842637848,99.4059732870284 +"368",88.277101235422,99.4059732870284 +"369",88.277101235422,99.3279853363731 +"370",88.3177579544092,99.3279853363731 +"371",88.3177579544092,99.279637774068 +"372",88.4205976150739,99.279637774068 +"373",88.4205976150739,99.7611170351762 +"374",88.6210975355555,99.7611170351762 +"375",88.6210975355555,99.8782534816936 +"376",89.243340310932,99.8782534816936 +"377",89.243340310932,99.443275094903 +"378",89.2633935939794,99.443275094903 +"379",89.2633935939794,99.4399250943675 +"380",89.2767027802172,99.4399250943675 +"381",89.2767027802172,99.3460175819969 +"382",90.0914098715853,99.3460175819969 +"383",90.0914098715853,99.2051212549512 +"384",90.6276993346357,99.2051212549512 +"385",90.6276993346357,98.6530343602227 +"386",90.6584691879107,98.6530343602227 +"387",90.6584691879107,98.8393806202156 +"388",91.0643678430386,98.8393806202156 +"389",91.0643678430386,98.7143354478185 +"390",91.0914954950617,98.7143354478185 +"391",91.0914954950617,99.0674078436602 +"392",91.210035276755,99.0674078436602 +"393",91.210035276755,99.3918801209606 +"394",91.7357804227938,99.3918801209606 +"395",91.7357804227938,99.4083534488491 +"396",92.2216993329872,99.4083534488491 +"397",92.2216993329872,99.2370905983619 +"398",92.3772594641926,99.2370905983619 +"399",92.3772594641926,98.8124874692134 +"400",92.7670254467996,98.8124874692134 +"401",92.7670254467996,98.4635820100942 +"402",92.8529516737822,98.4635820100942 +"403",92.8529516737822,98.0553832074172 +"404",92.8702422032107,98.0553832074172 +"405",92.8702422032107,98.3352033778181 +"406",93.1801875845089,98.3352033778181 +"407",93.1801875845089,98.1455414221113 +"408",94.1969403924719,98.1455414221113 +"409",94.1969403924719,97.6428746438881 +"410",94.1971167068395,97.6428746438881 +"411",94.1971167068395,97.7503484340218 +"412",94.8497168224117,97.7503484340218 +"413",94.8497168224117,98.0251323171179 +"414",94.8771461520192,98.0251323171179 +"415",94.8771461520192,98.2684508678668 +"416",94.8830943929778,98.2684508678668 +"417",94.8830943929778,98.4202654378264 +"418",95.2340203677926,98.4202654378264 +"419",95.2340203677926,98.4855842541695 +"420",95.5203235544225,98.4855842541695 +"421",95.5203235544225,98.6362854615594 +"422",95.6816670058475,98.6362854615594 +"423",95.6816670058475,98.3002867258694 +"424",95.971788168377,98.3002867258694 +"425",95.971788168377,98.2747328502649 +"426",96.0438769758354,98.2747328502649 +"427",96.0438769758354,98.0109132385454 +"428",96.3496615013265,98.0109132385454 +"429",96.3496615013265,97.7280886216879 +"430",96.8030909957999,97.7280886216879 +"431",96.8030909957999,97.4089869761394 +"432",96.8761758471887,97.4089869761394 +"433",96.8761758471887,97.5029387042908 +"434",98.5073237177379,97.5029387042908 +"435",98.5073237177379,97.5550294780917 +"436",98.7785030491102,97.5550294780917 +"437",98.7785030491102,97.8918734811326 +"438",98.8046646675605,97.8918734811326 +"439",98.8046646675605,98.0605091382505 +"440",99.4342865673039,98.0605091382505 +"441",99.4342865673039,98.0462024789532 +"442",99.4720817502467,98.0462024789532 +"443",99.4720817502467,98.1720651359499 +"444",99.6990245922173,98.1720651359499 +"445",99.6990245922173,98.0872480286976 +"446",102.098925864519,98.0872480286976 +"447",102.098925864519,98.3448987816382 +"448",104.058320450381,98.3448987816382 +"449",104.058320450381,98.2492071510407 +"450",104.315477661837,98.2492071510407 +"451",104.315477661837,98.1826530512132 +"452",104.351762673599,98.1826530512132 +"453",104.351762673599,98.0396807922561 +"454",104.677105517787,98.0396807922561 +"455",104.677105517787,98.3654955647314 +"456",104.721287183944,98.3654955647314 +"457",104.721287183944,98.6070537582168 +"458",104.869796213489,98.6070537582168 +"459",104.869796213489,98.6168110082516 +"460",105.324562956628,98.6168110082516 +"461",105.324562956628,98.4442879560059 +"462",107.179952411698,98.4442879560059 +"463",107.179952411698,98.0411672863409 +"464",107.18004494582,98.0411672863409 +"465",107.18004494582,98.2365465435797 +"466",108.590997290154,98.2365465435797 +"467",108.590997290154,98.3653155149375 +"468",109.097189064543,98.3653155149375 +"469",109.097189064543,98.3011384075768 +"470",112.161063379946,98.3011384075768 +"471",112.161063379946,98.2581689289623 +"472",112.895042656757,98.2581689289623 +"473",112.895042656757,98.4912514599809 +"474",113.138831919023,98.4912514599809 +"475",113.138831919023,98.8086887983949 +"476",113.977564689128,98.8086887983949 +"477",113.977564689128,98.7713944981491 +"478",115.583466012258,98.7713944981491 +"479",115.583466012258,98.4470343469847 +"480",115.623160133264,98.4470343469847 +"481",115.623160133264,98.4533281189535 +"482",115.652546873544,98.4533281189535 +"483",115.652546873544,98.5553157637173 +"484",115.780040885226,98.5553157637173 +"485",115.780040885226,98.8661274600851 +"486",115.836288855926,98.8661274600851 +"487",115.836288855926,98.903504006226 +"488",115.90054969864,98.903504006226 +"489",115.90054969864,98.7561604409004 +"490",116.207272603835,98.7561604409004 +"491",116.207272603835,98.6343930791721 +"492",116.223388211284,98.6343930791721 +"493",116.223388211284,98.9499598550182 +"494",117.155634705175,98.9499598550182 +"495",117.155634705175,99.227048922711 +"496",118.302293497267,99.227048922711 +"497",118.302293497267,99.117181969537 +"498",118.90765517412,99.117181969537 +"499",118.90765517412,99.0264393981211 +"500",119.464384825011,99.0264393981211 +"501",119.464384825011,98.7353250963945 +"502",119.492439854309,98.7353250963945 +"503",119.492439854309,98.496790625096 +"504",119.536909308008,98.496790625096 +"505",119.536909308008,98.3863008367223 +"506",119.803197260555,98.3863008367223 +"507",119.803197260555,98.6669060237861 +"508",122.275230102678,98.6669060237861 +"509",122.275230102678,98.6092178333121 +"510",122.906091559617,98.6092178333121 +"511",122.906091559617,98.8679819825408 +"512",122.96496821594,98.8679819825408 +"513",122.96496821594,99.1048695065317 +"514",123.165289180607,99.1048695065317 +"515",123.165289180607,98.7307730795272 +"516",123.360077118967,98.7307730795272 +"517",123.360077118967,98.6158356034952 +"518",124.29429398382,98.6158356034952 +"519",124.29429398382,98.2412707854087 +"520",124.977480029493,98.2412707854087 +"521",124.977480029493,98.3449637846965 +"522",124.99258049361,98.3449637846965 +"523",124.99258049361,97.9784827086003 +"524",125.401275544574,97.9784827086003 +"525",125.401275544574,98.2447686263156 +"526",125.63416742795,98.2447686263156 +"527",125.63416742795,98.3491266431356 +"528",126.039951380262,98.3491266431356 +"529",126.039951380262,98.4633570855577 +"530",126.226854301072,98.4633570855577 +"531",126.226854301072,98.7707020341304 +"532",129.37195020363,98.7707020341304 +"533",129.37195020363,98.6309772375366 +"534",129.475054926061,98.6309772375366 +"535",129.475054926061,98.7021151242603 +"536",129.788282070421,98.7021151242603 +"537",129.788282070421,98.8753651674275 +"538",129.791483080332,98.8753651674275 +"539",129.791483080332,98.8413384182126 +"540",130.286291227645,98.8413384182126 +"541",130.286291227645,98.9285572068687 +"542",133.133813975701,98.9285572068687 +"543",133.133813975701,99.3697502552305 +"544",133.360178545607,99.3697502552305 +"545",133.360178545607,99.5348685697559 +"546",133.505799818941,99.5348685697559 +"547",133.505799818941,99.4341860462885 +"548",133.548605280437,99.4341860462885 +"549",133.548605280437,99.4277076884499 +"550",134.768084730912,99.4277076884499 +"551",134.768084730912,99.5582028969521 +"552",135.02449545613,99.5582028969521 +"553",135.02449545613,99.4834561764454 +"554",135.728410150866,99.4834561764454 +"555",135.728410150866,99.2314209790219 +"556",135.999416833479,99.2314209790219 +"557",135.999416833479,99.0837734090399 +"558",136.123722937078,99.0837734090399 +"559",136.123722937078,99.3978414726494 +"560",137.758773081106,99.3978414726494 +"561",137.758773081106,99.4768847186724 +"562",137.870361007006,99.4768847186724 +"563",137.870361007006,99.119576522338 +"564",137.942005588682,99.119576522338 +"565",137.942005588682,98.9752394365639 +"566",138.000044515792,98.9752394365639 +"567",138.000044515792,98.9234023993412 +"568",138.465518761998,98.9234023993412 +"569",138.465518761998,99.039346669005 +"570",138.509108543843,99.039346669005 +"571",138.509108543843,98.5093346244395 +"572",138.812821721565,98.5093346244395 +"573",138.812821721565,98.3497455853244 +"574",139.551339052816,98.3497455853244 +"575",139.551339052816,98.7315751273369 +"576",139.580526719006,98.7315751273369 +"577",139.580526719006,98.8077296313706 +"578",141.980338547147,98.8077296313706 +"579",141.980338547147,98.6087132938833 +"580",142.638104041543,98.6087132938833 +"581",142.638104041543,98.4754663757733 +"582",142.675407790235,98.4754663757733 +"583",142.675407790235,98.2065966579351 +"584",142.675589032599,98.2065966579351 +"585",142.675589032599,98.2907414927258 +"586",143.210991212001,98.2907414927258 +"587",143.210991212001,98.3030688931575 +"588",143.664522665438,98.3030688931575 +"589",143.664522665438,98.2528639806531 +"590",143.897468877108,98.2528639806531 +"591",143.897468877108,98.1996736871124 +"592",144.197450874563,98.1996736871124 +"593",144.197450874563,98.3822971219119 +"594",146.207172158264,98.3822971219119 +"595",146.207172158264,98.5515734003268 +"596",147.180961428077,98.5515734003268 +"597",147.180961428077,98.7672154996229 +"598",147.21513382825,98.7672154996229 +"599",147.21513382825,98.9291374793926 +"600",147.215970336151,98.9291374793926 +"601",147.215970336151,98.876313404927 +"602",147.957088158948,98.876313404927 +"603",147.957088158948,98.5145974512429 +"604",148.06200330851,98.5145974512429 +"605",148.06200330851,98.7302529054662 +"606",149.543350145453,98.7302529054662 +"607",149.543350145453,98.4421128867958 +"608",151.044724096488,98.4421128867958 +"609",151.044724096488,98.3942112624113 +"610",154.749447735061,98.3942112624113 +"611",154.749447735061,98.3804708246973 +"612",154.7701995383,98.3804708246973 +"613",154.7701995383,98.8578295991866 +"614",154.807746758134,98.8578295991866 +"615",154.807746758134,98.9392483606608 +"616",154.879674567461,98.9392483606608 +"617",154.879674567461,98.7892539178956 +"618",155.970475079284,98.7892539178956 +"619",155.970475079284,98.882003325355 +"620",156.712906860453,98.882003325355 +"621",156.712906860453,98.6262039518153 +"622",157.35206104455,98.6262039518153 +"623",157.35206104455,98.6553343482794 +"624",157.624084468961,98.6553343482794 +"625",157.624084468961,98.4067665589072 +"626",158.085230880626,98.4067665589072 +"627",158.085230880626,98.6438736031917 +"628",158.100102538331,98.6438736031917 +"629",158.100102538331,98.8075075539538 +"630",158.162921657234,98.8075075539538 +"631",158.162921657234,99.0931863058419 +"632",158.293652643956,99.0931863058419 +"633",158.293652643956,99.1483560110716 +"634",158.343709120482,99.1483560110716 +"635",158.343709120482,98.7339627016704 +"636",158.390170494393,98.7339627016704 +"637",158.390170494393,99.001926713623 +"638",159.019658170229,99.001926713623 +"639",159.019658170229,99.3149554414833 +"640",160.325036556163,99.3149554414833 +"641",160.325036556163,99.1267191633822 +"642",160.838998091913,99.1267191633822 +"643",160.838998091913,99.162388857281 +"644",161.181780446552,99.162388857281 +"645",161.181780446552,99.4291012036941 +"646",161.648806089079,99.4291012036941 +"647",161.648806089079,99.4652394093362 +"648",162.298703758793,99.4652394093362 +"649",162.298703758793,99.2404909875618 +"650",162.689157386293,99.2404909875618 +"651",162.689157386293,99.4181147697991 +"652",162.689210022629,99.4181147697991 +"653",162.689210022629,99.1108398413446 +"654",162.776043301023,99.1108398413446 +"655",162.776043301023,98.8898355112988 +"656",163.884113462471,98.8898355112988 +"657",163.884113462471,98.6068259877873 +"658",164.07924785645,98.6068259877873 +"659",164.07924785645,98.7265842206392 +"660",164.535332595952,98.7265842206392 +"661",164.535332595952,98.6329127311049 +"662",164.566240329569,98.6329127311049 +"663",164.566240329569,98.6213120275934 +"664",164.784798746077,98.6213120275934 +"665",164.784798746077,98.8031395920901 +"666",164.787235046648,98.8031395920901 +"667",164.787235046648,98.9417814002695 +"668",165.013003241002,98.9417814002695 +"669",165.013003241002,98.7962630065119 +"670",166.006255874819,98.7962630065119 +"671",166.006255874819,98.8691707271507 +"672",166.115054003326,98.8691707271507 +"673",166.115054003326,99.027622896196 +"674",166.225004222289,99.027622896196 +"675",166.225004222289,99.1237635968435 +"676",166.226872442255,99.1237635968435 +"677",166.226872442255,99.0029461543319 +"678",166.252176450123,99.0029461543319 +"679",166.252176450123,98.951177890322 +"680",166.385263606125,98.951177890322 +"681",166.385263606125,98.3915549851899 +"682",166.522104201915,98.3915549851899 +"683",166.522104201915,98.783533924963 +"684",166.612767581871,98.783533924963 +"685",166.612767581871,98.7770114944986 +"686",167.462711394935,98.7770114944986 +"687",167.462711394935,99.0924173103675 +"688",167.608113852157,99.0924173103675 +"689",167.608113852157,99.2568334876752 +"690",168.456873938715,99.2568334876752 +"691",168.456873938715,99.5717310882375 +"692",169.014957179749,99.5717310882375 +"693",169.014957179749,99.7260957676333 +"694",169.037057484695,99.7260957676333 +"695",169.037057484695,99.8334658738118 +"696",169.116061018615,99.8334658738118 +"697",169.116061018615,99.867173104398 +"698",169.296411861912,99.867173104398 +"699",169.296411861912,99.7799968304527 +"700",170.789148950156,99.7799968304527 +"701",170.789148950156,99.7844041403902 +"702",171.14069797483,99.7844041403902 +"703",171.14069797483,99.8788777823705 +"704",171.227259853452,99.8788777823705 +"705",171.227259853452,99.7588677629184 +"706",171.235063050955,99.7588677629184 +"707",171.235063050955,99.7487012317388 +"708",171.448782295916,99.7487012317388 +"709",171.448782295916,99.5896923268465 +"710",171.44928199671,99.5896923268465 +"711",171.44928199671,99.3712800515492 +"712",171.468054509857,99.3712800515492 +"713",171.468054509857,99.0354719965802 +"714",171.529833468331,99.0354719965802 +"715",171.529833468331,99.0670177862845 +"716",172.321225041042,99.0670177862845 +"717",172.321225041042,99.03114342375 +"718",172.513462657477,99.03114342375 +"719",172.513462657477,98.6685765608159 +"720",173.0890785037,98.6685765608159 +"721",173.0890785037,98.5116603680547 +"722",173.980397088982,98.5116603680547 +"723",173.980397088982,98.3783417800062 +"724",174.933530539049,98.3783417800062 +"725",174.933530539049,98.48460850603 +"726",175.299902673538,98.48460850603 +"727",175.299902673538,98.3945138278905 +"728",175.315674914526,98.3945138278905 +"729",175.315674914526,98.5221015182717 +"730",175.35797971591,98.5221015182717 +"731",175.35797971591,98.5104383747457 +"732",175.368222599639,98.5104383747457 +"733",175.368222599639,98.7797284515395 +"734",176.254656503681,98.7797284515395 +"735",176.254656503681,98.8681180631604 +"736",176.451846964214,98.8681180631604 +"737",176.451846964214,99.3049278491883 +"738",176.464096140277,99.3049278491883 +"739",176.464096140277,99.0667590458512 +"740",176.468567832942,99.0667590458512 +"741",176.468567832942,98.5508082139531 +"742",176.483475221495,98.5508082139531 +"743",176.483475221495,98.4200304377287 +"744",177.01684718786,98.4200304377287 +"745",177.01684718786,98.2946806894175 +"746",177.152653935166,98.2946806894175 +"747",177.152653935166,98.0015221384835 +"748",178.035837659981,98.0015221384835 +"749",178.035837659981,97.7575046573398 +"750",179.635961697606,97.7575046573398 +"751",179.635961697606,98.0208879594937 +"752",179.760958999852,98.0208879594937 +"753",179.760958999852,97.7200578468522 +"754",179.770270094055,97.7200578468522 +"755",179.770270094055,97.9917085096786 +"756",181.547565606849,97.9917085096786 +"757",181.547565606849,97.8141962932306 +"758",182.300316875708,97.8141962932306 +"759",182.300316875708,97.9861695388214 +"760",182.301595787394,97.9861695388214 +"761",182.301595787394,97.5871388214659 +"762",182.472936101311,97.5871388214659 +"763",182.472936101311,97.5259560067576 +"764",182.492293019239,97.5259560067576 +"765",182.492293019239,97.5465409217246 +"766",182.555801539423,97.5465409217246 +"767",182.555801539423,97.4304692735906 +"768",182.584474143299,97.4304692735906 +"769",182.584474143299,97.8981369091239 +"770",183.934437540517,97.8981369091239 +"771",183.934437540517,97.9664769290091 +"772",183.961736864248,97.9664769290091 +"773",183.961736864248,98.0482527576504 +"774",184.678921617641,98.0482527576504 +"775",184.678921617641,98.112268818547 +"776",184.842090214081,98.112268818547 +"777",184.842090214081,98.6149224481111 +"778",185.663750781148,98.6149224481111 +"779",185.663750781148,98.6155547497989 +"780",186.352739461648,98.6155547497989 +"781",186.352739461648,98.1545677350383 +"782",188.516368777823,98.1545677350383 +"783",188.516368777823,98.6414054076187 +"784",190.610809107705,98.6414054076187 +"785",190.610809107705,98.3587638069812 +"786",190.920091912811,98.3587638069812 +"787",190.920091912811,98.4204397048117 +"788",191.037851172814,98.4204397048117 +"789",191.037851172814,98.5054219968441 +"790",194.837343632051,98.5054219968441 +"791",194.837343632051,98.5997115784385 +"792",197.27874918621,98.5997115784385 +"793",197.27874918621,98.6643571497594 +"794",198.697251075263,98.6643571497594 +"795",198.697251075263,99.1804228812821 +"796",199.39131083669,99.1804228812821 +"797",199.39131083669,98.9846293761305 +"798",199.739688637661,98.9846293761305 +"799",199.739688637661,99.0604883583346 +"800",200.766473909818,99.0604883583346 +"801",200.766473909818,98.9563859029752 +"802",200.78123721406,98.9563859029752 +"803",200.78123721406,99.355959568936 +"804",200.813435176323,99.355959568936 +"805",200.813435176323,99.5681950498984 +"806",202.2224893135,99.5681950498984 +"807",202.2224893135,99.5983806839523 +"808",202.558160463701,99.5983806839523 +"809",202.558160463701,99.4314840096284 +"810",202.608291227321,99.4314840096284 +"811",202.608291227321,99.6450056028323 +"812",202.646187230477,99.6450056028323 +"813",202.646187230477,99.732542037975 +"814",203.860678600087,99.732542037975 +"815",203.860678600087,99.6093033664572 +"816",204.74883521067,99.6093033664572 +"817",204.74883521067,99.3467153129521 +"818",204.898608842838,99.3467153129521 +"819",204.898608842838,98.7504350930671 +"820",204.900249093986,98.7504350930671 +"821",204.900249093986,98.7970039742502 +"822",205.707545438716,98.7970039742502 +"823",205.707545438716,98.9669298386876 +"824",206.447654210507,98.9669298386876 +"825",206.447654210507,99.1825341352944 +"826",207.038933696317,99.1825341352944 +"827",207.038933696317,99.0639924248145 +"828",207.853571858898,99.0639924248145 +"829",207.853571858898,99.1759887229995 +"830",208.207730176549,99.1759887229995 +"831",208.207730176549,98.8481386142265 +"832",208.613426838653,98.8481386142265 +"833",208.613426838653,98.9727899134121 +"834",209.362942595695,98.9727899134121 +"835",209.362942595695,99.2465628407643 +"836",209.395446017149,99.2465628407643 +"837",209.395446017149,99.479101319575 +"838",209.45152979587,99.479101319575 +"839",209.45152979587,99.3602534307713 +"840",211.750734453285,99.3602534307713 +"841",211.750734453285,98.99617043319 +"842",211.885603218456,98.99617043319 +"843",211.885603218456,98.8442919574591 +"844",212.821539139418,98.8442919574591 +"845",212.821539139418,98.599555221695 +"846",212.883792575592,98.599555221695 +"847",212.883792575592,98.2548421432749 +"848",213.092307310818,98.2548421432749 +"849",213.092307310818,98.4280485375657 +"850",214.880287449842,98.4280485375657 +"851",214.880287449842,98.3583305832688 +"852",214.883571117518,98.3583305832688 +"853",214.883571117518,97.8775759551154 +"854",216.146167347762,97.8775759551154 +"855",216.146167347762,97.5572153808458 +"856",216.206360910061,97.5572153808458 +"857",216.206360910061,97.1758079388879 +"858",218.394074461718,97.1758079388879 +"859",218.394074461718,97.0319104855879 +"860",218.407921862007,97.0319104855879 +"861",218.407921862007,96.7691942314372 +"862",221.05869149425,96.7691942314372 +"863",221.05869149425,96.8929469307365 +"864",221.813013528082,96.8929469307365 +"865",221.813013528082,96.8342757666065 +"866",224.092583019944,96.8342757666065 +"867",224.092583019944,96.7606082854581 +"868",224.718617682499,96.7606082854581 +"869",224.718617682499,96.5888330406101 +"870",229.972642318003,96.5888330406101 +"871",229.972642318003,96.57442218607 +"872",230.138774706646,96.57442218607 +"873",230.138774706646,96.7301372409338 +"874",231.65786166935,96.7301372409338 +"875",231.65786166935,97.3445731122476 +"876",231.865185828992,97.3445731122476 +"877",231.865185828992,97.2591707902805 +"878",231.870559712447,97.2591707902805 +"879",231.870559712447,97.3970604071055 +"880",231.918288207536,97.3970604071055 +"881",231.918288207536,97.1644931594033 +"882",232.012265754826,97.1644931594033 +"883",232.012265754826,97.1403330915181 +"884",232.470397587601,97.1403330915181 +"885",232.470397587601,97.3088023674225 +"886",233.030098499607,97.3088023674225 +"887",233.030098499607,97.3784787840795 +"888",233.115736740437,97.3784787840795 +"889",233.115736740437,97.6166404583754 +"890",233.799970403829,97.6166404583754 +"891",233.799970403829,97.6185852952567 +"892",234.161287739393,97.6185852952567 +"893",234.161287739393,97.1177471958935 +"894",235.472923371308,97.1177471958935 +"895",235.472923371308,97.1753620515914 +"896",235.491640524452,97.1753620515914 +"897",235.491640524452,97.2163827849633 +"898",235.503885058575,97.2163827849633 +"899",235.503885058575,97.4897886420785 +"900",236.346997286426,97.4897886420785 +"901",236.346997286426,97.5921094894543 +"902",237.453118527223,97.5921094894543 +"903",237.453118527223,97.2380155696792 +"904",237.560150863554,97.2380155696792 +"905",237.560150863554,97.3502429500352 +"906",238.487395998901,97.3502429500352 +"907",238.487395998901,96.7362488260013 +"908",239.237657898076,96.7362488260013 +"909",239.237657898076,96.4095621496 +"910",239.604674805511,96.4095621496 +"911",239.604674805511,96.3047248216684 +"912",239.688564592431,96.3047248216684 +"913",239.688564592431,95.8667894882905 +"914",241.112640031265,95.8667894882905 +"915",241.112640031265,95.900777491278 +"916",241.159887345666,95.900777491278 +"917",241.159887345666,95.7113188789728 +"918",242.839445040693,95.7113188789728 +"919",242.839445040693,95.9489623980972 +"920",243.08626368566,95.9489623980972 +"921",243.08626368566,95.8568219424094 +"922",243.093944289318,95.8568219424094 +"923",243.093944289318,96.190155020117 +"924",243.382934927133,96.190155020117 +"925",243.382934927133,95.9355616633907 +"926",243.863077276542,95.9355616633907 +"927",243.863077276542,96.3129637126689 +"928",245.15915754303,96.3129637126689 +"929",245.15915754303,96.0563544621509 +"930",245.180300865581,96.0563544621509 +"931",245.180300865581,95.793664718757 +"932",247.406693036346,95.793664718757 +"933",247.406693036346,95.5277603343179 +"934",247.476575120801,95.5277603343179 +"935",247.476575120801,96.1805529069527 +"936",248.813448416846,96.1805529069527 +"937",248.813448416846,96.028962038436 +"938",249.870548657072,96.028962038436 +"939",249.870548657072,95.788332722361 +"940",249.960225241698,95.788332722361 +"941",249.960225241698,95.8364614980457 +"942",250.977096822475,95.8364614980457 +"943",250.977096822475,95.9250178805474 +"944",250.98630291116,95.9250178805474 +"945",250.98630291116,95.8150002977508 +"946",251.063695283738,95.8150002977508 +"947",251.063695283738,95.75775741616 +"948",251.521603966463,95.75775741616 +"949",251.521603966463,96.0164457887671 +"950",251.830547514586,96.0164457887671 +"951",251.830547514586,95.9237278769933 +"952",251.856182590252,95.9237278769933 +"953",251.856182590252,95.8165819105643 +"954",251.864257559659,95.8165819105643 +"955",251.864257559659,96.0355620905753 +"956",252.366079863849,96.0355620905753 +"957",252.366079863849,95.8485920300223 +"958",253.607340245459,95.8485920300223 +"959",253.607340245459,96.0491139566147 +"960",253.747663751034,96.0491139566147 +"961",253.747663751034,96.1858708362337 +"962",254.305692507665,96.1858708362337 +"963",254.305692507665,96.3374051105633 +"964",254.306828002464,96.3374051105633 +"965",254.306828002464,95.9018990360928 +"966",254.588330885682,95.9018990360928 +"967",254.588330885682,96.3426283164862 +"968",254.631974108932,96.3426283164862 +"969",254.631974108932,96.0474887940758 +"970",254.811224478268,96.0474887940758 +"971",254.811224478268,96.1828008781576 +"972",254.845475712289,96.1828008781576 +"973",254.845475712289,96.2893340786174 +"974",254.850428637048,96.2893340786174 +"975",254.850428637048,95.9527078778289 +"976",258.469893314797,95.9527078778289 +"977",258.469893314797,95.7396550689996 +"978",258.83235184893,95.7396550689996 +"979",258.83235184893,95.7841118045321 +"980",258.981316023425,95.7841118045321 +"981",258.981316023425,95.9432530885206 +"982",259.175659272793,95.9432530885206 +"983",259.175659272793,95.9871707819103 +"984",259.595253174423,95.9871707819103 +"985",259.595253174423,96.3602863499279 +"986",259.947092029701,96.3602863499279 +"987",259.947092029701,96.5660313074061 +"988",260.005357162195,96.5660313074061 +"989",260.005357162195,96.3585803681988 +"990",260.07047280988,96.3585803681988 +"991",260.07047280988,96.2517910009918 +"992",260.240150874351,96.2517910009918 +"993",260.240150874351,96.2267650599681 +"994",260.752500340111,96.2267650599681 +"995",260.752500340111,96.4605044427989 +"996",263.106827810825,96.4605044427989 +"997",263.106827810825,96.0487930873826 +"998",263.107524642567,96.0487930873826 +"999",263.107524642567,96.2261891735999 +"1000",263.265674995618,96.2261891735999 +"1001",263.265674995618,96.6171054529245 diff --git a/graphiques/CH3-SIMGAMMAGAUSS.pdf b/graphiques/CH3-SIMGAMMAGAUSS.pdf new file mode 100644 index 0000000..1fd4009 Binary files /dev/null and b/graphiques/CH3-SIMGAMMAGAUSS.pdf differ diff --git a/graphiques/CH3-SIMULGAL0121.pdf b/graphiques/CH3-SIMULGAL0121.pdf new file mode 100644 index 0000000..6d50442 Binary files /dev/null and b/graphiques/CH3-SIMULGAL0121.pdf differ diff --git a/graphiques/ch3.r b/graphiques/ch3.r new file mode 100644 index 0000000..254d60a --- /dev/null +++ b/graphiques/ch3.r @@ -0,0 +1,24 @@ +set.seed(908144032) + +## GRAPHIQUE 1 ## +s0 <- 100 +n <- 500 +## On simule des temps d'arrivée Gamma +t <- c(0,rep(cumsum(rgamma(n,0.5)),each=2)) +## On simule des sauts gaussiens +j <- head(c(0,0,rep(cumsum(rnorm(n,0.005/365,0.0025)),each=2)),-1) +## On trace le parcours +cbind(t,j) +pdf("CH3-SIMGAMMAGAUSS.pdf") +plot(t,s0*exp(j),type="l",xlab="t",ylab="Y(t)",main=" ") +dev.off() +write.csv(data.frame(Temps=t,Prix=s0*exp(j)),file="CH3-SIMGAMMAGAUSS.csv") + +## GRAPHIQUE 2 ## +source("../contenus/r/code/rGAL.r") + +d1 <- rGALkappa(2500,0,1,2,1) +pdf("CH3-SIMULGAL0121.pdf") +hist(d1,prob=TRUE,col="grey",breaks=25,xlab="y",ylab="f(y)",main="") +lines(density(d1),lwd=3) +dev.off() diff --git a/graphiques/increment1.dia b/graphiques/increment1.dia new file mode 100644 index 0000000..de9e933 Binary files /dev/null and b/graphiques/increment1.dia differ diff --git a/graphiques/increment1.tex b/graphiques/increment1.tex new file mode 100644 index 0000000..76a5259 --- /dev/null +++ b/graphiques/increment1.tex @@ -0,0 +1,198 @@ +% Graphic for TeX using PGF +% Title: /home/francois/projet-de-maitrise/graphiques/increment1.dia +% Creator: Dia v0.97.2 +% CreationDate: Mon Jul 29 16:42:05 2013 +% For: francois +% \usepackage{tikz} +% The following commands are not supported in PSTricks at present +% We define them conditionally, so when they are implemented, +% this pgf file will use them. +\ifx\du\undefined + \newlength{\du} +\fi +\setlength{\du}{15\unitlength} +\begin{tikzpicture} +\pgftransformxscale{1.000000} +\pgftransformyscale{-1.000000} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\definecolor{dialinecolor}{rgb}{1.000000, 1.000000, 1.000000} +\pgfsetfillcolor{dialinecolor} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\pgfsetarrowsend{stealth} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (19.200000\du,10.800000\du)--(28.800000\du,10.800000\du); +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\pgfsetarrowsend{stealth} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (19.200000\du,10.800000\du)--(19.200000\du,3.600000\du); +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (20.400000\du,10.600000\du)--(20.400000\du,11.000000\du); +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (26.400000\du,10.600000\du)--(26.400000\du,11.000000\du); +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (19.000000\du,9.600000\du)--(19.400000\du,9.600000\du); +} +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (20.200000\du,12.000000\du){0}; +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (26.200000\du,12.000000\du){$T_1$}; +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (18.200000\du,9.800000\du){0}; +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (17.400000\du,6.200000\du){$X_1$}; +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (19.000000\du,6.000000\du)--(19.400000\du,6.000000\du); +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (26.200000\du,6.000000\du)--(27.400000\du,6.000000\du); +} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (26.200000\du,6.000000\du)--(27.400000\du,6.000000\du); +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{26.200000\du}{6.000000\du}} +\pgfpathcurveto{\pgfpoint{26.200000\du}{5.875000\du}}{\pgfpoint{26.325000\du}{5.750000\du}}{\pgfpoint{26.450000\du}{5.750000\du}} +\pgfpathcurveto{\pgfpoint{26.575000\du}{5.750000\du}}{\pgfpoint{26.700000\du}{5.875000\du}}{\pgfpoint{26.700000\du}{6.000000\du}} +\pgfpathcurveto{\pgfpoint{26.700000\du}{6.125000\du}}{\pgfpoint{26.575000\du}{6.250000\du}}{\pgfpoint{26.450000\du}{6.250000\du}} +\pgfpathcurveto{\pgfpoint{26.325000\du}{6.250000\du}}{\pgfpoint{26.200000\du}{6.125000\du}}{\pgfpoint{26.200000\du}{6.000000\du}} +\pgfusepath{fill} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{26.200000\du}{6.000000\du}} +\pgfpathcurveto{\pgfpoint{26.200000\du}{5.875000\du}}{\pgfpoint{26.325000\du}{5.750000\du}}{\pgfpoint{26.450000\du}{5.750000\du}} +\pgfpathcurveto{\pgfpoint{26.575000\du}{5.750000\du}}{\pgfpoint{26.700000\du}{5.875000\du}}{\pgfpoint{26.700000\du}{6.000000\du}} +\pgfpathcurveto{\pgfpoint{26.700000\du}{6.125000\du}}{\pgfpoint{26.575000\du}{6.250000\du}}{\pgfpoint{26.450000\du}{6.250000\du}} +\pgfpathcurveto{\pgfpoint{26.325000\du}{6.250000\du}}{\pgfpoint{26.200000\du}{6.125000\du}}{\pgfpoint{26.200000\du}{6.000000\du}} +\pgfusepath{stroke} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (20.200000\du,9.600000\du)--(26.050000\du,9.600000\du); +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{20.200000\du}{9.600000\du}} +\pgfpathcurveto{\pgfpoint{20.200000\du}{9.475000\du}}{\pgfpoint{20.325000\du}{9.350000\du}}{\pgfpoint{20.450000\du}{9.350000\du}} +\pgfpathcurveto{\pgfpoint{20.575000\du}{9.350000\du}}{\pgfpoint{20.700000\du}{9.475000\du}}{\pgfpoint{20.700000\du}{9.600000\du}} +\pgfpathcurveto{\pgfpoint{20.700000\du}{9.725000\du}}{\pgfpoint{20.575000\du}{9.850000\du}}{\pgfpoint{20.450000\du}{9.850000\du}} +\pgfpathcurveto{\pgfpoint{20.325000\du}{9.850000\du}}{\pgfpoint{20.200000\du}{9.725000\du}}{\pgfpoint{20.200000\du}{9.600000\du}} +\pgfusepath{fill} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{20.200000\du}{9.600000\du}} +\pgfpathcurveto{\pgfpoint{20.200000\du}{9.475000\du}}{\pgfpoint{20.325000\du}{9.350000\du}}{\pgfpoint{20.450000\du}{9.350000\du}} +\pgfpathcurveto{\pgfpoint{20.575000\du}{9.350000\du}}{\pgfpoint{20.700000\du}{9.475000\du}}{\pgfpoint{20.700000\du}{9.600000\du}} +\pgfpathcurveto{\pgfpoint{20.700000\du}{9.725000\du}}{\pgfpoint{20.575000\du}{9.850000\du}}{\pgfpoint{20.450000\du}{9.850000\du}} +\pgfpathcurveto{\pgfpoint{20.325000\du}{9.850000\du}}{\pgfpoint{20.200000\du}{9.725000\du}}{\pgfpoint{20.200000\du}{9.600000\du}} +\pgfusepath{stroke} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +\definecolor{dialinecolor}{rgb}{1.000000, 1.000000, 1.000000} +\pgfsetfillcolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{26.550000\du}{9.600000\du}} +\pgfpathcurveto{\pgfpoint{26.550000\du}{9.725000\du}}{\pgfpoint{26.425000\du}{9.850000\du}}{\pgfpoint{26.300000\du}{9.850000\du}} +\pgfpathcurveto{\pgfpoint{26.175000\du}{9.850000\du}}{\pgfpoint{26.050000\du}{9.725000\du}}{\pgfpoint{26.050000\du}{9.600000\du}} +\pgfpathcurveto{\pgfpoint{26.050000\du}{9.475000\du}}{\pgfpoint{26.175000\du}{9.350000\du}}{\pgfpoint{26.300000\du}{9.350000\du}} +\pgfpathcurveto{\pgfpoint{26.425000\du}{9.350000\du}}{\pgfpoint{26.550000\du}{9.475000\du}}{\pgfpoint{26.550000\du}{9.600000\du}} +\pgfusepath{fill} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{26.550000\du}{9.600000\du}} +\pgfpathcurveto{\pgfpoint{26.550000\du}{9.725000\du}}{\pgfpoint{26.425000\du}{9.850000\du}}{\pgfpoint{26.300000\du}{9.850000\du}} +\pgfpathcurveto{\pgfpoint{26.175000\du}{9.850000\du}}{\pgfpoint{26.050000\du}{9.725000\du}}{\pgfpoint{26.050000\du}{9.600000\du}} +\pgfpathcurveto{\pgfpoint{26.050000\du}{9.475000\du}}{\pgfpoint{26.175000\du}{9.350000\du}}{\pgfpoint{26.300000\du}{9.350000\du}} +\pgfpathcurveto{\pgfpoint{26.425000\du}{9.350000\du}}{\pgfpoint{26.550000\du}{9.475000\du}}{\pgfpoint{26.550000\du}{9.600000\du}} +\pgfusepath{stroke} +\end{tikzpicture} diff --git a/graphiques/normaletronque.dia b/graphiques/normaletronque.dia new file mode 100644 index 0000000..d4ec654 Binary files /dev/null and b/graphiques/normaletronque.dia differ diff --git a/graphiques/normaletronque.log b/graphiques/normaletronque.log new file mode 100644 index 0000000..7396eda --- /dev/null +++ b/graphiques/normaletronque.log @@ -0,0 +1,1795 @@ +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.7.6) 30 JUL 2013 16:32 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**\input normaletronque.tex +(./normaletronque.tex +\du=\skip41 + + +! LaTeX Error: Environment tikzpicture undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.14 \begin{tikzpicture} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Undefined control sequence. +l.15 \pgftransformxscale + {1.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.15 \pgftransformxscale{1 + .000000} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.16 \pgftransformyscale + {-1.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no - in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.17 \definecolor + {dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.18 \pgfsetstrokecolor + {dialinecolor} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +! Undefined control sequence. +l.19 \definecolor + {dialinecolor}{rgb}{1.000000, 1.000000, 1.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.20 \pgfsetfillcolor + {dialinecolor} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +! Undefined control sequence. +l.21 \pgfsetlinewidth + {0.100000\du} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.21 \pgfsetlinewidth{0.100000\du} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.21 \pgfsetlinewidth{0.100000\du} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Undefined control sequence. +l.22 \pgfsetdash + {}{0pt} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.23 \pgfsetdash + {}{0pt} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.24 \pgfsetbuttcap + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.26 \definecolor + {dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.27 \pgfsetfillcolor + {dialinecolor} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +! Undefined control sequence. +l.29 \pgfsetarrowsstart + {stealth} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +! Undefined control sequence. +l.30 \pgfsetarrowsend + {stealth} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +! Undefined control sequence. +l.31 \definecolor + {dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.32 \pgfsetstrokecolor + {dialinecolor} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +! Undefined control sequence. +l.33 \draw + (18.000000\du,24.000000\du)--(36.000000\du,24.000000\du); +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no ( in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Illegal unit of measure (pt inserted). + + . +l.33 \draw (18.000000\du,24. + 000000\du)--(36.000000\du,24.000000\du); +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + ) +l.33 \draw (18.000000\du,24.000000\du) + --(36.000000\du,24.000000\du); +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + ) +l.33 \draw (18.000000\du,24.000000\du) + --(36.000000\du,24.000000\du); +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no ) in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Illegal unit of measure (pt inserted). + + . +l.33 ...000000\du,24.000000\du)--(36.000000\du,24. + 000000\du); +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + ) +l.33 ...24.000000\du)--(36.000000\du,24.000000\du) + ; +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + ) +l.33 ...24.000000\du)--(36.000000\du,24.000000\du) + ; +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.35 \pgfsetlinewidth + {0.100000\du} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.35 \pgfsetlinewidth{0.100000\du} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.35 \pgfsetlinewidth{0.100000\du} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Undefined control sequence. +l.36 \pgfsetdash + {}{0pt} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.37 \pgfsetdash + {}{0pt} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.38 \pgfsetbuttcap + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.40 \definecolor + {dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.41 \pgfsetfillcolor + {dialinecolor} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +! Undefined control sequence. +l.43 \pgfsetarrowsend + {stealth} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +! Undefined control sequence. +l.44 \definecolor + {dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.45 \pgfsetstrokecolor + {dialinecolor} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +! Undefined control sequence. +l.46 \draw + (27.000000\du,24.000000\du)--(27.000000\du,15.000000\du); +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no ( in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Illegal unit of measure (pt inserted). + + . +l.46 \draw (27.000000\du,24. + 000000\du)--(27.000000\du,15.000000\du); +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + ) +l.46 \draw (27.000000\du,24.000000\du) + --(27.000000\du,15.000000\du); +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + ) +l.46 \draw (27.000000\du,24.000000\du) + --(27.000000\du,15.000000\du); +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no ) in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Illegal unit of measure (pt inserted). + + . +l.46 ...000000\du,24.000000\du)--(27.000000\du,15. + 000000\du); +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + ) +l.46 ...24.000000\du)--(27.000000\du,15.000000\du) + ; +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + ) +l.46 ...24.000000\du)--(27.000000\du,15.000000\du) + ; +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.48 \pgfsetlinewidth + {0.100000\du} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.48 \pgfsetlinewidth{0.100000\du} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.48 \pgfsetlinewidth{0.100000\du} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Undefined control sequence. +l.49 \pgfsetdash + {}{0pt} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.50 \pgfsetdash + {}{0pt} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.51 \pgfsetmiterjoin + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.52 \pgfsetbuttcap + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.54 \definecolor + {dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.55 \pgfsetfillcolor + {dialinecolor} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +! Undefined control sequence. +l.57 \definecolor + {dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.58 \pgfsetstrokecolor + {dialinecolor} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +! Undefined control sequence. +l.59 \pgfpathmoveto + {\pgfpoint{21.000000\du}{24.000000\du}} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.59 \pgfpathmoveto{\pgfpoint + {21.000000\du}{24.000000\du}} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 2 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.59 \pgfpathmoveto{\pgfpoint{21.000000\du} + {24.000000\du}} +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.59 \pgfpathmoveto{\pgfpoint{21.000000\du} + {24.000000\du}} +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.59 ...veto{\pgfpoint{21.000000\du}{24.000000\du} + } +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.59 ...veto{\pgfpoint{21.000000\du}{24.000000\du} + } +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Undefined control sequence. +l.60 \pgfpathcurveto + {\pgfpoint{24.000000\du}{24.000000\du}}{\pgfpoint{25.008... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.60 \pgfpathcurveto{\pgfpoint + {24.000000\du}{24.000000\du}}{\pgfpoint{25.008... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.60 \pgfpathcurveto{\pgfpoint{24.000000\du} + {24.000000\du}}{\pgfpoint{25.008... +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.60 \pgfpathcurveto{\pgfpoint{24.000000\du} + {24.000000\du}}{\pgfpoint{25.008... +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.60 ...veto{\pgfpoint{24.000000\du}{24.000000\du} + }{\pgfpoint{25.008000\du}{... +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.60 ...veto{\pgfpoint{24.000000\du}{24.000000\du} + }{\pgfpoint{25.008000\du}{... +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Undefined control sequence. +l.60 ...int{24.000000\du}{24.000000\du}}{\pgfpoint + {25.008000\du}{18.000000\d... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.60 ...du}{24.000000\du}}{\pgfpoint{25.008000\du} + {18.000000\du}}{\pgfpoint{... +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.60 ...du}{24.000000\du}}{\pgfpoint{25.008000\du} + {18.000000\du}}{\pgfpoint{... +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no 1 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.60 ...du}}{\pgfpoint{25.008000\du}{18.000000\du} + }{\pgfpoint{27.000000\du}{... +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.60 ...du}}{\pgfpoint{25.008000\du}{18.000000\du} + }{\pgfpoint{27.000000\du}{... +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Undefined control sequence. +l.60 ...int{25.008000\du}{18.000000\du}}{\pgfpoint + {27.000000\du}{18.000000\du}} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 2 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.60 ...du}{18.000000\du}}{\pgfpoint{27.000000\du} + {18.000000\du}} +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.60 ...du}{18.000000\du}}{\pgfpoint{27.000000\du} + {18.000000\du}} +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no 1 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.60 ...du}}{\pgfpoint{27.000000\du}{18.000000\du} + } +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.60 ...du}}{\pgfpoint{27.000000\du}{18.000000\du} + } +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Undefined control sequence. +l.61 \pgfusepath + {stroke} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.63 \pgfsetlinewidth + {0.100000\du} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.63 \pgfsetlinewidth{0.100000\du} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.63 \pgfsetlinewidth{0.100000\du} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Undefined control sequence. +l.64 \pgfsetdash + {}{0pt} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.65 \pgfsetdash + {}{0pt} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.66 \pgfsetmiterjoin + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.67 \pgfsetbuttcap + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.69 \definecolor + {dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.70 \pgfsetfillcolor + {dialinecolor} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +! Undefined control sequence. +l.72 \definecolor + {dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.73 \pgfsetstrokecolor + {dialinecolor} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +! Undefined control sequence. +l.74 \pgfpathmoveto + {\pgfpoint{27.000000\du}{18.000000\du}} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.74 \pgfpathmoveto{\pgfpoint + {27.000000\du}{18.000000\du}} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 2 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.74 \pgfpathmoveto{\pgfpoint{27.000000\du} + {18.000000\du}} +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.74 \pgfpathmoveto{\pgfpoint{27.000000\du} + {18.000000\du}} +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no 1 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.74 ...veto{\pgfpoint{27.000000\du}{18.000000\du} + } +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.74 ...veto{\pgfpoint{27.000000\du}{18.000000\du} + } +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Undefined control sequence. +l.75 \pgfpathcurveto + {\pgfpoint{29.324000\du}{18.000000\du}}{\pgfpoint{30.000... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.75 \pgfpathcurveto{\pgfpoint + {29.324000\du}{18.000000\du}}{\pgfpoint{30.000... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 2 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +! Missing number, treated as zero. + + } +l.75 \pgfpathcurveto{\pgfpoint{29.324000\du} + {18.000000\du}}{\pgfpoint{30.000... +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.75 \pgfpathcurveto{\pgfpoint{29.324000\du} + {18.000000\du}}{\pgfpoint{30.000... +(That makes 100 errors; please try again.) +Here is how much of TeX's memory you used: + 7 strings out of 493485 + 174 string characters out of 3143527 + 47818 words of memory out of 3000000 + 3414 multiletter control sequences out of 15000+200000 + 3640 words of font info for 14 fonts, out of 3000000 for 9000 + 957 hyphenation exceptions out of 8191 + 6i,0n,10p,160b,18s stack positions out of 5000i,500n,10000p,200000b,50000s + +! ==> Fatal error occurred, no output PDF file produced! diff --git a/graphiques/normaletronque.tex b/graphiques/normaletronque.tex new file mode 100644 index 0000000..59b64c2 --- /dev/null +++ b/graphiques/normaletronque.tex @@ -0,0 +1,112 @@ +% Graphic for TeX using PGF +% Title: /home/francois/projet-de-maitrise/graphiques/normaletronque.dia +% Creator: Dia v0.97.2 +% CreationDate: Tue Jul 30 16:24:46 2013 +% For: francois +% \usepackage{tikz} +% The following commands are not supported in PSTricks at present +% We define them conditionally, so when they are implemented, +% this pgf file will use them. +\ifx\du\undefined + \newlength{\du} +\fi +\setlength{\du}{15\unitlength} +\begin{tikzpicture} +\pgftransformxscale{1.000000} +\pgftransformyscale{-1.000000} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\definecolor{dialinecolor}{rgb}{1.000000, 1.000000, 1.000000} +\pgfsetfillcolor{dialinecolor} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\pgfsetarrowsstart{stealth} +\pgfsetarrowsend{stealth} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (18.000000\du,24.000000\du)--(36.000000\du,24.000000\du); +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\pgfsetarrowsend{stealth} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (27.000000\du,24.000000\du)--(27.000000\du,15.000000\du); +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{21.000000\du}{24.000000\du}} +\pgfpathcurveto{\pgfpoint{24.000000\du}{24.000000\du}}{\pgfpoint{25.008000\du}{18.000000\du}}{\pgfpoint{27.000000\du}{18.000000\du}} +\pgfusepath{stroke} +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{27.000000\du}{18.000000\du}} +\pgfpathcurveto{\pgfpoint{29.324000\du}{18.000000\du}}{\pgfpoint{30.000000\du}{24.000000\du}}{\pgfpoint{33.000000\du}{24.000000\du}} +\pgfusepath{stroke} +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\definecolor{dialinecolor}{rgb}{1.000000, 1.000000, 1.000000} +\pgfsetfillcolor{dialinecolor} +\fill (31.287500\du,22.975000\du)--(31.287500\du,23.900000\du)--(33.087500\du,23.900000\du)--(33.087500\du,22.975000\du)--cycle; +\definecolor{dialinecolor}{rgb}{1.000000, 1.000000, 1.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (31.287500\du,22.975000\du)--(31.287500\du,23.900000\du)--(33.087500\du,23.900000\du)--(33.087500\du,22.975000\du)--cycle; +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (31.185444\du,23.681250\du)--(31.185444\du,24.312500\du); +} +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (30.821374\du,24.943750\du){$w_0$}; +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (34.881350\du,24.743750\du){$z_{w_0}$}; +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (24.484382\du,16.243750\du){$f(z_{w_0})$}; +\end{tikzpicture} diff --git a/graphiques/probdroite.dia b/graphiques/probdroite.dia new file mode 100644 index 0000000..57b977f Binary files /dev/null and b/graphiques/probdroite.dia differ diff --git a/graphiques/probdroite.tex b/graphiques/probdroite.tex new file mode 100644 index 0000000..0f63f84 --- /dev/null +++ b/graphiques/probdroite.tex @@ -0,0 +1,252 @@ +% Graphic for TeX using PGF +% Title: /home/francois/projet-de-maitrise/graphiques/probdroite.dia +% Creator: Dia v0.97.2 +% CreationDate: Tue Jul 30 16:06:31 2013 +% For: francois +% \usepackage{tikz} +% The following commands are not supported in PSTricks at present +% We define them conditionally, so when they are implemented, +% this pgf file will use them. +\ifx\du\undefined + \newlength{\du} +\fi +\setlength{\du}{15\unitlength} +\begin{tikzpicture} +\pgftransformxscale{1.000000} +\pgftransformyscale{-1.000000} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\definecolor{dialinecolor}{rgb}{1.000000, 1.000000, 1.000000} +\pgfsetfillcolor{dialinecolor} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\pgfsetarrowsend{latex} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (12.000000\du,0.000000\du)--(24.000000\du,0.000000\du); +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\pgfsetarrowsend{latex} +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (12.000000\du,0.000000\du)--(12.000000\du,-12.000000\du); +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{23.400000\du}{0.000000\du}} +\pgfpathcurveto{\pgfpoint{12.400000\du}{0.000000\du}}{\pgfpoint{16.000000\du}{-10.000000\du}}{\pgfpoint{12.000000\du}{-5.000000\du}} +\pgfusepath{stroke} +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (19.000000\du,-0.700000\du)--(19.000000\du,0.000000\du); +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +{\pgfsetcornersarced{\pgfpoint{0.000000\du}{0.000000\du}}\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (19.032329\du,-0.056240\du)--(19.700000\du,-0.400000\du); +}} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +{\pgfsetcornersarced{\pgfpoint{0.000000\du}{0.000000\du}}\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (19.700000\du,-0.500000\du)--(19.700000\du,0.000000\du); +}} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +{\pgfsetcornersarced{\pgfpoint{0.000000\du}{0.000000\du}}\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (19.700000\du,0.000000\du)--(20.100000\du,-0.300000\du); +}} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (20.100000\du,-0.400000\du)--(20.100000\du,0.000000\du); +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +{\pgfsetcornersarced{\pgfpoint{0.000000\du}{0.000000\du}}\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (20.100000\du,0.000000\du)--(20.500000\du,-0.200000\du); +}} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +{\pgfsetcornersarced{\pgfpoint{0.000000\du}{0.000000\du}}\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (20.500000\du,-0.300000\du)--(20.500000\du,0.000000\du); +}} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +{\pgfsetcornersarced{\pgfpoint{0.000000\du}{0.000000\du}}\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (20.900000\du,-0.200000\du)--(20.476079\du,-0.012490\du); +}} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{19.000000\du}{0.025000\du}} +\pgfpathcurveto{\pgfpoint{19.468950\du}{0.025000\du}}{\pgfpoint{19.200000\du}{0.925000\du}}{\pgfpoint{20.500000\du}{0.925000\du}} +\pgfusepath{stroke} +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{21.500000\du}{0.925000\du}} +\pgfpathcurveto{\pgfpoint{22.800000\du}{0.925000\du}}{\pgfpoint{22.558025\du}{0.025000\du}}{\pgfpoint{23.000000\du}{0.025000\du}} +\pgfusepath{stroke} +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{20.500000\du}{0.925000\du}} +\pgfpathcurveto{\pgfpoint{20.700000\du}{0.925000\du}}{\pgfpoint{21.000000\du}{1.125000\du}}{\pgfpoint{21.000000\du}{1.425000\du}} +\pgfusepath{stroke} +} +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetmiterjoin +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\pgfpathmoveto{\pgfpoint{21.500000\du}{0.925000\du}} +\pgfpathcurveto{\pgfpoint{21.300000\du}{0.925000\du}}{\pgfpoint{21.000000\du}{1.125000\du}}{\pgfpoint{21.000000\du}{1.425000\du}} +\pgfusepath{stroke} +} +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (19.800000\du,2.175000\du){P(X>y)}; +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (10.300000\du,-11.500000\du){f(x)}; +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (23.400000\du,1.000000\du){x}; +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (22.000000\du,1.000000\du){}; +\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (20.900000\du,-0.200000\du)--(20.900000\du,0.000000\du); +} +\end{tikzpicture} diff --git a/memoire/908144032.aux b/memoire/908144032.aux new file mode 100644 index 0000000..38ccd23 --- /dev/null +++ b/memoire/908144032.aux @@ -0,0 +1,119 @@ +\relax +\providecommand*{\memsetcounter}[2]{} +\catcode`:\active +\catcode`;\active +\catcode`!\active +\catcode`?\active +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} +\bibstyle{plainnatmod} +\select@language{french} +\@writefile{toc}{\select@language{french}} +\@writefile{lof}{\select@language{french}} +\@writefile{lot}{\select@language{french}} +\select@language{french} +\@writefile{toc}{\select@language{french}} +\@writefile{lof}{\select@language{french}} +\@writefile{lot}{\select@language{french}} +\@input{resume.aux} +\@input{abstract.aux} +\@writefile{toc}{\contentsline {chapter}{Table des mati{\`e}res}{vii}{section*.5}} +\citation{madan1990variance} +\@writefile{toc}{\contentsline {chapter}{Liste des tableaux}{xi}{section*.6}} +\citation{mitchell1916critique} +\@writefile{toc}{\contentsline {chapter}{Liste des figures}{xiii}{section*.7}} +\@input{remerciements.aux} +\@input{introduction.aux} +\@input{chapitre1.aux} +\@input{chapitre2.aux} +\@input{chapitre3.aux} +\@input{chapitre4.aux} +\@input{chapitre5.aux} +\@input{chapitre6.aux} +\@input{chapitre7.aux} +\@input{chapitre8.aux} +\@input{chapitre9.aux} +\@input{conclusion.aux} +\@input{annexe1.aux} +\@input{annexe2.aux} +\@input{annexe3.aux} +\bibdata{memoire} +\bibcite{abramowitz1965handbook}{{1}{1965}{{Abramowitz et Stegun}}{{}}} +\bibcite{applebaum2004levy}{{2}{2004}{{Applebaum}}{{}}} +\bibcite{bachelier1900theorie}{{3}{1900}{{Bachelier}}{{}}} +\bibcite{barndorff2001levy}{{4}{2001}{{Barndorff-Nielsen et~al.}}{{Barndorff-Nielsen, Mikosch, et Resnick}}} +\bibcite{berkson1980minimum}{{5}{1980}{{Berkson}}{{}}} +\bibcite{bingham2004risk}{{6}{2004}{{Bingham et Kiesel}}{{}}} +\bibcite{black1976pricing}{{7}{1976}{{Black}}{{}}} +\bibcite{black1973pricing}{{8}{1973}{{Black et Scholes}}{{}}} +\bibcite{buckle1995bayesian}{{9}{1995}{{Buckle}}{{}}} +\bibcite{butler2007saddlepoint}{{10}{2007}{{Butler}}{{}}} +\bibcite{carr1999option}{{11}{1999}{{Carr et Madan}}{{}}} +\bibcite{crowder1986consistency}{{12}{1986}{{Crowder}}{{}}} +\bibcite{crowder1987linear}{{13}{1987}{{Crowder}}{{}}} +\@writefile{toc}{\contentsline {chapter}{Bibliographie}{131}{section*.62}} +\bibcite{daniels1954saddlepoint}{{14}{1954}{{Daniels}}{{}}} +\bibcite{derman1996modelrisk}{{15}{1996}{{Derman}}{{}}} +\bibcite{dodge2004statistique}{{16}{2004}{{Dodge}}{{}}} +\bibcite{epps2007pricing}{{17}{2007}{{Epps}}{{}}} +\bibcite{epps1983test}{{18}{1983}{{Epps et Pulley}}{{}}} +\bibcite{everitt2006cambridge}{{19}{2006}{{Everitt et Skrondal}}{{}}} +\bibcite{fama1993common}{{20}{1993}{{Fama et French}}{{}}} +\bibcite{feuerverger1981efficiency}{{21}{1981}{{Feuerverger et McDunnough}}{{}}} +\bibcite{fox1986large}{{22}{1986}{{Fox et Taqqu}}{{}}} +\bibcite{gil1951note}{{23}{1951}{{Gil-Pelaez}}{{}}} +\bibcite{gourieroux1989statistique}{{24}{1989}{{Gourieroux et Monfort}}{{}}} +\bibcite{hall2005generalized}{{25}{2005}{{Hall}}{{}}} +\bibcite{hamilton1994time}{{26}{1994}{{Hamilton}}{{}}} +\bibcite{hansen1982large}{{27}{1982}{{Hansen}}{{}}} +\bibcite{henze1990approximation}{{28}{1990}{{Henze}}{{}}} +\bibcite{heston1993closed}{{29}{1993}{{Heston}}{{}}} +\bibcite{hinkley1977estimation}{{30}{1977}{{Hinkley et Revankar}}{{}}} +\bibcite{hogg1978introduction}{{31}{1978}{{Hogg et Craig}}{{}}} +\bibcite{itkin2005pricing}{{32}{2005}{{Itkin}}{{}}} +\bibcite{kotz2001laplace}{{33}{2001}{{Kotz et~al.}}{{Kotz, Kozubowski, et Podg{\'o}rski}}} +\bibcite{KOUTROUVELIS01011980}{{34}{1980}{{Koutrouvelis}}{{}}} +\bibcite{kozubowski1999class}{{35}{1999}{{Kozubowski et Podg{\'o}rski}}{{}}} +\bibcite{kozubowski2001asymmetric}{{36}{2001}{{Kozubowski et Podg{\'o}rski}}{{}}} +\bibcite{kyprianou2007introductory}{{37}{2007}{{Kyprianou}}{{}}} +\bibcite{lugannani1980saddle}{{38}{1980}{{Lugannani et Rice}}{{}}} +\bibcite{lukacs1960characteristic}{{39}{1960}{{Lukacs}}{{}}} +\bibcite{luong1987minimum}{{40}{1987}{{Luong et Thompson}}{{}}} +\bibcite{madan1990variance}{{41}{1990}{{Madan et Seneta}}{{}}} +\bibcite{madan1998variance}{{42}{1998}{{Madan et~al.}}{{Madan, Carr, et Chang}}} +\bibcite{mandelbrot1963variation}{{43}{1963}{{Mandelbrot}}{{}}} +\bibcite{merton1976option}{{44}{1976}{{Merton}}{{}}} +\bibcite{mitchell1916critique}{{45}{1916}{{Mitchell}}{{}}} +\bibcite{newey1994large}{{46}{1994}{{Newey et McFadden}}{{}}} +\bibcite{newey1987hypothesis}{{47}{1987}{{Newey et West}}{{}}} +\bibcite{praetz1972distribution}{{48}{1972}{{Praetz}}{{}}} +\bibcite{press1967compound}{{49}{1967}{{Press}}{{}}} +\bibcite{sato1999levy}{{50}{1999}{{Sato}}{{}}} +\bibcite{schoutens2003levy}{{51}{2003}{{Schoutens}}{{}}} +\bibcite{RpackageVarianceGamma}{{52}{2012}{{Scott et Dong}}{{}}} +\bibcite{seneta2004fitting}{{53}{2004}{{Seneta}}{{}}} +\bibcite{shapiro1965analysis}{{54}{1965}{{Shapiro et Wilk}}{{}}} +\bibcite{shephard1991characteristic}{{55}{1991}{{Shephard}}{{}}} +\bibcite{spiegel1999schaum}{{56}{1999}{{Spiegel et Liu}}{{}}} +\bibcite{stuart1987kendall}{{57}{1987}{{Stuart et Ord}}{{}}} +\bibcite{teschl2004topics}{{58}{2004}{{Teschl}}{{}}} +\bibcite{wang2003evaluating}{{59}{2003}{{Wang et~al.}}{{Wang, Tsang, et Marsaglia}}} +\bibcite{wendel1961non}{{60}{1961}{{Wendel}}{{}}} +\bibcite{wolfowitz1957minimum}{{61}{1957}{{Wolfowitz}}{{}}} +\bibcite{wooldridge2001econometric}{{62}{2001}{{Wooldridge}}{{}}} +\@input{deed.aux} +\memsetcounter{lastsheet}{155} +\memsetcounter{lastpage}{137} diff --git a/memoire/908144032.bbl b/memoire/908144032.bbl new file mode 100644 index 0000000..398d9f7 --- /dev/null +++ b/memoire/908144032.bbl @@ -0,0 +1,361 @@ +\begin{thebibliography}{62} +\providecommand{\natexlab}[1]{#1} +\providecommand{\url}[1]{\texttt{#1}} +\expandafter\ifx\csname urlstyle\endcsname\relax + \providecommand{\doi}[1]{doi: #1}\else + \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi + +\bibitem[Abramowitz et Stegun(1965)]{abramowitz1965handbook} +Milton Abramowitz et Irene~A Stegun. +\newblock \emph{Handbook of mathematical functions: with formulas, graphs, and + mathematical tables}, volume~55. +\newblock Dover publications, 1965. + +\bibitem[Applebaum(2004)]{applebaum2004levy} +David Applebaum. +\newblock L{\'e}vy processes: From probability to finance and quantum groups. +\newblock \emph{Notices of the AMS}, 51\penalty0 (11):\penalty0 1336--1347, + 2004. + +\bibitem[Bachelier(1900)]{bachelier1900theorie} +Louis Bachelier. +\newblock \emph{Th{\'e}orie de la sp{\'e}culation}. +\newblock Gauthier-Villars, 1900. + +\bibitem[Barndorff-Nielsen et~al.(2001)Barndorff-Nielsen, Mikosch, et + Resnick]{barndorff2001levy} +O.E.E. Barndorff-Nielsen, T.E. Mikosch, et S.I.E. Resnick. +\newblock \emph{L{\'e}vy Processes: Theory and Applications}. +\newblock Birkhäuser, 2001. +\newblock ISBN 9780817641672. +\newblock URL \url{http://books.google.ca/books?id=ExpTdTauXMwC}. + +\bibitem[Berkson(1980)]{berkson1980minimum} +Joseph Berkson. +\newblock Minimum chi-square, not maximum likelihood! +\newblock \emph{The Annals of Statistics}, pages 457--487, 1980. + +\bibitem[Bingham et Kiesel(2004)]{bingham2004risk} +Nicholas~H Bingham et R{\"u}diger Kiesel. +\newblock \emph{Risk-neutral valuation: Pricing and hedging of financial + derivatives}. +\newblock Springer, 2004. + +\bibitem[Black(1976)]{black1976pricing} +Fischer Black. +\newblock The pricing of commodity contracts. +\newblock \emph{Journal of financial economics}, 3\penalty0 (1):\penalty0 + 167--179, 1976. + +\bibitem[Black et Scholes(1973)]{black1973pricing} +Fischer Black et Myron Scholes. +\newblock The pricing of options and corporate liabilities. +\newblock \emph{The journal of political economy}, pages 637--654, 1973. + +\bibitem[Buckle(1995)]{buckle1995bayesian} +DJ~Buckle. +\newblock Bayesian inference for stable distributions. +\newblock \emph{Journal of the American Statistical Association}, 90\penalty0 + (430):\penalty0 605--613, 1995. + +\bibitem[Butler(2007)]{butler2007saddlepoint} +Ronald~W Butler. +\newblock \emph{Saddlepoint approximations with applications}, volume~22. +\newblock Cambridge University Press, 2007. + +\bibitem[Carr et Madan(1999)]{carr1999option} +Peter Carr et Dilip Madan. +\newblock Option valuation using the fast fourier transform. +\newblock \emph{Journal of Computational Finance}, 2\penalty0 (4):\penalty0 + 61--73, 1999. + +\bibitem[Crowder(1986)]{crowder1986consistency} +Martin Crowder. +\newblock On consistency and inconsistency of estimating equations. +\newblock \emph{Econometric Theory}, pages 305--330, 1986. + +\bibitem[Crowder(1987)]{crowder1987linear} +Martin Crowder. +\newblock On linear and quadratic estimating functions. +\newblock \emph{Biometrika}, 74\penalty0 (3):\penalty0 591--597, 1987. + +\bibitem[Daniels(1954)]{daniels1954saddlepoint} +Henry~E Daniels. +\newblock Saddlepoint approximations in statistics. +\newblock \emph{The Annals of Mathematical Statistics}, pages 631--650, 1954. + +\bibitem[Derman(1996)]{derman1996modelrisk} +Emanuel Derman. +\newblock Model risk. +\newblock Technical report, Goldman Sachs, 1996. + +\bibitem[Dodge(2004)]{dodge2004statistique} +Yadolah Dodge. +\newblock \emph{Statistique: dictionnaire encyclop{\'e}dique}. +\newblock Springer Verlag France, 2004. + +\bibitem[Epps(2007)]{epps2007pricing} +Thomas~W Epps. +\newblock \emph{Pricing derivative securities}. +\newblock World Scientific Publishing Company Incorporated, 2007. + +\bibitem[Epps et Pulley(1983)]{epps1983test} +Thomas~W Epps et Lawrence~B Pulley. +\newblock A test for normality based on the empirical characteristic function. +\newblock \emph{Biometrika}, 70\penalty0 (3):\penalty0 723--726, 1983. + +\bibitem[Everitt et Skrondal(2006)]{everitt2006cambridge} +Brian Everitt et Anders Skrondal. +\newblock \emph{The Cambridge dictionary of statistics}, volume~4. +\newblock Cambridge University Press Cambridge, 2006. + +\bibitem[Fama et French(1993)]{fama1993common} +Eugene~F Fama et Kenneth~R French. +\newblock Common risk factors in the returns on stocks and bonds. +\newblock \emph{Journal of financial economics}, 33\penalty0 (1):\penalty0 + 3--56, 1993. + +\bibitem[Feuerverger et McDunnough(1981)]{feuerverger1981efficiency} +Andrey Feuerverger et Philip McDunnough. +\newblock On the efficiency of empirical characteristic function procedures. +\newblock \emph{Journal of the Royal Statistical Society. Series B + (Methodological)}, pages 20--27, 1981. + +\bibitem[Fox et Taqqu(1986)]{fox1986large} +Robert Fox et Murad~S Taqqu. +\newblock Large-sample properties of parameter estimates for strongly dependent + stationary gaussian time series. +\newblock \emph{The Annals of Statistics}, 14\penalty0 (2):\penalty0 517--532, + 1986. + +\bibitem[Gil-Pelaez(1951)]{gil1951note} +J~Gil-Pelaez. +\newblock Note on the inversion theorem. +\newblock \emph{Biometrika}, 38\penalty0 (3-4):\penalty0 481--482, 1951. + +\bibitem[Gourieroux et Monfort(1989)]{gourieroux1989statistique} +Christian Gourieroux et Alain Monfort. +\newblock \emph{Statistique et mod{\`e}les {\'e}conom{\'e}triques: Notions + g{\'e}n{\'e}rales, estimation, pr{\'e}vision, algorithmes}, volume~1. +\newblock Economica, 1989. + +\bibitem[Hall(2005)]{hall2005generalized} +Alastair~R Hall. +\newblock \emph{Generalized method of moments}. +\newblock Oxford University Press Oxford, 2005. + +\bibitem[Hamilton(1994)]{hamilton1994time} +James~Douglas Hamilton. +\newblock \emph{Time series analysis}, volume~2. +\newblock Cambridge Univ Press, 1994. + +\bibitem[Hansen(1982)]{hansen1982large} +Lars~Peter Hansen. +\newblock Large sample properties of generalized method of moments estimators. +\newblock \emph{Econometrica: Journal of the Econometric Society}, pages + 1029--1054, 1982. + +\bibitem[Henze(1990)]{henze1990approximation} +N~Henze. +\newblock An approximation to the limit distribution of the epps-pulley test + statistic for normality. +\newblock \emph{Metrika}, 37\penalty0 (1):\penalty0 7--18, 1990. + +\bibitem[Heston(1993)]{heston1993closed} +Steven~L Heston. +\newblock A closed-form solution for options with stochastic volatility with + applications to bond and currency options. +\newblock \emph{Review of financial studies}, 6\penalty0 (2):\penalty0 + 327--343, 1993. + +\bibitem[Hinkley et Revankar(1977)]{hinkley1977estimation} +David~V Hinkley et Nagesh~S Revankar. +\newblock Estimation of the pareto law from underreported data: A further + analysis. +\newblock \emph{Journal of Econometrics}, 5\penalty0 (1):\penalty0 1--11, 1977. + +\bibitem[Hogg et Craig(1978)]{hogg1978introduction} +R.V. Hogg et A.T. Craig. +\newblock \emph{Introduction to mathematical statistics}. +\newblock Macmillan, 1978. +\newblock ISBN 9780029789902. +\newblock URL \url{http://books.google.ca/books?id=OZYQAQAAIAAJ}. + +\bibitem[Itkin(2005)]{itkin2005pricing} +Andrey Itkin. +\newblock Pricing options with vg model using fft. +\newblock \emph{arXiv preprint physics/0503137}, 2005. + +\bibitem[Kotz et~al.(2001)Kotz, Kozubowski, et Podg{\'o}rski]{kotz2001laplace} +S.~Kotz, T.J. Kozubowski, et K.~Podg{\'o}rski. +\newblock \emph{The Laplace Distribution and Generalizations: A Revisit With + Applications to Communications, Exonomics, Engineering, and Finance}. +\newblock Progress in Mathematics Series. Birkh{\"a}user, 2001. +\newblock ISBN 9780817641665. +\newblock URL \url{http://books.google.ca/books?id=cb8B07hwULUC}. + +\bibitem[Koutrouvelis(1980)]{KOUTROUVELIS01011980} +I.~A. Koutrouvelis. +\newblock A goodness-of-fit test of simple hypotheses based on the empirical + characteristic function. +\newblock \emph{Biometrika}, 67\penalty0 (1):\penalty0 238--240, 1980. +\newblock \doi{10.1093/biomet/67.1.238}. +\newblock URL \url{http://biomet.oxfordjournals.org/content/67/1/238.abstract}. + +\bibitem[Kozubowski et Podg{\'o}rski(1999)]{kozubowski1999class} +Tomasz~J Kozubowski et Krzysztof Podg{\'o}rski. +\newblock A class of asymmetric distributions. +\newblock \emph{Actuarial Research Clearing House}, 1:\penalty0 113--134, 1999. + +\bibitem[Kozubowski et Podg{\'o}rski(2001)]{kozubowski2001asymmetric} +Tomasz~J Kozubowski et Krzysztof Podg{\'o}rski. +\newblock Asymmetric laplace laws and modeling financial data. +\newblock \emph{Mathematical and computer modelling}, 34\penalty0 (9):\penalty0 + 1003--1021, 2001. + +\bibitem[Kyprianou(2007)]{kyprianou2007introductory} +Andreas~E Kyprianou. +\newblock \emph{Introductory lectures on fluctuations of L{\'e}vy processes + with applications}. +\newblock Springer, 2007. + +\bibitem[Lugannani et Rice(1980)]{lugannani1980saddle} +Robert Lugannani et Stephen Rice. +\newblock Saddle point approximation for the distribution of the sum of + independent random variables. +\newblock \emph{Advances in applied probability}, pages 475--490, 1980. + +\bibitem[Lukacs(1960)]{lukacs1960characteristic} +Eugene Lukacs. +\newblock \emph{Characteristic functions}, volume~4. +\newblock Griffin London, 1960. + +\bibitem[Luong et Thompson(1987)]{luong1987minimum} +A~Luong et ME~Thompson. +\newblock Minimum-distance methods based on quadratic distances for transforms. +\newblock \emph{Canadian Journal of Statistics}, 15\penalty0 (3):\penalty0 + 239--251, 1987. + +\bibitem[Madan et Seneta(1990)]{madan1990variance} +Dilip~B Madan et Eugene Seneta. +\newblock The variance gamma (vg) model for share market returns. +\newblock \emph{Journal of business}, pages 511--524, 1990. + +\bibitem[Madan et~al.(1998)Madan, Carr, et Chang]{madan1998variance} +Dilip~B Madan, Peter~P Carr, et Eric~C Chang. +\newblock The variance gamma process and option pricing. +\newblock \emph{European Finance Review}, 2\penalty0 (1):\penalty0 79--105, + 1998. + +\bibitem[Mandelbrot(1963)]{mandelbrot1963variation} +B.~Mandelbrot. +\newblock The variation of certain speculative prices. +\newblock \emph{Journal of business}, pages 394--419, 1963. + +\bibitem[Merton(1976)]{merton1976option} +Robert~C Merton. +\newblock Option pricing when underlying stock returns are discontinuous. +\newblock \emph{Journal of financial economics}, 3\penalty0 (1):\penalty0 + 125--144, 1976. + +\bibitem[Mitchell(1916)]{mitchell1916critique} +Wesley~C Mitchell. +\newblock A critique of index numbers of the prices of stocks. +\newblock \emph{The Journal of Political Economy}, pages 625--693, 1916. + +\bibitem[Newey et McFadden(1994)]{newey1994large} +Whitney~K Newey et Daniel McFadden. +\newblock Large sample estimation and hypothesis testing. +\newblock \emph{Handbook of econometrics}, 4:\penalty0 2111--2245, 1994. + +\bibitem[Newey et West(1987)]{newey1987hypothesis} +Whitney~K Newey et Kenneth~D West. +\newblock Hypothesis testing with efficient method of moments estimation. +\newblock \emph{International Economic Review}, 28\penalty0 (3):\penalty0 + 777--787, 1987. + +\bibitem[Praetz(1972)]{praetz1972distribution} +Peter~D Praetz. +\newblock The distribution of share price changes. +\newblock \emph{Journal of business}, pages 49--55, 1972. + +\bibitem[Press(1967)]{press1967compound} +S~James Press. +\newblock A compound events model for security prices. +\newblock \emph{Journal of Business}, pages 317--335, 1967. + +\bibitem[Sato(1999)]{sato1999levy} +K.~Sato. +\newblock \emph{L{\'e}vy Processes and Infinitely Divisible Distributions}. +\newblock Cambridge Studies in Advanced Mathematics. Cambridge University + Press, 1999. +\newblock ISBN 9780521553025. +\newblock URL \url{http://books.google.ca/books?id=CwT5BNG0-owC}. + +\bibitem[Schoutens(2003)]{schoutens2003levy} +Wim Schoutens. +\newblock \emph{L{\'e}vy processes in Finance}. +\newblock Wiley, 2003. + +\bibitem[Scott et Dong(2012)]{RpackageVarianceGamma} +David Scott et Christine~Yang Dong. +\newblock \emph{VarianceGamma: The Variance Gamma Distribution}, 2012. +\newblock URL \url{http://CRAN.R-project.org/package=VarianceGamma}. +\newblock R package version 0.3-1. + +\bibitem[Seneta(2004)]{seneta2004fitting} +Eugene Seneta. +\newblock Fitting the variance-gamma model to financial data. +\newblock \emph{Journal of Applied Probability}, pages 177--187, 2004. + +\bibitem[Shapiro et Wilk(1965)]{shapiro1965analysis} +Samuel~Sanford Shapiro et Martin~B Wilk. +\newblock An analysis of variance test for normality (complete samples). +\newblock \emph{Biometrika}, 52\penalty0 (3/4):\penalty0 591--611, 1965. + +\bibitem[Shephard(1991)]{shephard1991characteristic} +Neil~G Shephard. +\newblock From characteristic function to distribution function: a simple + framework for the theory. +\newblock \emph{Econometric Theory}, 7\penalty0 (04):\penalty0 519--529, 1991. + +\bibitem[Spiegel et Liu(1999)]{spiegel1999schaum} +Murray~R Spiegel et John Liu. +\newblock \emph{Schaum's mathematical handbook of formulas and tables}, volume + 1000. +\newblock McGraw-Hill, 1999. + +\bibitem[Stuart et Ord(1987)]{stuart1987kendall} +Alan Stuart et J~Keith Ord. +\newblock \emph{Kendall’s advanced theory of statistics, Vol. 1}. +\newblock Oxford University Press, New York, 1987. + +\bibitem[Teschl(2004)]{teschl2004topics} +Gerald Teschl. +\newblock Topics in real and functional analysis. +\newblock \emph{unpublished, available online at + \url{http://www.mat.univie.ac.at/\~gerald}}, 2004. + +\bibitem[Wang et~al.(2003)Wang, Tsang, et Marsaglia]{wang2003evaluating} +Jingbo Wang, Wai~Wan Tsang, et George Marsaglia. +\newblock Evaluating kolmogorov's distribution. +\newblock \emph{Journal of Statistical Software}, 8\penalty0 (18):\penalty0 + 1--4, 2003. + +\bibitem[Wendel(1961)]{wendel1961non} +JG~Wendel. +\newblock The non-absolute convergence of gil-pelaez'inversion integral. +\newblock \emph{The Annals of Mathematical Statistics}, 32\penalty0 + (1):\penalty0 338--339, 1961. + +\bibitem[Wolfowitz(1957)]{wolfowitz1957minimum} +Jacob Wolfowitz. +\newblock The minimum distance method. +\newblock \emph{The Annals of Mathematical Statistics}, pages 75--88, 1957. + +\bibitem[Wooldridge(2001)]{wooldridge2001econometric} +Jeffrey~M Wooldridge. +\newblock \emph{Econometric analysis of cross section and panel data}. +\newblock MIT press, 2001. + +\end{thebibliography} diff --git a/memoire/908144032.blg b/memoire/908144032.blg new file mode 100644 index 0000000..0f43762 --- /dev/null +++ b/memoire/908144032.blg @@ -0,0 +1,64 @@ +This is BibTeX, Version 0.99d (TeX Live 2012/Debian) +Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 +The top-level auxiliary file: 908144032.aux +The style file: plainnatmod.bst +A level-1 auxiliary file: resume.aux +A level-1 auxiliary file: abstract.aux +A level-1 auxiliary file: remerciements.aux +A level-1 auxiliary file: introduction.aux +A level-1 auxiliary file: chapitre1.aux +A level-1 auxiliary file: chapitre2.aux +A level-1 auxiliary file: chapitre3.aux +A level-1 auxiliary file: chapitre4.aux +A level-1 auxiliary file: chapitre5.aux +A level-1 auxiliary file: chapitre6.aux +A level-1 auxiliary file: chapitre7.aux +A level-1 auxiliary file: chapitre8.aux +A level-1 auxiliary file: chapitre9.aux +A level-1 auxiliary file: conclusion.aux +A level-1 auxiliary file: annexe1.aux +A level-1 auxiliary file: annexe2.aux +A level-1 auxiliary file: annexe3.aux +A level-1 auxiliary file: deed.aux +Database file #1: memoire.bib +You've used 62 entries, + 2773 wiz_defined-function locations, + 971 strings with 13495 characters, +and the built_in function-call counts, 25173 in all, are: += -- 2351 +> -- 953 +< -- 79 ++ -- 341 +- -- 279 +* -- 1701 +:= -- 3797 +add.period$ -- 199 +call.type$ -- 62 +change.case$ -- 259 +chr.to.int$ -- 62 +cite$ -- 124 +duplicate$ -- 1356 +empty$ -- 2325 +format.name$ -- 387 +if$ -- 5400 +int.to.chr$ -- 1 +int.to.str$ -- 1 +missing$ -- 80 +newline$ -- 329 +num.names$ -- 248 +pop$ -- 578 +preamble$ -- 1 +purify$ -- 217 +quote$ -- 0 +skip$ -- 1031 +stack$ -- 0 +substring$ -- 1122 +swap$ -- 191 +text.length$ -- 21 +text.prefix$ -- 0 +top$ -- 0 +type$ -- 582 +warning$ -- 0 +while$ -- 275 +width$ -- 0 +write$ -- 821 diff --git a/memoire/908144032.lof b/memoire/908144032.lof new file mode 100644 index 0000000..2e29e7d --- /dev/null +++ b/memoire/908144032.lof @@ -0,0 +1,30 @@ +\select@language {french} +\select@language {french} +\addvspace {10pt} +\contentsline {figure}{\numberline {1.1}{\ignorespaces Mod\IeC {\`e}le de Bachelier: probabilit\IeC {\'e} compos\IeC {\'e}e\relax }}{8}{figure.caption.12} +\contentsline {figure}{\numberline {1.2}{\ignorespaces Distribution des rendements annuels de 40 titres boursiers, de 1890 \IeC {\`a} 1915, Table XVIII de \cite {mitchell1916critique}\relax }}{10}{figure.caption.13} +\addvspace {10pt} +\contentsline {figure}{\numberline {2.1}{\ignorespaces Premier incr\IeC {\'e}ment d'un processus subordonn\IeC {\'e}\relax }}{23}{figure.caption.16} +\contentsline {figure}{\numberline {2.2}{\ignorespaces Simulation d'un processus de Wiener subordonn\IeC {\'e} par un processus gamma\relax }}{24}{figure.caption.17} +\contentsline {figure}{\numberline {2.3}{\ignorespaces Fonction de densit\IeC {\'e} de la distribution Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e avec diff\IeC {\'e}rents param\IeC {\`e}tres: $GAL(y;\theta ,\sigma ,\kappa ,\tau )$\relax }}{30}{figure.caption.19} +\contentsline {figure}{\numberline {2.4}{\ignorespaces Histogramme et estimateur de densit\IeC {\'e} par noyau de 2500 r\IeC {\'e}alisations de la variable al\IeC {\'e}atoire $Y\sim GAL(\theta =0,\sigma =1,\kappa =2,\tau =1)$\relax }}{34}{figure.caption.20} +\addvspace {10pt} +\contentsline {figure}{\numberline {3.1}{\ignorespaces Probabilit\IeC {\'e} \IeC {\`a} l'extr\IeC {\'e}mit\IeC {\'e} du support\relax }}{40}{figure.caption.23} +\contentsline {figure}{\numberline {3.2}{\ignorespaces Distribution normale tronqu\IeC {\'e}e \IeC {\`a} $w_0$\relax }}{41}{figure.caption.24} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\contentsline {figure}{\numberline {9.1}{\ignorespaces Repr\IeC {\'e}sentation en s\IeC {\'e}rie chronologique de l'\IeC {\'e}chantillon $R_1$\relax }}{98}{figure.caption.28} +\contentsline {figure}{\numberline {9.2}{\ignorespaces Distribution de la variable al\IeC {\'e}atoire $R_1$\relax }}{99}{figure.caption.31} +\contentsline {figure}{\numberline {9.3}{\ignorespaces Graphique Quantile-Quantile\relax }}{100}{figure.caption.33} +\contentsline {figure}{\numberline {9.4}{\ignorespaces \IeC {\'E}quation du point de selle pour $r=0.01$\relax }}{104}{figure.caption.37} +\contentsline {figure}{\numberline {9.5}{\ignorespaces Densit\IeC {\'e} de $R_1^{*}$ selon la m\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{106}{figure.caption.42} +\contentsline {figure}{\numberline {9.6}{\ignorespaces Densit\IeC {\'e} de $R_1^{*}$ selon la m\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale\relax }}{107}{figure.caption.43} +\contentsline {figure}{\numberline {9.7}{\ignorespaces Prix de l'option selon les param\IeC {\`e}tres estim\IeC {\'e}s avec la m\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{110}{figure.caption.49} +\contentsline {figure}{\numberline {9.8}{\ignorespaces Prix de l'option selon les param\IeC {\`e}tres estim\IeC {\'e}s avec la m\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale\relax }}{111}{figure.caption.50} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} diff --git a/memoire/908144032.log b/memoire/908144032.log new file mode 100644 index 0000000..8de3ce7 --- /dev/null +++ b/memoire/908144032.log @@ -0,0 +1,2277 @@ +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.10.25) 21 NOV 2013 20:00 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**\input 908144032.tex +(./908144032.tex (./ulthese.cls +Document Class: ulthese 2012/09/30 v1.0 Classe pour les theses et memoires de l +'Universite Laval +(/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty +Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC) +) +(/usr/share/texlive/texmf-dist/tex/latex/memoir/memoir.cls +Document Class: memoir 2011/03/06 v3.6j configurable book, report, article docu +ment class +\onelineskip=\skip41 +\lxvchars=\skip42 +\xlvchars=\skip43 +\@memcnta=\count79 +\c@@memmarkcntra=\count80 +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifpdf.sty +Package: ifpdf 2011/01/30 v2.3 Provides the ifpdf switch (HO) +Package ifpdf Info: pdfTeX in PDF mode is detected. +) +(/usr/share/texlive/texmf-dist/tex/latex/ifetex/ifetex.sty +Package: ifetex 2011/12/15 v1.2 ifetex + +(/usr/share/texlive/texmf-dist/tex/plain/ifetex/ifetex.tex)) +(/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional +) +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifluatex.sty +Package: ifluatex 2010/03/01 v1.3 Provides the ifluatex switch (HO) +Package ifluatex Info: LuaTeX not detected. +) +\stockheight=\skip44 +\stockwidth=\skip45 +\trimtop=\skip46 +\trimedge=\skip47 + +(/usr/share/texlive/texmf-dist/tex/latex/etex-pkg/etex.sty +Package: etex 1998/03/26 v2.0 eTeX basic definition package (PEB) +\et@xins=\count81 +) +(/usr/share/texlive/texmf-dist/tex/latex/memoir/mem11.clo +File: mem11.clo 2008/01/30 v0.3 memoir class 11pt size option +) +\binding=\skip48 +\spinemargin=\skip49 +\foremargin=\skip50 +\uppermargin=\skip51 +\lowermargin=\skip52 +\headdrop=\skip53 +\normalrulethickness=\skip54 +\mem@maxheadheight=\skip55 +\mem@maxfootheight=\skip56 +\headwidth=\skip57 +\c@storedpagenumber=\count82 +\memPD=\dimen102 +\m@mabparskip=\skip58 +\thanksmarkwidth=\skip59 +\thanksmarksep=\skip60 +\droptitle=\skip61 +\c@book=\count83 +\c@part=\count84 +\c@chapter=\count85 +\c@section=\count86 +\c@subsection=\count87 +\c@subsubsection=\count88 +\c@paragraph=\count89 +\c@subparagraph=\count90 +\beforechapskip=\skip62 +\midchapskip=\skip63 +\afterchapskip=\skip64 +\chapindent=\skip65 +\bottomsectionskip=\skip66 +\secindent=\skip67 +\beforesecskip=\skip68 +\aftersecskip=\skip69 +\subsecindent=\skip70 +\beforesubsecskip=\skip71 +\aftersubsecskip=\skip72 +\subsubsecindent=\skip73 +\beforesubsubsecskip=\skip74 +\aftersubsubsecskip=\skip75 +\paraindent=\skip76 +\beforeparaskip=\skip77 +\afterparaskip=\skip78 +\subparaindent=\skip79 +\beforesubparaskip=\skip80 +\aftersubparaskip=\skip81 +\pfbreakskip=\skip82 +\c@@ppsavesec=\count91 +\c@@ppsaveapp=\count92 +\ragrparindent=\dimen103 +\everylistparindent=\dimen104 +\parsepi=\skip83 +\topsepi=\skip84 +\itemsepi=\skip85 +\parsepii=\skip86 +\topsepii=\skip87 +\topsepiii=\skip88 +\itemsepii=\skip89 +\itemsepiii=\skip90 +\partopsepii=\skip91 +\partopsepiii=\skip92 +\m@msavetopsep=\skip93 +\m@msavepartopsep=\skip94 +\@enLab=\toks14 +\abstitleskip=\skip95 +\absleftindent=\skip96 +\abs@leftindent=\dimen105 +\absrightindent=\skip97 +\absparindent=\skip98 +\absparsep=\skip99 +\c@vslineno=\count93 +\c@poemline=\count94 +\c@modulo@vs=\count95 +\c@memfvsline=\count96 +\vleftskip=\skip100 +\vrightskip=\skip101 +\stanzaskip=\skip102 +\versewidth=\skip103 +\vgap=\skip104 +\vindent=\skip105 +\vleftmargin=\dimen106 +\c@verse=\count97 +\c@chrsinstr=\count98 +\beforepoemtitleskip=\skip106 +\afterpoemtitleskip=\skip107 +\c@poem=\count99 +\beforePoemTitleskip=\skip108 +\midPoemTitleskip=\skip109 +\afterPoemTitleskip=\skip110 +\col@sep=\dimen107 +\extrarowheight=\dimen108 +\NC@list=\toks15 +\extratabsurround=\skip111 +\backup@length=\skip112 +\TX@col@width=\dimen109 +\TX@old@table=\dimen110 +\TX@old@col=\dimen111 +\TX@target=\dimen112 +\TX@delta=\dimen113 +\TX@cols=\count100 +\TX@ftn=\toks16 +\heavyrulewidth=\dimen114 +\lightrulewidth=\dimen115 +\cmidrulewidth=\dimen116 +\belowrulesep=\dimen117 +\belowbottomsep=\dimen118 +\aboverulesep=\dimen119 +\abovetopsep=\dimen120 +\cmidrulesep=\dimen121 +\cmidrulekern=\dimen122 +\defaultaddspace=\dimen123 +\@cmidla=\count101 +\@cmidlb=\count102 +\@aboverulesep=\dimen124 +\@belowrulesep=\dimen125 +\@thisruleclass=\count103 +\@lastruleclass=\count104 +\@thisrulewidth=\dimen126 +\ctableftskip=\skip113 +\ctabrightskip=\skip114 +\abovecolumnspenalty=\count105 +\@linestogo=\count106 +\@cellstogo=\count107 +\@cellsincolumn=\count108 +\crtok=\toks17 +\@mincolumnwidth=\dimen127 +\c@newflo@tctr=\count109 +\mem@margin@floatbox=\box26 +\@contcwidth=\skip115 +\@contindw=\skip116 +\abovecaptionskip=\skip117 +\belowcaptionskip=\skip118 +\subfloattopskip=\skip119 +\subfloatcapskip=\skip120 +\subfloatcaptopadj=\skip121 +\subfloatbottomskip=\skip122 +\subfloatlabelskip=\skip123 +\subfloatcapmargin=\dimen128 +\c@@contsubnum=\count110 +\m@mscap@capbox=\box27 +\m@mscap@fbox=\box28 +\sidecapsep=\dimen129 +\sidecapwidth=\dimen130 +\m@m@tempdima=\dimen131 +\m@mscapraise=\dimen132 +\sidecapraise=\dimen133 +\m@mscapmainwidth=\dimen134 +\m@mscaplkern=\dimen135 +\beforeepigraphskip=\skip124 +\afterepigraphskip=\skip125 +\epigraphwidth=\skip126 +\epigraphrule=\skip127 +LaTeX Info: Redefining \em on input line 7377. +LaTeX Info: Redefining \emph on input line 7385. +\tocentryskip=\skip128 +\tocbaseline=\skip129 +\cftparskip=\skip130 +\cftbeforebookskip=\skip131 +\cftbookindent=\dimen136 +\cftbooknumwidth=\dimen137 +\cftbeforepartskip=\skip132 +\cftpartindent=\skip133 +\cftpartnumwidth=\skip134 +\cftbeforechapterskip=\skip135 +\cftchapterindent=\skip136 +\cftchapternumwidth=\skip137 +\cftbeforesectionskip=\skip138 +\cftsectionindent=\skip139 +\cftsectionnumwidth=\skip140 +\cftbeforesubsectionskip=\skip141 +\cftsubsectionindent=\skip142 +\cftsubsectionnumwidth=\skip143 +\cftbeforesubsubsectionskip=\skip144 +\cftsubsubsectionindent=\skip145 +\cftsubsubsectionnumwidth=\skip146 +\cftbeforeparagraphskip=\skip147 +\cftparagraphindent=\skip148 +\cftparagraphnumwidth=\skip149 +\cftbeforesubparagraphskip=\skip150 +\cftsubparagraphindent=\skip151 +\cftsubparagraphnumwidth=\skip152 +\prechapterprecisshift=\dimen138 +\c@maxsecnumdepth=\count111 +\bibindent=\dimen139 +\bibitemsep=\skip153 +\indexcolsep=\skip154 +\indexrule=\skip155 +\indexmarkstyle=\toks18 +\@indexbox=\insert233 +\glossarycolsep=\dimen140 +\glossaryrule=\dimen141 +\sideparvshift=\skip156 +\sideins=\insert232 +\sidebartopsep=\skip157 +\sidebarhsep=\skip158 +\sidebarvsep=\skip159 +\sidebarwidth=\skip160 +\footmarkwidth=\skip161 +\footmarksep=\skip162 +\footparindent=\skip163 +\footinsdim=\skip164 +\footinsv@r=\insert231 +\@mpfootinsv@r=\insert230 +\m@m@k=\count112 +\m@m@h=\dimen142 +\m@mipn@skip=\skip165 +\sidefootins=\insert229 +\sidefootadjust=\skip166 +\sidefootheight=\skip167 +\sidefoothsep=\skip168 +\sidefootvsep=\skip169 +\sidefootwidth=\skip170 +\m@mdownsf=\skip171 +\c@sidefootnote=\count113 +\sidefootmarkwidth=\skip172 +\sidefootmarksep=\skip173 +\sidefootparindent=\skip174 +\c@pagenote=\count114 +\c@pagenoteshadow=\count115 +\mem@pn@lastkern=\skip175 +\every@verbatim=\toks19 +\afterevery@verbatim=\toks20 +\verbatim@line=\toks21 +\tab@position=\count116 +\verbatim@in@stream=\read1 +\verbatimindent=\skip176 +\verbatim@out=\write3 +\bvboxsep=\skip177 +\c@memfbvline=\count117 +\c@bvlinectr=\count118 +\bvnumlength=\skip178 +\fb@frw=\dimen143 +\fb@frh=\dimen144 +\FrameRule=\dimen145 +\FrameSep=\dimen146 +\c@cp@cntr=\count119 +LaTeX Info: Redefining \: on input line 11807. +LaTeX Info: Redefining \! on input line 11809. +\c@ism@mctr=\count120 +\c@xsm@mctr=\count121 +\c@csm@mctr=\count122 +\c@ksm@mctr=\count123 +\c@xksm@mctr=\count124 +\c@cksm@mctr=\count125 +\c@msm@mctr=\count126 +\c@xmsm@mctr=\count127 +\c@cmsm@mctr=\count128 +\c@bsm@mctr=\count129 +\c@workm@mctr=\count130 +\c@sheetsequence=\count131 +\c@lastsheet=\count132 +\c@lastpage=\count133 +\c@figure=\count134 +\c@lofdepth=\count135 +\cftbeforefigureskip=\skip179 +\cftfigureindent=\skip180 +\cftfigurenumwidth=\skip181 +\c@table=\count136 +\c@lotdepth=\count137 +\cftbeforetableskip=\skip182 +\cfttableindent=\skip183 +\cfttablenumwidth=\skip184 +Package abstract [2008/07/23] emulated by memoir. +Package appendix [2008/07/23] emulated by memoir. +Package array [2008/07/23] emulated by memoir. +Package booktabs [2008/07/23] emulated by memoir. +Package ccaption [2008/07/23] emulated by memoir. +Package changepage [2008/07/23] emulated by memoir. +Package chngcntr [2008/07/23] emulated by memoir. +Package chngpage [2008/07/23] emulated by memoir. +Package crop emulated by memoir. +Package dcolumn [2008/07/23] emulated by memoir. +Package delarray [2008/07/23] emulated by memoir. +Package enumerate [2008/07/23] emulated by memoir. +Package epigraph [2008/07/23] emulated by memoir. +Package ifmtarg [2008/07/23] emulated by memoir. +Package index [2008/07/23] emulated by memoir. +Package makeidx [2008/07/23] emulated by memoir. +Package moreverb [2008/07/23] emulated by memoir. +Package mparhack [2008/07/23] emulated by memoir. +Package needspace [2008/07/23] emulated by memoir. +Package newfile [2008/07/23] emulated by memoir. +Package nextpage [2008/07/23] emulated by memoir. +Package pagenote [2008/07/23] emulated by memoir. +Package parskip [2008/07/23] emulated by memoir. +Package patchcmd [2008/07/23] emulated by memoir. +Package setspace [2008/07/23] emulated by memoir. +Package shortvrb [2008/07/23] emulated by memoir. +Package showidx [2008/07/23] emulated by memoir. +Package tabularx [2008/07/23] emulated by memoir. +Package titleref [2008/07/23] emulated by memoir. +Package titling [2008/07/23] emulated by memoir. +Package tocbibind [2008/07/23] emulated by memoir. +Package tocloft [2008/07/23] emulated by memoir. +Package tocvsec2 [2008/07/23] emulated by memoir. +Package verbatim [2008/07/23] emulated by memoir. +Package verse [2008/07/23] emulated by memoir. + +(/usr/share/texlive/texmf-dist/tex/latex/memoir/mempatch.sty +Package: mempatch 2009/07/24 v6.0f Patches for memoir class v1.6180339 +)) +(/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +Package: fontenc 2005/09/27 v1.99g Standard LaTeX package + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.def +File: t1enc.def 2005/09/27 v1.99g Standard LaTeX file +LaTeX Font Info: Redeclaring font encoding T1 on input line 43. +)) +(/usr/share/texlive/texmf-dist/tex/latex/natbib/natbib.sty +Package: natbib 2010/09/13 8.31b (PWD, AO) +\bibhang=\skip185 +\bibsep=\skip186 +LaTeX Info: Redefining \cite on input line 694. +\c@NAT@ctr=\count138 +) +(/var/lib/texmf/tex/generic/babel/babel.sty +Package: babel 2008/07/08 v3.8m The Babel package + +(/usr/share/texlive/texmf-dist/tex/generic/babel/english.ldf +Language: english 2005/03/30 v3.3o English support from the babel system + +(/usr/share/texlive/texmf-dist/tex/generic/babel/babel.def +File: babel.def 2008/07/08 v3.8m Babel common definitions +\babel@savecnt=\count139 +\U@D=\dimen147 +) +\l@canadian = a dialect from \language\l@american +\l@australian = a dialect from \language\l@british +\l@newzealand = a dialect from \language\l@british +) +(/usr/share/texlive/texmf-dist/tex/generic/babel/frenchb.ldf +Language: frenchb 2010/08/21 v2.5a French support from the babel system +Package babel Info: Making : an active character on input line 234. +Package babel Info: Making ; an active character on input line 235. +Package babel Info: Making ! an active character on input line 236. +Package babel Info: Making ? an active character on input line 237. +\FB@Mht=\dimen148 +\std@mcc=\count140 +\dec@mcc=\count141 +\parindentFFN=\dimen149 + +************************************* +* Local config file frenchb.cfg used +* +(/usr/share/texlive/texmf-dist/tex/generic/babel/frenchb.cfg))) +(/usr/share/texlive/texmf-dist/tex/latex/carlisle/scalefnt.sty) +(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +Package: keyval 1999/03/16 v1.13 key=value parser (DPC) +\KV@toks@=\toks22 +) +(/usr/share/texlive/texmf-dist/tex/latex/numprint/numprint.sty +Package: numprint 2008/02/17 v1.38 Print numbers (HH) +\c@nprt@mantissa@digitsbefore=\count142 +\c@nprt@mantissa@digitsafter=\count143 +\c@nprt@exponent@digitsbefore=\count144 +\c@nprt@exponent@digitsafter=\count145 +\nprt@digitwidth=\skip187 +\nprt@sepwidth=\skip188 +\nprt@decimalwidth=\skip189 +\nprt@blockwidth=\skip190 +\nprt@digittoks=\toks23 +\nprt@pretoks=\toks24 +\nprt@posttoks=\toks25 +\nprt@thisdigit=\count146 +\nprt@curpos=\count147 +\nprt@rndpos=\count148 +\c@nprt@digitsfirstblock=\count149 +\c@nprt@blockcnt=\count150 +\c@nprt@cntprint=\count151 + +No configuration file `numprint.cfg' found.) +(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty +Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR) + +(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty +Package: graphics 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR) + +(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty +Package: trig 1999/03/16 v1.09 sin cos tan (DPC) +) +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/graphics.cfg +File: graphics.cfg 2010/04/23 v1.9 graphics configuration of TeX Live +) +Package graphics Info: Driver file: pdftex.def on input line 91. + +(/usr/share/texlive/texmf-dist/tex/latex/pdftex-def/pdftex.def +File: pdftex.def 2011/05/27 v0.06d Graphics/color for pdfTeX + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty +Package: infwarerr 2010/04/08 v1.3 Providing info/warning/error messages (HO) +) +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty +Package: ltxcmds 2011/11/09 v1.22 LaTeX kernel commands for general use (HO) +) +\Gread@gobject=\count152 +)) +\Gin@req@height=\dimen150 +\Gin@req@width=\dimen151 +) +(/usr/share/texmf/tex/latex/xcolor/xcolor.sty +Package: xcolor 2007/01/21 v2.11 LaTeX color extensions (UK) + +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg +File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive +) +Package xcolor Info: Driver file: pdftex.def on input line 225. +Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1337. +Package xcolor Info: Model `hsb' substituted by `rgb' on input line 1341. +Package xcolor Info: Model `RGB' extended on input line 1353. +Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1355. +Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1356. +Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1357. +Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1358. +Package xcolor Info: Model `Gray' substituted by `gray' on input line 1359. +Package xcolor Info: Model `wave' substituted by `hsb' on input line 1360. +) +(/usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty +Package: textcomp 2005/09/27 v1.99g Standard LaTeX package +Package textcomp Info: Sub-encoding information: +(textcomp) 5 = only ISO-Adobe without \textcurrency +(textcomp) 4 = 5 + \texteuro +(textcomp) 3 = 4 + \textohm +(textcomp) 2 = 3 + \textestimated + \textcurrency +(textcomp) 1 = TS1 - \textcircled - \t +(textcomp) 0 = TS1 (full) +(textcomp) Font families with sub-encoding setting implement +(textcomp) only a restricted character set as indicated. +(textcomp) Family '?' is the default used for unknown fonts. +(textcomp) See the documentation for details. +Package textcomp Info: Setting ? sub-encoding to TS1/1 on input line 71. + +(/usr/share/texlive/texmf-dist/tex/latex/base/ts1enc.def +File: ts1enc.def 2001/06/05 v3.0e (jk/car/fm) Standard LaTeX file +) +LaTeX Info: Redefining \oldstylenums on input line 266. +Package textcomp Info: Setting cmr sub-encoding to TS1/0 on input line 281. +Package textcomp Info: Setting cmss sub-encoding to TS1/0 on input line 282. +Package textcomp Info: Setting cmtt sub-encoding to TS1/0 on input line 283. +Package textcomp Info: Setting cmvtt sub-encoding to TS1/0 on input line 284. +Package textcomp Info: Setting cmbr sub-encoding to TS1/0 on input line 285. +Package textcomp Info: Setting cmtl sub-encoding to TS1/0 on input line 286. +Package textcomp Info: Setting ccr sub-encoding to TS1/0 on input line 287. +Package textcomp Info: Setting ptm sub-encoding to TS1/4 on input line 288. +Package textcomp Info: Setting pcr sub-encoding to TS1/4 on input line 289. +Package textcomp Info: Setting phv sub-encoding to TS1/4 on input line 290. +Package textcomp Info: Setting ppl sub-encoding to TS1/3 on input line 291. +Package textcomp Info: Setting pag sub-encoding to TS1/4 on input line 292. +Package textcomp Info: Setting pbk sub-encoding to TS1/4 on input line 293. +Package textcomp Info: Setting pnc sub-encoding to TS1/4 on input line 294. +Package textcomp Info: Setting pzc sub-encoding to TS1/4 on input line 295. +Package textcomp Info: Setting bch sub-encoding to TS1/4 on input line 296. +Package textcomp Info: Setting put sub-encoding to TS1/5 on input line 297. +Package textcomp Info: Setting uag sub-encoding to TS1/5 on input line 298. +Package textcomp Info: Setting ugq sub-encoding to TS1/5 on input line 299. +Package textcomp Info: Setting ul8 sub-encoding to TS1/4 on input line 300. +Package textcomp Info: Setting ul9 sub-encoding to TS1/4 on input line 301. +Package textcomp Info: Setting augie sub-encoding to TS1/5 on input line 302. +Package textcomp Info: Setting dayrom sub-encoding to TS1/3 on input line 303. +Package textcomp Info: Setting dayroms sub-encoding to TS1/3 on input line 304. + +Package textcomp Info: Setting pxr sub-encoding to TS1/0 on input line 305. +Package textcomp Info: Setting pxss sub-encoding to TS1/0 on input line 306. +Package textcomp Info: Setting pxtt sub-encoding to TS1/0 on input line 307. +Package textcomp Info: Setting txr sub-encoding to TS1/0 on input line 308. +Package textcomp Info: Setting txss sub-encoding to TS1/0 on input line 309. +Package textcomp Info: Setting txtt sub-encoding to TS1/0 on input line 310. +Package textcomp Info: Setting lmr sub-encoding to TS1/0 on input line 311. +Package textcomp Info: Setting lmdh sub-encoding to TS1/0 on input line 312. +Package textcomp Info: Setting lmss sub-encoding to TS1/0 on input line 313. +Package textcomp Info: Setting lmssq sub-encoding to TS1/0 on input line 314. +Package textcomp Info: Setting lmvtt sub-encoding to TS1/0 on input line 315. +Package textcomp Info: Setting qhv sub-encoding to TS1/0 on input line 316. +Package textcomp Info: Setting qag sub-encoding to TS1/0 on input line 317. +Package textcomp Info: Setting qbk sub-encoding to TS1/0 on input line 318. +Package textcomp Info: Setting qcr sub-encoding to TS1/0 on input line 319. +Package textcomp Info: Setting qcs sub-encoding to TS1/0 on input line 320. +Package textcomp Info: Setting qpl sub-encoding to TS1/0 on input line 321. +Package textcomp Info: Setting qtm sub-encoding to TS1/0 on input line 322. +Package textcomp Info: Setting qzc sub-encoding to TS1/0 on input line 323. +Package textcomp Info: Setting qhvc sub-encoding to TS1/0 on input line 324. +Package textcomp Info: Setting futs sub-encoding to TS1/4 on input line 325. +Package textcomp Info: Setting futx sub-encoding to TS1/4 on input line 326. +Package textcomp Info: Setting futj sub-encoding to TS1/4 on input line 327. +Package textcomp Info: Setting hlh sub-encoding to TS1/3 on input line 328. +Package textcomp Info: Setting hls sub-encoding to TS1/3 on input line 329. +Package textcomp Info: Setting hlst sub-encoding to TS1/3 on input line 330. +Package textcomp Info: Setting hlct sub-encoding to TS1/5 on input line 331. +Package textcomp Info: Setting hlx sub-encoding to TS1/5 on input line 332. +Package textcomp Info: Setting hlce sub-encoding to TS1/5 on input line 333. +Package textcomp Info: Setting hlcn sub-encoding to TS1/5 on input line 334. +Package textcomp Info: Setting hlcw sub-encoding to TS1/5 on input line 335. +Package textcomp Info: Setting hlcf sub-encoding to TS1/5 on input line 336. +Package textcomp Info: Setting pplx sub-encoding to TS1/3 on input line 337. +Package textcomp Info: Setting pplj sub-encoding to TS1/3 on input line 338. +Package textcomp Info: Setting ptmx sub-encoding to TS1/4 on input line 339. +Package textcomp Info: Setting ptmj sub-encoding to TS1/4 on input line 340. +) +(/usr/share/texlive/texmf-dist/tex/latex/psnfss/helvet.sty +Package: helvet 2005/04/12 PSNFSS-v9.2a (WaS) +) + +****************************************************** +Stock height and width: 794.96999pt by 614.295pt +Top and edge trims: 0pt and 0pt +Page height and width: 794.96999pt by 614.295pt +Text height and width: 636.60028pt by 443pt +Spine and edge margins: 99.58464pt and 71.13188pt +Upper and lower margins: 71.13188pt and 87.23782pt +Headheight and headsep: 13.6pt and 19.8738pt +Footskip: 56.9055pt +Columnsep and columnseprule: 10pt and 0pt +Marginparsep and marginparwidth: 7pt and 111pt +Sidecapsep and sidecapwidth: 7pt and 111pt +Sidebarhsep and sidebarwidth: 7pt and 111pt +Sidebarvsep and sidebartopsep: 13.6pt and 0pt +Sidebarheight: 539.9202pt +Sidefoothsep and sidefootwidth: 7pt and 111pt +Sidefootvsep and sidefootheight: 13.6pt and 539.9202pt +****************************************************** + +) (/usr/share/texlive/texmf-dist/tex/latex/koma-script/scrtime.sty +Package: scrtime 2012/05/15 v3.11 KOMA-Script package (time of LaTeX run) + +(/usr/share/texlive/texmf-dist/tex/latex/koma-script/scrkbase.sty +Package: scrkbase 2012/05/15 v3.11 KOMA-Script package (KOMA-Script-dependent b +asics and keyval usage) + +(/usr/share/texlive/texmf-dist/tex/latex/koma-script/scrbase.sty +Package: scrbase 2012/05/15 v3.11 KOMA-Script package (KOMA-Script-independent +basics and keyval usage) + +(/usr/share/texlive/texmf-dist/tex/latex/koma-script/scrlfile.sty +Package: scrlfile 2011/03/09 v3.09 KOMA-Script package (loading files) + +Package scrlfile, 2011/03/09 v3.09 KOMA-Script package (loading files) + Copyright (C) Markus Kohm + + +LaTeX Warning: Command \InputIfFileExists has changed. + Check if current package is valid. + +)))) (/usr/share/texmf/tex/latex/pgf/frontendlayer/tikz.sty +(/usr/share/texmf/tex/latex/pgf/basiclayer/pgf.sty +(/usr/share/texmf/tex/latex/pgf/utilities/pgfrcs.sty +(/usr/share/texmf/tex/generic/pgf/utilities/pgfutil-common.tex +\pgfutil@everybye=\toks26 +) +(/usr/share/texmf/tex/generic/pgf/utilities/pgfutil-latex.def +\pgfutil@abb=\box29 + +(/usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty +Package: everyshi 2001/05/15 v3.00 EveryShipout Package (MS) +)) +(/usr/share/texmf/tex/generic/pgf/utilities/pgfrcs.code.tex +Package: pgfrcs 2010/10/25 v2.10 (rcs-revision 1.24) +)) +Package: pgf 2008/01/15 v2.10 (rcs-revision 1.12) + +(/usr/share/texmf/tex/latex/pgf/basiclayer/pgfcore.sty +(/usr/share/texmf/tex/latex/pgf/systemlayer/pgfsys.sty +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys.code.tex +Package: pgfsys 2010/06/30 v2.10 (rcs-revision 1.37) + +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeys.code.tex +\pgfkeys@pathtoks=\toks27 +\pgfkeys@temptoks=\toks28 + +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex +\pgfkeys@tmptoks=\toks29 +)) +\pgf@x=\dimen152 +\pgf@y=\dimen153 +\pgf@xa=\dimen154 +\pgf@ya=\dimen155 +\pgf@xb=\dimen156 +\pgf@yb=\dimen157 +\pgf@xc=\dimen158 +\pgf@yc=\dimen159 +\w@pgf@writea=\write4 +\r@pgf@reada=\read2 +\c@pgf@counta=\count153 +\c@pgf@countb=\count154 +\c@pgf@countc=\count155 +\c@pgf@countd=\count156 + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgf.cfg +File: pgf.cfg 2008/05/14 (rcs-revision 1.7) +) +Package pgfsys Info: Driver file for pgf: pgfsys-pdftex.def on input line 900. + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys-pdftex.def +File: pgfsys-pdftex.def 2009/05/22 (rcs-revision 1.26) + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def +File: pgfsys-common-pdf.def 2008/05/19 (rcs-revision 1.10) +))) +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +File: pgfsyssoftpath.code.tex 2008/07/18 (rcs-revision 1.7) +\pgfsyssoftpath@smallbuffer@items=\count157 +\pgfsyssoftpath@bigbuffer@items=\count158 +) +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +File: pgfsysprotocol.code.tex 2006/10/16 (rcs-revision 1.4) +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcore.code.tex +Package: pgfcore 2010/04/11 v2.10 (rcs-revision 1.7) + +(/usr/share/texmf/tex/generic/pgf/math/pgfmath.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathcalc.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathutil.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathparser.code.tex +\pgfmath@dimen=\dimen160 +\pgfmath@count=\count159 +\pgfmath@box=\box30 +\pgfmath@toks=\toks30 +\pgfmath@stack@operand=\toks31 +\pgfmath@stack@operation=\toks32 +) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.random.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.base.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.round.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex))) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfloat.code.tex +\c@pgfmathroundto@lastzeros=\count160 +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex +File: pgfcorepoints.code.tex 2010/04/09 (rcs-revision 1.20) +\pgf@picminx=\dimen161 +\pgf@picmaxx=\dimen162 +\pgf@picminy=\dimen163 +\pgf@picmaxy=\dimen164 +\pgf@pathminx=\dimen165 +\pgf@pathmaxx=\dimen166 +\pgf@pathminy=\dimen167 +\pgf@pathmaxy=\dimen168 +\pgf@xx=\dimen169 +\pgf@xy=\dimen170 +\pgf@yx=\dimen171 +\pgf@yy=\dimen172 +\pgf@zx=\dimen173 +\pgf@zy=\dimen174 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex +File: pgfcorepathconstruct.code.tex 2010/08/03 (rcs-revision 1.24) +\pgf@path@lastx=\dimen175 +\pgf@path@lasty=\dimen176 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex +File: pgfcorepathusage.code.tex 2008/04/22 (rcs-revision 1.12) +\pgf@shorten@end@additional=\dimen177 +\pgf@shorten@start@additional=\dimen178 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex +File: pgfcorescopes.code.tex 2010/09/08 (rcs-revision 1.34) +\pgfpic=\box31 +\pgf@hbox=\box32 +\pgf@layerbox@main=\box33 +\pgf@picture@serial@count=\count161 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex +File: pgfcoregraphicstate.code.tex 2008/04/22 (rcs-revision 1.9) +\pgflinewidth=\dimen179 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex +File: pgfcoretransformations.code.tex 2009/06/10 (rcs-revision 1.11) +\pgf@pt@x=\dimen180 +\pgf@pt@y=\dimen181 +\pgf@pt@temp=\dimen182 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorequick.code.tex +File: pgfcorequick.code.tex 2008/10/09 (rcs-revision 1.3) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex +File: pgfcoreobjects.code.tex 2006/10/11 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex +File: pgfcorepathprocessing.code.tex 2008/10/09 (rcs-revision 1.8) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex +File: pgfcorearrows.code.tex 2008/04/23 (rcs-revision 1.11) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex +File: pgfcoreshade.code.tex 2008/11/23 (rcs-revision 1.13) +\pgf@max=\dimen183 +\pgf@sys@shading@range@num=\count162 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex +File: pgfcoreimage.code.tex 2010/03/25 (rcs-revision 1.16) + +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex +File: pgfcoreexternal.code.tex 2010/09/01 (rcs-revision 1.17) +\pgfexternal@startupbox=\box34 +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex +File: pgfcorelayers.code.tex 2010/08/27 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex +File: pgfcoretransparency.code.tex 2008/01/17 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex +File: pgfcorepatterns.code.tex 2009/07/02 (rcs-revision 1.3) +))) +(/usr/share/texmf/tex/generic/pgf/modules/pgfmoduleshapes.code.tex +File: pgfmoduleshapes.code.tex 2010/09/09 (rcs-revision 1.13) +\pgfnodeparttextbox=\box35 +) +(/usr/share/texmf/tex/generic/pgf/modules/pgfmoduleplot.code.tex +File: pgfmoduleplot.code.tex 2010/10/22 (rcs-revision 1.8) +) +(/usr/share/texmf/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +Package: pgfcomp-version-0-65 2007/07/03 v2.10 (rcs-revision 1.7) +\pgf@nodesepstart=\dimen184 +\pgf@nodesepend=\dimen185 +) +(/usr/share/texmf/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +Package: pgfcomp-version-1-18 2007/07/23 v2.10 (rcs-revision 1.1) +)) +(/usr/share/texmf/tex/latex/pgf/utilities/pgffor.sty +(/usr/share/texmf/tex/latex/pgf/utilities/pgfkeys.sty +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeys.code.tex)) +(/usr/share/texmf/tex/generic/pgf/utilities/pgffor.code.tex +Package: pgffor 2010/03/23 v2.10 (rcs-revision 1.18) +\pgffor@iter=\dimen186 +\pgffor@skip=\dimen187 +\pgffor@stack=\toks33 +\pgffor@toks=\toks34 +)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +Package: tikz 2010/10/13 v2.10 (rcs-revision 1.76) + +(/usr/share/texmf/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex +File: pgflibraryplothandlers.code.tex 2010/05/31 v2.10 (rcs-revision 1.15) +\pgf@plot@mark@count=\count163 +\pgfplotmarksize=\dimen188 +) +\tikz@lastx=\dimen189 +\tikz@lasty=\dimen190 +\tikz@lastxsaved=\dimen191 +\tikz@lastysaved=\dimen192 +\tikzleveldistance=\dimen193 +\tikzsiblingdistance=\dimen194 +\tikz@figbox=\box36 +\tikz@tempbox=\box37 +\tikztreelevel=\count164 +\tikznumberofchildren=\count165 +\tikznumberofcurrentchild=\count166 +\tikz@fig@count=\count167 + +(/usr/share/texmf/tex/generic/pgf/modules/pgfmodulematrix.code.tex +File: pgfmodulematrix.code.tex 2010/08/24 (rcs-revision 1.4) +\pgfmatrixcurrentrow=\count168 +\pgfmatrixcurrentcolumn=\count169 +\pgf@matrix@numberofcolumns=\count170 +) +\tikz@expandcount=\count171 + +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopat +hs.code.tex +File: tikzlibrarytopaths.code.tex 2008/06/17 v2.10 (rcs-revision 1.2) +))) (/usr/share/texlive/texmf-dist/tex/latex/multirow/multirow.sty +\bigstrutjot=\dimen195 +) +(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +Package: inputenc 2008/03/30 v1.1d Input encoding file +\inpenc@prehook=\toks35 +\inpenc@posthook=\toks36 + +(/usr/share/texlive/texmf-dist/tex/latex/base/utf8.def +File: utf8.def 2008/04/05 v1.1m UTF-8 support for inputenc +Now handling font encoding OML ... +... no UTF-8 mapping file for font encoding OML +Now handling font encoding T1 ... +... processing UTF-8 mapping file for font encoding T1 + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu +File: t1enc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A1 (decimal 161) + defining Unicode char U+00A3 (decimal 163) + defining Unicode char U+00AB (decimal 171) + defining Unicode char U+00BB (decimal 187) + defining Unicode char U+00BF (decimal 191) + defining Unicode char U+00C0 (decimal 192) + defining Unicode char U+00C1 (decimal 193) + defining Unicode char U+00C2 (decimal 194) + defining Unicode char U+00C3 (decimal 195) + defining Unicode char U+00C4 (decimal 196) + defining Unicode char U+00C5 (decimal 197) + defining Unicode char U+00C6 (decimal 198) + defining Unicode char U+00C7 (decimal 199) + defining Unicode char U+00C8 (decimal 200) + defining Unicode char U+00C9 (decimal 201) + defining Unicode char U+00CA (decimal 202) + defining Unicode char U+00CB (decimal 203) + defining Unicode char U+00CC (decimal 204) + defining Unicode char U+00CD (decimal 205) + defining Unicode char U+00CE (decimal 206) + defining Unicode char U+00CF (decimal 207) + defining Unicode char U+00D0 (decimal 208) + defining Unicode char U+00D1 (decimal 209) + defining Unicode char U+00D2 (decimal 210) + defining Unicode char U+00D3 (decimal 211) + defining Unicode char U+00D4 (decimal 212) + defining Unicode char U+00D5 (decimal 213) + defining Unicode char U+00D6 (decimal 214) + defining Unicode char U+00D8 (decimal 216) + defining Unicode char U+00D9 (decimal 217) + defining Unicode char U+00DA (decimal 218) + defining Unicode char U+00DB (decimal 219) + defining Unicode char U+00DC (decimal 220) + defining Unicode char U+00DD (decimal 221) + defining Unicode char U+00DE (decimal 222) + defining Unicode char U+00DF (decimal 223) + defining Unicode char U+00E0 (decimal 224) + defining Unicode char U+00E1 (decimal 225) + defining Unicode char U+00E2 (decimal 226) + defining Unicode char U+00E3 (decimal 227) + defining Unicode char U+00E4 (decimal 228) + defining Unicode char U+00E5 (decimal 229) + defining Unicode char U+00E6 (decimal 230) + defining Unicode char U+00E7 (decimal 231) + defining Unicode char U+00E8 (decimal 232) + defining Unicode char U+00E9 (decimal 233) + defining Unicode char U+00EA (decimal 234) + defining Unicode char U+00EB (decimal 235) + defining Unicode char U+00EC (decimal 236) + defining Unicode char U+00ED (decimal 237) + defining Unicode char U+00EE (decimal 238) + defining Unicode char U+00EF (decimal 239) + defining Unicode char U+00F0 (decimal 240) + defining Unicode char U+00F1 (decimal 241) + defining Unicode char U+00F2 (decimal 242) + defining Unicode char U+00F3 (decimal 243) + defining Unicode char U+00F4 (decimal 244) + defining Unicode char U+00F5 (decimal 245) + defining Unicode char U+00F6 (decimal 246) + defining Unicode char U+00F8 (decimal 248) + defining Unicode char U+00F9 (decimal 249) + defining Unicode char U+00FA (decimal 250) + defining Unicode char U+00FB (decimal 251) + defining Unicode char U+00FC (decimal 252) + defining Unicode char U+00FD (decimal 253) + defining Unicode char U+00FE (decimal 254) + defining Unicode char U+00FF (decimal 255) + defining Unicode char U+0102 (decimal 258) + defining Unicode char U+0103 (decimal 259) + defining Unicode char U+0104 (decimal 260) + defining Unicode char U+0105 (decimal 261) + defining Unicode char U+0106 (decimal 262) + defining Unicode char U+0107 (decimal 263) + defining Unicode char U+010C (decimal 268) + defining Unicode char U+010D (decimal 269) + defining Unicode char U+010E (decimal 270) + defining Unicode char U+010F (decimal 271) + defining Unicode char U+0110 (decimal 272) + defining Unicode char U+0111 (decimal 273) + defining Unicode char U+0118 (decimal 280) + defining Unicode char U+0119 (decimal 281) + defining Unicode char U+011A (decimal 282) + defining Unicode char U+011B (decimal 283) + defining Unicode char U+011E (decimal 286) + defining Unicode char U+011F (decimal 287) + defining Unicode char U+0130 (decimal 304) + defining Unicode char U+0131 (decimal 305) + defining Unicode char U+0132 (decimal 306) + defining Unicode char U+0133 (decimal 307) + defining Unicode char U+0139 (decimal 313) + defining Unicode char U+013A (decimal 314) + defining Unicode char U+013D (decimal 317) + defining Unicode char U+013E (decimal 318) + defining Unicode char U+0141 (decimal 321) + defining Unicode char U+0142 (decimal 322) + defining Unicode char U+0143 (decimal 323) + defining Unicode char U+0144 (decimal 324) + defining Unicode char U+0147 (decimal 327) + defining Unicode char U+0148 (decimal 328) + defining Unicode char U+014A (decimal 330) + defining Unicode char U+014B (decimal 331) + defining Unicode char U+0150 (decimal 336) + defining Unicode char U+0151 (decimal 337) + defining Unicode char U+0152 (decimal 338) + defining Unicode char U+0153 (decimal 339) + defining Unicode char U+0154 (decimal 340) + defining Unicode char U+0155 (decimal 341) + defining Unicode char U+0158 (decimal 344) + defining Unicode char U+0159 (decimal 345) + defining Unicode char U+015A (decimal 346) + defining Unicode char U+015B (decimal 347) + defining Unicode char U+015E (decimal 350) + defining Unicode char U+015F (decimal 351) + defining Unicode char U+0160 (decimal 352) + defining Unicode char U+0161 (decimal 353) + defining Unicode char U+0162 (decimal 354) + defining Unicode char U+0163 (decimal 355) + defining Unicode char U+0164 (decimal 356) + defining Unicode char U+0165 (decimal 357) + defining Unicode char U+016E (decimal 366) + defining Unicode char U+016F (decimal 367) + defining Unicode char U+0170 (decimal 368) + defining Unicode char U+0171 (decimal 369) + defining Unicode char U+0178 (decimal 376) + defining Unicode char U+0179 (decimal 377) + defining Unicode char U+017A (decimal 378) + defining Unicode char U+017B (decimal 379) + defining Unicode char U+017C (decimal 380) + defining Unicode char U+017D (decimal 381) + defining Unicode char U+017E (decimal 382) + defining Unicode char U+200C (decimal 8204) + defining Unicode char U+2013 (decimal 8211) + defining Unicode char U+2014 (decimal 8212) + defining Unicode char U+2018 (decimal 8216) + defining Unicode char U+2019 (decimal 8217) + defining Unicode char U+201A (decimal 8218) + defining Unicode char U+201C (decimal 8220) + defining Unicode char U+201D (decimal 8221) + defining Unicode char U+201E (decimal 8222) + defining Unicode char U+2030 (decimal 8240) + defining Unicode char U+2031 (decimal 8241) + defining Unicode char U+2039 (decimal 8249) + defining Unicode char U+203A (decimal 8250) + defining Unicode char U+2423 (decimal 9251) +) +Now handling font encoding OT1 ... +... processing UTF-8 mapping file for font encoding OT1 + +(/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu +File: ot1enc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A1 (decimal 161) + defining Unicode char U+00A3 (decimal 163) + defining Unicode char U+00B8 (decimal 184) + defining Unicode char U+00BF (decimal 191) + defining Unicode char U+00C5 (decimal 197) + defining Unicode char U+00C6 (decimal 198) + defining Unicode char U+00D8 (decimal 216) + defining Unicode char U+00DF (decimal 223) + defining Unicode char U+00E6 (decimal 230) + defining Unicode char U+00EC (decimal 236) + defining Unicode char U+00ED (decimal 237) + defining Unicode char U+00EE (decimal 238) + defining Unicode char U+00EF (decimal 239) + defining Unicode char U+00F8 (decimal 248) + defining Unicode char U+0131 (decimal 305) + defining Unicode char U+0141 (decimal 321) + defining Unicode char U+0142 (decimal 322) + defining Unicode char U+0152 (decimal 338) + defining Unicode char U+0153 (decimal 339) + defining Unicode char U+2013 (decimal 8211) + defining Unicode char U+2014 (decimal 8212) + defining Unicode char U+2018 (decimal 8216) + defining Unicode char U+2019 (decimal 8217) + defining Unicode char U+201C (decimal 8220) + defining Unicode char U+201D (decimal 8221) +) +Now handling font encoding OMS ... +... processing UTF-8 mapping file for font encoding OMS + +(/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu +File: omsenc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A7 (decimal 167) + defining Unicode char U+00B6 (decimal 182) + defining Unicode char U+00B7 (decimal 183) + defining Unicode char U+2020 (decimal 8224) + defining Unicode char U+2021 (decimal 8225) + defining Unicode char U+2022 (decimal 8226) +) +Now handling font encoding OMX ... +... no UTF-8 mapping file for font encoding OMX +Now handling font encoding U ... +... no UTF-8 mapping file for font encoding U +Now handling font encoding TS1 ... +... processing UTF-8 mapping file for font encoding TS1 + +(/usr/share/texlive/texmf-dist/tex/latex/base/ts1enc.dfu +File: ts1enc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A2 (decimal 162) + defining Unicode char U+00A3 (decimal 163) + defining Unicode char U+00A4 (decimal 164) + defining Unicode char U+00A5 (decimal 165) + defining Unicode char U+00A6 (decimal 166) + defining Unicode char U+00A7 (decimal 167) + defining Unicode char U+00A8 (decimal 168) + defining Unicode char U+00A9 (decimal 169) + defining Unicode char U+00AA (decimal 170) + defining Unicode char U+00AC (decimal 172) + defining Unicode char U+00AE (decimal 174) + defining Unicode char U+00AF (decimal 175) + defining Unicode char U+00B0 (decimal 176) + defining Unicode char U+00B1 (decimal 177) + defining Unicode char U+00B2 (decimal 178) + defining Unicode char U+00B3 (decimal 179) + defining Unicode char U+00B4 (decimal 180) + defining Unicode char U+00B5 (decimal 181) + defining Unicode char U+00B6 (decimal 182) + defining Unicode char U+00B7 (decimal 183) + defining Unicode char U+00B9 (decimal 185) + defining Unicode char U+00BA (decimal 186) + defining Unicode char U+00BC (decimal 188) + defining Unicode char U+00BD (decimal 189) + defining Unicode char U+00BE (decimal 190) + defining Unicode char U+00D7 (decimal 215) + defining Unicode char U+00F7 (decimal 247) + defining Unicode char U+0192 (decimal 402) + defining Unicode char U+02C7 (decimal 711) + defining Unicode char U+02D8 (decimal 728) + defining Unicode char U+02DD (decimal 733) + defining Unicode char U+0E3F (decimal 3647) + defining Unicode char U+2016 (decimal 8214) + defining Unicode char U+2020 (decimal 8224) + defining Unicode char U+2021 (decimal 8225) + defining Unicode char U+2022 (decimal 8226) + defining Unicode char U+2030 (decimal 8240) + defining Unicode char U+2031 (decimal 8241) + defining Unicode char U+203B (decimal 8251) + defining Unicode char U+203D (decimal 8253) + defining Unicode char U+2044 (decimal 8260) + defining Unicode char U+204E (decimal 8270) + defining Unicode char U+2052 (decimal 8274) + defining Unicode char U+20A1 (decimal 8353) + defining Unicode char U+20A4 (decimal 8356) + defining Unicode char U+20A6 (decimal 8358) + defining Unicode char U+20A9 (decimal 8361) + defining Unicode char U+20AB (decimal 8363) + defining Unicode char U+20AC (decimal 8364) + defining Unicode char U+20B1 (decimal 8369) + defining Unicode char U+2103 (decimal 8451) + defining Unicode char U+2116 (decimal 8470) + defining Unicode char U+2117 (decimal 8471) + defining Unicode char U+211E (decimal 8478) + defining Unicode char U+2120 (decimal 8480) + defining Unicode char U+2122 (decimal 8482) + defining Unicode char U+2126 (decimal 8486) + defining Unicode char U+2127 (decimal 8487) + defining Unicode char U+212E (decimal 8494) + defining Unicode char U+2190 (decimal 8592) + defining Unicode char U+2191 (decimal 8593) + defining Unicode char U+2192 (decimal 8594) + defining Unicode char U+2193 (decimal 8595) + defining Unicode char U+2329 (decimal 9001) + defining Unicode char U+232A (decimal 9002) + defining Unicode char U+2422 (decimal 9250) + defining Unicode char U+25E6 (decimal 9702) + defining Unicode char U+25EF (decimal 9711) + defining Unicode char U+266A (decimal 9834) +) + defining Unicode char U+00A9 (decimal 169) + defining Unicode char U+00AA (decimal 170) + defining Unicode char U+00AE (decimal 174) + defining Unicode char U+00BA (decimal 186) + defining Unicode char U+02C6 (decimal 710) + defining Unicode char U+02DC (decimal 732) + defining Unicode char U+200C (decimal 8204) + defining Unicode char U+2026 (decimal 8230) + defining Unicode char U+2122 (decimal 8482) + defining Unicode char U+2423 (decimal 9251) +)) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2000/07/18 v2.13 AMS math features +\@mathmargin=\skip191 + +For additional information on amsmath, use the `?' option. +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2000/06/29 v2.01 + +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 +\@emptytoks=\toks37 +\ex@=\dimen196 +)) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d +\pmbraise@=\dimen197 +) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +Package: amsopn 1999/12/14 v2.01 operator names +) +\inf@bad=\count172 +LaTeX Info: Redefining \frac on input line 211. +\uproot@=\count173 +\leftroot@=\count174 +LaTeX Info: Redefining \overline on input line 307. +\classnum@=\count175 +\DOTSCASE@=\count176 +LaTeX Info: Redefining \ldots on input line 379. +LaTeX Info: Redefining \dots on input line 382. +LaTeX Info: Redefining \cdots on input line 467. +\Mathstrutbox@=\box38 +\strutbox@=\box39 +\big@size=\dimen198 +LaTeX Font Info: Redeclaring font encoding OML on input line 567. +LaTeX Font Info: Redeclaring font encoding OMS on input line 568. +\macc@depth=\count177 +\c@MaxMatrixCols=\count178 +\dotsspace@=\muskip10 +\c@parentequation=\count179 +\dspbrk@lvl=\count180 +\tag@help=\toks38 +\row@=\count181 +\column@=\count182 +\maxfields@=\count183 +\andhelp@=\toks39 +\eqnshift@=\dimen199 +\alignsep@=\dimen200 +\tagshift@=\dimen201 +\tagwidth@=\dimen202 +\totwidth@=\dimen203 +\lineht@=\dimen204 +\@envbody=\toks40 +\multlinegap=\skip192 +\multlinetaggap=\skip193 +\mathdisplay@stack=\toks41 +LaTeX Info: Redefining \[ on input line 2666. +LaTeX Info: Redefining \] on input line 2667. +) +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +Package: amsfonts 2009/06/22 v3.00 Basic AMSFonts support +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 96. +) +(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty +Package: amsthm 2009/07/02 v2.20.1 +\thm@style=\toks42 +\thm@bodyfont=\toks43 +\thm@headfont=\toks44 +\thm@notefont=\toks45 +\thm@headpunct=\toks46 +\thm@preskip=\skip194 +\thm@postskip=\skip195 +\thm@headsep=\skip196 +\dth@everypar=\toks47 +) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thmtools.sty +Package: thmtools 2012/05/04 v63 +\thmt@toks=\toks48 +\c@thmt@dummyctr=\count184 + +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-patch.sty +Package: thm-patch 2012/05/04 v63 + +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/parseargs.sty +Package: parseargs 2012/05/04 v63 +\@parsespec=\toks49 +)) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-kv.sty +Package: thm-kv 2012/05/04 v63 +Package thm-kv Info: Theorem names will be uppercased on input line 40. + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvsetkeys.sty +Package: kvsetkeys 2012/04/25 v1.16 Key value parser (HO) + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/etexcmds.sty +Package: etexcmds 2011/02/16 v1.5 Avoid name clashes with e-TeX commands (HO) +Package etexcmds Info: Could not find \expanded. +(etexcmds) That can mean that you are not using pdfTeX 1.50 or +(etexcmds) that some package has redefined \expanded. +(etexcmds) In the latter case, load this package earlier. +)) +Package thm-kv Info: kvsetkeys patch (v1.16 or later) on input line 156. +) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-autoref.sty +Package: thm-autoref 2012/05/04 v63 + +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/aliasctr.sty +Package: aliasctr 2012/05/04 v63 + +(/usr/share/texlive/texmf-dist/tex/latex/carlisle/remreset.sty))) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-listof.sty +Package: thm-listof 2012/05/04 v63 +) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-restate.sty +Package: thm-restate 2012/05/04 v63 +) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-amsthm.sty +Package: thm-amsthm 2012/05/04 v63 +\thmt@style@headstyle=\toks50 +)) +(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty +Package: hyperref 2012/05/13 v6.82q Hypertext links for LaTeX + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty +Package: hobsub-hyperref 2012/05/28 v1.13 Bundle oberdiek, subset hyperref (HO) + + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty +Package: hobsub-generic 2012/05/28 v1.13 Bundle oberdiek, subset generic (HO) +Package: hobsub 2012/05/28 v1.13 Construct package bundles (HO) +Package hobsub Info: Skipping package `infwarerr' (already loaded). +Package hobsub Info: Skipping package `ltxcmds' (already loaded). +Package hobsub Info: Skipping package `ifluatex' (already loaded). +Package: ifvtex 2010/03/01 v1.5 Detect VTeX and its facilities (HO) +Package ifvtex Info: VTeX not detected. +Package: intcalc 2007/09/27 v1.1 Expandable calculations with integers (HO) +Package hobsub Info: Skipping package `ifpdf' (already loaded). +Package hobsub Info: Skipping package `etexcmds' (already loaded). +Package hobsub Info: Skipping package `kvsetkeys' (already loaded). +Package: kvdefinekeys 2011/04/07 v1.3 Define keys (HO) +Package: pdftexcmds 2011/11/29 v0.20 Utility functions of pdfTeX for LuaTeX (HO +) +Package pdftexcmds Info: LuaTeX not detected. +Package pdftexcmds Info: \pdf@primitive is available. +Package pdftexcmds Info: \pdf@ifprimitive is available. +Package pdftexcmds Info: \pdfdraftmode found. +Package: pdfescape 2011/11/25 v1.13 Implements pdfTeX's escape features (HO) +Package: bigintcalc 2012/04/08 v1.3 Expandable calculations on big integers (HO +) +Package: bitset 2011/01/30 v1.1 Handle bit-vector datatype (HO) +Package: uniquecounter 2011/01/30 v1.2 Provide unlimited unique counter (HO) +) +Package hobsub Info: Skipping package `hobsub' (already loaded). +Package: letltxmacro 2010/09/02 v1.4 Let assignment for LaTeX macros (HO) +Package: hopatch 2012/05/28 v1.2 Wrapper for package hooks (HO) +Package: xcolor-patch 2011/01/30 xcolor patch +Package: atveryend 2011/06/30 v1.8 Hooks at the very end of document (HO) +Package atveryend Info: \enddocument detected (standard20110627). +Package: atbegshi 2011/10/05 v1.16 At begin shipout hook (HO) +Package: refcount 2011/10/16 v3.4 Data extraction from label references (HO) +Package: hycolor 2011/01/30 v1.7 Color options for hyperref/bookmark (HO) +) +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty +Package: kvoptions 2011/06/30 v3.11 Key value format for package options (HO) +) +\@linkdim=\dimen205 +\Hy@linkcounter=\count185 +\Hy@pagecounter=\count186 + +(/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def +File: pd1enc.def 2012/05/13 v6.82q Hyperref: PDFDocEncoding definition (HO) +Now handling font encoding PD1 ... +... no UTF-8 mapping file for font encoding PD1 +) +\Hy@SavedSpaceFactor=\count187 + +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg +File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive +) +Package hyperref Info: Hyper figures OFF on input line 4062. +Package hyperref Info: Link nesting OFF on input line 4067. +Package hyperref Info: Hyper index ON on input line 4070. +Package hyperref Info: Plain pages OFF on input line 4077. +Package hyperref Info: Backreferencing OFF on input line 4082. +Package hyperref Info: Implicit mode ON; LaTeX internals redefined. +Package hyperref Info: Bookmarks ON on input line 4300. +\c@Hy@tempcnt=\count188 + +(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty +\Urlmuskip=\muskip11 +Package: url 2006/04/12 ver 3.3 Verb mode for urls, etc. +) +LaTeX Info: Redefining \url on input line 4653. +\Fld@menulength=\count189 +\Field@Width=\dimen206 +\Fld@charsize=\dimen207 +Package hyperref Info: Hyper figures OFF on input line 5773. +Package hyperref Info: Link nesting OFF on input line 5778. +Package hyperref Info: Hyper index ON on input line 5781. +Package hyperref Info: backreferencing OFF on input line 5788. +Package hyperref Info: Link coloring OFF on input line 5793. +Package hyperref Info: Link coloring with OCG OFF on input line 5798. +Package hyperref Info: PDF/A mode OFF on input line 5803. +LaTeX Info: Redefining \ref on input line 5843. +LaTeX Info: Redefining \pageref on input line 5847. +\Hy@abspage=\count190 +\c@Item=\count191 +\c@Hfootnote=\count192 +) +(/usr/share/texlive/texmf-dist/tex/latex/memoir/memhfixc.sty +Package: memhfixc 2010/08/17 v1.15 nameref/hyperref package fixes for memoir cl +ass +\c@memhycontfloat=\count193 +) + +Package hyperref Message: Driver (autodetected): hpdftex. + +(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def +File: hpdftex.def 2012/05/13 v6.82q Hyperref driver for pdfTeX +\Fld@listcount=\count194 +\c@bookmark@seq@number=\count195 + +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty +Package: rerunfilecheck 2011/04/15 v1.7 Rerun checks for auxiliary files (HO) +Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2 +82. +) +\Hy@SectionHShift=\skip197 +) +Package hyperref Info: Option `colorlinks' set `true' on input line 26. + +(/usr/share/texlive/texmf-dist/tex/latex/cleveref/cleveref.sty +Package: cleveref 2012/03/07{} v0.18.5{} Intelligent cross-referencing +LaTeX Info: Redefining \Cref on input line 897. +Package cleveref Info: `hyperref' support loaded on input line 2166. +LaTeX Info: Redefining \cref on input line 2166. +LaTeX Info: Redefining \Cref on input line 2166. +LaTeX Info: Redefining \crefrange on input line 2166. +LaTeX Info: Redefining \Crefrange on input line 2166. +LaTeX Info: Redefining \cpageref on input line 2166. +LaTeX Info: Redefining \Cpageref on input line 2166. +LaTeX Info: Redefining \cpagerefrange on input line 2166. +LaTeX Info: Redefining \Cpagerefrange on input line 2166. +LaTeX Info: Redefining \labelcref on input line 2166. +LaTeX Info: Redefining \labelcpageref on input line 2166. +Package cleveref Info: `amsthm' support loaded on input line 2581. +Package cleveref Info: loaded `english' language definitions on input line 6907 +. +Package cleveref Info: loaded `english' language definitions on input line 6908 +. +) +(/usr/share/texlive/texmf-dist/tex/latex/caption/caption.sty +Package: caption 2012/02/19 v3.2f Customizing captions (AR) + +(/usr/share/texlive/texmf-dist/tex/latex/caption/caption3.sty +Package: caption3 2012/01/12 v1.4b caption3 kernel (AR) +\captionmargin=\dimen208 +\captionmargin@=\dimen209 +\captionwidth=\dimen210 +\caption@indent=\dimen211 +\caption@parindent=\dimen212 +\caption@hangindent=\dimen213 +Package caption Info: The definition of \@makecaption has been changed, +(caption) frenchb will NOT customize it. +Package caption Info: Unknown document class (or package), +(caption) standard defaults will be used. +) + +Package caption Warning: Unsupported document class (or package) detected, +(caption) usage of the caption package is not recommended. +See the caption package documentation for explanation. + +Package caption Info: \@makecaption = \long macro:#1#2->\let \@memtempa \relax +\ifdim \prevdepth >-99\p@ \vskip \abovecaptionskip \else \def \@memtempa {\vbox + to\topskip {}}\fi \let \@contfnote \footnote \renewcommand {\footnote }[2][]{} + \let \@contfmark \footnotemark \renewcommand {\footnotemark }[1][]{} \let \@co +ntpnote \pagenote \renewcommand \pagenote [2][]{} \sbox \@tempboxa {\@contnfont + #1\@contdelim \@conttfont #2\@contfinal } \let \footnote \@contfnote \let \foo +tnotemark \@contfmark \let \pagenote \@contpnote \ifdim \wd \@tempboxa <\linewi +dth \centering \fi \if@contcw \centering \parbox [t]{\@contcwidth }{\ifdim \wd +\@tempboxa <\@contcwidth \centering \fi \fi \ifdim \wd \@tempboxa <\linewidth \ +@contpre {\@contnfont #1\@contdelim }\@memtempa {\@contcshortstyle \@conttfont +#2\@contfinal \par } \else \if@conthang \sbox \@tempboxa {\@contnfont #1\@contd +elim } \@contpre {\@contcstyle \hangindent =\wd \@tempboxa \noindent \box \@tem +pboxa \@memtempa \@conttfont #2\@contfinal \par } \else \if@contindent \@contpr +e {\@contnfont #1\@contdelim }\@memtempa {\@contcstyle \hangindent =\@contindw +\hangafter =\@ne \@conttfont #2\@contfinal \par }\else \@contpre {\@contnfont # +1\@contdelim }\@memtempa {\@contcstyle \@conttfont #2\@contfinal \par } \fi \fi + \fi \@contpost \if@contcw \par } \fi \vskip \belowcaptionskip . +\c@ContinuedFloat=\count196 +Package caption Info: hyperref package is loaded. +) (/usr/share/texlive/texmf-dist/tex/latex/auto-pst-pdf/auto-pst-pdf.sty +Package: auto-pst-pdf 2009/04/26 v0.6 Wrapper for pst-pdf + +(/usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty +Package: xkeyval 2008/08/13 v2.6a package option processing (HA) + +(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex +\XKV@toks=\toks51 +\XKV@tempa@toks=\toks52 +\XKV@depth=\count197 +File: xkeyval.tex 2008/08/13 v2.6a key=value parser (HA) +)) +(/usr/share/texlive/texmf-dist/tex/latex/ifplatform/ifplatform.sty +Package: ifplatform 2010/10/22 v0.4 Testing for the operating system + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/catchfile.sty +Package: catchfile 2011/03/01 v1.6 Catch the contents of a file (HO) +) + +Package ifplatform Warning: + shell escape is disabled, so I can only detect \ifwindows. + +) +\c@app@runs=\count198 + (/usr/share/texlive/texmf-dist/tex/latex/pst-pdf/pst-pdf.sty +Package: pst-pdf 2008/10/09 v1.1v PS graphics for pdfLaTeX (RN,HjG) +\c@pspicture=\count199 +Package pst-pdf Info: MODE: 1 (pdfTeX mode) on input line 214. + +(/usr/share/texlive/texmf-dist/tex/latex/pstricks/pstricks.sty +Package: pstricks 2011/10/31 v0.52 LaTeX wrapper for `PSTricks' (RN,HV) + +(/usr/share/texlive/texmf-dist/tex/generic/pstricks/pstricks.tex +(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/pst-xkey.tex +File: pst-xkey.tex 2005/11/25 v1.6 PSTricks specialization of xkeyval (HA) +) +(/usr/share/texlive/texmf-dist/tex/generic/pstricks/pst-fp.tex +`pst-fp' v0.05, 2010/01/17 (hv) +\pstFP@xs=\count200 +\pstFP@xia=\count201 +\pstFP@xib=\count202 +\pstFP@xfa=\count203 +\pstFP@xfb=\count204 +\pstFP@rega=\count205 +\pstFP@regb=\count206 +\pstFP@regs=\count207 +\pstFP@times=\count208 +) +\psLoopIndex=\count209 + +`PSTricks' v2.29 <2012/06/04> (tvz) +\pst@dima=\dimen214 +\pst@dimb=\dimen215 +\pst@dimc=\dimen216 +\pst@dimd=\dimen217 +\pst@dimg=\dimen218 +\pst@dimh=\dimen219 +\pst@dimm=\dimen220 +\pst@dimn=\dimen221 +\pst@dimo=\dimen222 +\pst@dimp=\dimen223 +\pst@hbox=\box40 +\pst@ibox=\box41 +\pst@boxg=\box42 +\pst@cnta=\count210 +\pst@cntb=\count211 +\pst@cntc=\count212 +\pst@cntd=\count213 +\pst@cntg=\count214 +\pst@cnth=\count215 +\pst@cntm=\count216 +\pst@cntn=\count217 +\pst@cnto=\count218 +\pst@cntp=\count219 +\@zero=\count220 +\pst@toks=\toks53 +(/usr/share/texlive/texmf-dist/tex/generic/pstricks/pstricks.con) +\psunit=\dimen224 +\psxunit=\dimen225 +\psyunit=\dimen226 +\pst@C@@rType=\count221 +\pslinewidth=\dimen227 +\psk@startLW=\dimen228 +Normal \dimen register pool exhausted, switching to extended pool. +\psk@endLW=\dimen256 +\pst@customdefs=\toks54 +\pslinearc=\dimen257 +\pst@symbolStep=\dimen258 +\pst@symbolWidth=\dimen259 +\pst@symbolLinewidth=\dimen260 +\everypsbox=\toks55 +\psframesep=\dimen261 +\pslabelsep=\dimen262 +\pst@shift=\dimen263 +\theoverlaybox=\box43 +) +File: pstricks.tex 2012/06/04 v2.29 `PSTricks' (tvz,hv) + +(/usr/share/texlive/texmf-dist/tex/generic/pstricks/pst-fp.tex) +File: pst-fp.tex 2012/06/04 v2.29 `PST-fp' (hv) +) +\ppf@output=\toks56 + +(/usr/share/texmf/tex/latex/preview/preview.sty +Package: preview 2010/02/14 11.87 (AUCTeX/preview-latex) +\pr@snippet=\count222 +\pr@box=\box44 +\pr@output=\toks57 +) +(/usr/share/texlive/texmf-dist/tex/latex/graphics/dvips.def +File: dvips.def 1999/02/16 v3.0i Driver-dependant file (DPC,SPQR) +) +(/usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +Package: environ 2008/06/18 v0.2 A new way to define environments +))) +(./908144032.aux (./resume.aux) (./abstract.aux) (./remerciements.aux) +(./introduction.aux) (./chapitre1.aux) (./chapitre2.aux) (./chapitre3.aux) +(./chapitre4.aux) (./chapitre5.aux) (./chapitre6.aux) (./chapitre7.aux + +LaTeX Warning: Label `eq:2' multiply defined. + + +LaTeX Warning: Label `eq:2@cref' multiply defined. + +) (./chapitre8.aux) (./chapitre9.aux) (./conclusion.aux) (./annexe1.aux) +(./annexe2.aux) (./annexe3.aux) (./deed.aux)) +\openout1 = `908144032.aux'. + +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 58. +LaTeX Font Info: Try loading font information for TS1+cmr on input line 58. + +(/usr/share/texlive/texmf-dist/tex/latex/base/ts1cmr.fd +File: ts1cmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions +) +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Info: Redefining \degres on input line 58. + + +Package frenchb.ldf Warning: The definition of \@makecaption has been changed, +(frenchb.ldf) frenchb will NOT customise it; +(frenchb.ldf) reported on input line 58. + +LaTeX Info: Redefining \dots on input line 58. +LaTeX Info: Redefining \up on input line 58. +(/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii +[Loading MPS to PDF converter (version 2006.09.02).] +\scratchcounter=\count223 +\scratchdimen=\dimen264 +\scratchbox=\box45 +\nofMPsegments=\count224 +\nofMParguments=\count225 +\everyMPshowfont=\toks58 +\MPscratchCnt=\count226 +\MPscratchDim=\dimen265 +\MPnumerator=\count227 +\makeMPintoPDFobject=\count228 +\everyMPtoPDFconversion=\toks59 +) ABD: EveryShipout initializing macros +\AtBeginShipoutBox=\box46 +Package hyperref Info: Link coloring ON on input line 58. + +(/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty +Package: nameref 2010/04/30 v2.40 Cross-referencing by name of section + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty +Package: gettitlestring 2010/12/03 v1.4 Cleanup title references (HO) +) +Normal \count register pool exhausted, switching to extended pool. +\c@section@level=\count277 +) +LaTeX Info: Redefining \ref on input line 58. +LaTeX Info: Redefining \pageref on input line 58. +LaTeX Info: Redefining \nameref on input line 58. + +(./908144032.out) (./908144032.out) +\@outlinefile=\write5 +\openout5 = `908144032.out'. + +Package caption Info: Begin \AtBeginDocument code. +Package caption Info: End \AtBeginDocument code. + +Preview: Fontsize 10.95pt +Preview: PDFoutput 1 +\c@theo=\count278 +\c@hypothese=\count279 +LaTeX Font Info: Try loading font information for T1+phv on input line 64. +(/usr/share/texlive/texmf-dist/tex/latex/psnfss/t1phv.fd +File: t1phv.fd 2001/06/04 scalable font definitions for T1/phv. +) +LaTeX Font Info: Font shape `T1/phv/m/n' will be +(Font) scaled to size 13.24796pt on input line 64. + + +File: ul_p.pdf Graphic file (type pdf) + +Package pdftex.def Info: ul_p.pdf used on input line 64. +(pdftex.def) Requested size: 97.10558pt x 40.15015pt. +LaTeX Font Info: Font shape `T1/phv/bx/n' in size <20.74> not available +(Font) Font shape `T1/phv/b/n' tried instead on input line 64. +LaTeX Font Info: Font shape `T1/phv/b/n' will be +(Font) scaled to size 19.08076pt on input line 64. +LaTeX Font Info: Font shape `T1/phv/bx/n' in size <17.28> not available +(Font) Font shape `T1/phv/b/n' tried instead on input line 64. +LaTeX Font Info: Font shape `T1/phv/b/n' will be +(Font) scaled to size 15.89755pt on input line 64. +LaTeX Font Info: Font shape `T1/phv/bx/n' in size <14.4> not available +(Font) Font shape `T1/phv/b/n' tried instead on input line 64. +LaTeX Font Info: Font shape `T1/phv/b/n' will be +(Font) scaled to size 13.24796pt on input line 64. +LaTeX Font Info: Try loading font information for TS1+phv on input line 64. + +(/usr/share/texlive/texmf-dist/tex/latex/psnfss/ts1phv.fd +File: ts1phv.fd 2001/06/04 scalable font definitions for TS1/phv. +) +LaTeX Font Info: Font shape `TS1/phv/m/n' will be +(Font) scaled to size 13.24796pt on input line 64. + [1 + + +Non-PDF special ignored! +Non-PDF special ignored! + + +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} <./ul_p.pdf>] [2 + +] +\openout2 = `resume.aux'. + + (./resume.tex) [3 + + +] +\openout2 = `abstract.aux'. + + (./abstract.tex [4 + + + +]) [5] [6 + + +] (./908144032.toc +LaTeX Font Info: Try loading font information for U+msa on input line 3. +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2009/06/22 v3.00 AMS symbols A +) +LaTeX Font Info: Try loading font information for U+msb on input line 3. + +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2009/06/22 v3.00 AMS symbols B +) +Overfull \hbox (0.15294pt too wide) detected at line 7 + \T1/cmr/bx/n/10.95 xiii + [] + + +Overfull \hbox (3.27959pt too wide) detected at line 8 + \T1/cmr/bx/n/10.95 xvii + [] + +[7] [8] +Overfull \hbox (1.88837pt too wide) detected at line 106 + \T1/cmr/bx/n/10.95 113 + [] + + +Overfull \hbox (1.88837pt too wide) detected at line 107 + \T1/cmr/bx/n/10.95 115 + [] + +[9] +Overfull \hbox (1.88837pt too wide) detected at line 129 + \T1/cmr/bx/n/10.95 125 + [] + + +Overfull \hbox (1.88837pt too wide) detected at line 135 + \T1/cmr/bx/n/10.95 129 + [] + + +Overfull \hbox (1.88837pt too wide) detected at line 136 + \T1/cmr/bx/n/10.95 131 + [] + + +Overfull \hbox (1.88837pt too wide) detected at line 137 + \T1/cmr/bx/n/10.95 137 + [] + +) [10] (./908144032.lot) [11 + +] [12 + +] (./908144032.lof) [13] [14 + +] [15 + +] [16 + +] +\openout2 = `remerciements.aux'. + + +(./remerciements.tex) [17 + + +] [18 + + +] +\openout2 = `introduction.aux'. + + (./introduction.tex +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [1 + + +]) +[2] +\openout2 = `chapitre1.aux'. + + (./chapitre1.tex [3 + + + +] [4] [5] +Package hyperref Info: bookmark level for unknown hypothese defaults to 0 on in +put line 169. + [6] [7] [8] +<../varia/mitchell1.pdf, id=964, 505.89pt x 505.89pt> +File: ../varia/mitchell1.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../varia/mitchell1.pdf used on input line 388. +(pdftex.def) Requested size: 379.41655pt x 379.41655pt. + [9] [10 <../varia/mitchell1.pdf>] +Overfull \hbox (0.33621pt too wide) in paragraph at lines 421--423 +[]\T1/cmr/m/n/10.95 Il doit ad-mettre une pro-ba-bi-lité non nulle que plu-sieu +rs chan-ge-ments consé-cu-tifs semblent + [] + + +Underfull \hbox (badness 1424) in paragraph at lines 464--465 +[]|\T1/cmr/m/n/10.95 Défini seule-ment lorsque $\OML/cmm/m/it/10.95 \OMS/cmsy +/m/n/10.95 6\OT1/cmr/m/n/10.95 = 1$\T1/cmr/m/n/10.95 . + [] + +[11] [12] [13] [14] [15] [16]) [17] +\openout2 = `chapitre2.aux'. + + (./chapitre2.tex [18 + + + +] [19] [20] +Overfull \hbox (44.27849pt too wide) in paragraph at lines 106--106 +[] + [] + +[21] [22] (../graphiques/increment1.tex +\du=\skip198 +) +<"../graphiques/CH3-SIMGAMMAGAUSS".pdf, id=1156, 505.89pt x 505.89pt> +File: "../graphiques/CH3-SIMGAMMAGAUSS".pdf Graphic file (type pdf) + + +Package pdftex.def Info: "../graphiques/CH3-SIMGAMMAGAUSS".pdf used on input li +ne 232. +(pdftex.def) Requested size: 404.71254pt x 404.71254pt. + [23] +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [24 <../graphiques/CH3-SIMGAMMAGAUSS.pdf>] [25] [26] [27] [28] +Underfull \hbox (badness 1565) in paragraph at lines 533--534 +[]|\T1/cmr/m/n/10.95 Distribution asy-mé-trique à gauche + [] + + +Underfull \hbox (badness 2582) in paragraph at lines 533--534 +\T1/cmr/m/n/10.95 po-si-tif. Dé-place la moyenne dans + [] + +<../contenus/r/code/dGAL-exemples.pdf, id=1228, 505.89pt x 505.89pt> +File: ../contenus/r/code/dGAL-exemples.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/dGAL-exemples.pdf used on input lin +e 547. +(pdftex.def) Requested size: 404.71254pt x 404.71254pt. + [29] +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [30 <../contenus/r/code/dGAL-exemples.pdf>] [31] [32] +<"../graphiques/CH3-SIMULGAL0121".pdf, id=1280, 505.89pt x 505.89pt> +File: "../graphiques/CH3-SIMULGAL0121".pdf Graphic file (type pdf) + + +Package pdftex.def Info: "../graphiques/CH3-SIMULGAL0121".pdf used on input lin +e 731. +(pdftex.def) Requested size: 404.71254pt x 404.71254pt. + [33] +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [34 <../graphiques/CH3-SIMULGAL0121.pdf>] +Overfull \hbox (27.4862pt too wide) in paragraph at lines 800--824 + [][] + [] + +[35] +Underfull \vbox (badness 1371) has occurred while \output is active [] + + [36] + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\begingroup' on input line 924. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\-command' on input line 924. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\NAT@ctype' on input line 924. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\z@' on input line 924. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\let' on input line 924. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\let' on input line 924. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\new@ifnextchar' on input line 924. + + +LaTeX Warning: Command \textendash invalid in math mode on input line 960. + +) [37] [38] +\openout2 = `chapitre3.aux'. + + (./chapitre3.tex (../graphiques/probdroite.tex) [39 + + + +] [40] +(../graphiques/normaletronque.tex) [41] [42] [43] +Overfull \hbox (2.71681pt too wide) in paragraph at lines 292--300 +[]\T1/cmr/m/n/10.95 Les li-mites in-fé-rieures et su-pé-rieures de la fonc-tion + $\OML/cmm/m/it/10.95 M[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 s; y[]\OT1/cmr +/m/n/10.95 )$ \T1/cmr/m/n/10.95 sont res-pec-ti-ve-ment $\OML/cmm/m/it/10.95 M[ +]\OT1/cmr/m/n/10.95 (\OMS/cmsy/m/n/10.95 ^^@1\OML/cmm/m/it/10.95 ; y[]\OT1/cmr/ +m/n/10.95 ) = + [] + +[44] [45] [46]) [47] +\openout2 = `chapitre4.aux'. + + (./chapitre4.tex [48 + + + +] [49] [50] [51] [52] +Overfull \hbox (3.95502pt too wide) in paragraph at lines 270--274 +\T1/cmr/m/n/10.95 par dé-fi-ni-tion, l'es-pé-rance de la fonc-tion pour les vra +is pa-ra-mètres est de 0. ($\OML/cmm/m/it/10.95 E [] \OT1/cmr/m/n/10.95 = + [] + +[53] [54] [55] [56] [57] [58] [59] +Overfull \hbox (22.6247pt too wide) in paragraph at lines 777--786 +[]\T1/cmr/m/n/10.95 Comme le rang de la ma-trice $[]\OML/cmm/m/it/10.95 a\OT1/c +mr/m/n/10.95 (\OML/cmm/m/it/10.95 ^^R[]\OT1/cmr/m/n/10.95 )$ \T1/cmr/m/n/10.95 +est égal au nombre de contraintes $\OML/cmm/m/it/10.95 r$\T1/cmr/m/n/10.95 , al +ors $[]\OML/cmm/m/it/10.95 a\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 ^^R[]\OT1/c +mr/m/n/10.95 )\OMS/cmsy/m/n/10.95 J[][]\OML/cmm/m/it/10.95 a\OT1/cmr/m/n/10.95 +(\OML/cmm/m/it/10.95 ^^R[]\OT1/cmr/m/n/10.95 )[]$ + [] + +[60] [61] [62]) [63] +\openout2 = `chapitre5.aux'. + + (./chapitre5.tex [64 + + + +] [65] [66] [67] +Overfull \hbox (2.98198pt too wide) in paragraph at lines 236--236 +[] + [] + +[68] [69] +Overfull \hbox (21.20654pt too wide) in paragraph at lines 327--327 +[] + [] + +[70]) [71] +\openout2 = `chapitre6.aux'. + + (./chapitre6.tex [72 + + + +] +Underfull \vbox (badness 1354) has occurred while \output is active [] + + [73] +[74] +Overfull \hbox (40.77754pt too wide) in paragraph at lines 153--153 +[] + [] + + +Underfull \vbox (badness 4872) has occurred while \output is active [] + + [75] +Overfull \hbox (7.56583pt too wide) in paragraph at lines 182--186 +[]\T1/cmr/m/n/10.95 On ob-tient la variance-covariance asymp-to-tique des pa-ra +-mètres à par-tir de la variance-covariance + [] + +[76] [77] +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [78]) +[79] +\openout2 = `chapitre7.aux'. + + (./chapitre7.tex [80 + + + +] [81] [82] + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `math shift' on input line 145. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\chi' on input line 145. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `superscript' on input line 145. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `math shift' on input line 145. + +[83] + +Package amsmath Warning: Foreign command \over; +(amsmath) \frac or \genfrac should be used instead +(amsmath) on input line 226. + +[84] [85]) [86] +\openout2 = `chapitre8.aux'. + + (./chapitre8.tex [87 + + + +] [88] [89] [90] [91] [92] [93] [94]) +[95] +\openout2 = `chapitre9.aux'. + + (./chapitre9.tex [96 + + + +] +<../contenus/r/code/ABBEYN-chronologie.pdf, id=2150, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-chronologie.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-chronologie.pdf used on inpu +t line 21. +(pdftex.def) Requested size: 289.09473pt x 289.09473pt. + +<../contenus/r/code/ABBEYN-histogramme.pdf, id=2154, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-histogramme.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-histogramme.pdf used on inpu +t line 72. +(pdftex.def) Requested size: 289.09473pt x 289.09473pt. + [97] +<../contenus/r/code/ABBEYN-qq.pdf, id=2167, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-qq.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-qq.pdf used on input line 11 +2. +(pdftex.def) Requested size: 289.09473pt x 289.09473pt. + [98 <../contenus/r/code/ABBEYN-chronologie.pdf>] [99 <../contenus/r/code/ABBEY +N-histogramme.pdf>] [100 <../contenus/r/code/ABBEYN-qq.pdf>] +Underfull \hbox (badness 10000) in paragraph at lines 200--211 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 200--211 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 212--223 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 212--223 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 224--225 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 224--225 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 226--236 + + [] + +[101] +Underfull \hbox (badness 10000) in paragraph at lines 319--330 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 319--330 + + [] + +[102] +Underfull \hbox (badness 10000) in paragraph at lines 330--341 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 330--341 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 341--352 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 341--352 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 352--363 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 352--363 + + [] + + +Underfull \vbox (badness 2189) has occurred while \output is active [] + + [103] +<../contenus/maxima/pointdeselleGMM.png, id=2238, 501.875pt x 301.125pt> +File: ../contenus/maxima/pointdeselleGMM.png Graphic file (type png) + + +Package pdftex.def Info: ../contenus/maxima/pointdeselleGMM.png used on input l +ine 399. +(pdftex.def) Requested size: 250.93687pt x 150.56212pt. + +Underfull \vbox (badness 1642) has occurred while \output is active [] + + [104 <../contenus/maxima/pointdeselleGMM.png>] +<../contenus/r/code/ABBEYN-densiteGALmu-7.pdf, id=2254, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-densiteGALmu-7.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-densiteGALmu-7.pdf used on i +nput line 517. +(pdftex.def) Requested size: 433.63878pt x 433.63878pt. + +<../contenus/r/code/ABBEYN-densiteGALmu-5.pdf, id=2255, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-densiteGALmu-5.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-densiteGALmu-5.pdf used on i +nput line 526. +(pdftex.def) Requested size: 433.63878pt x 433.63878pt. + [105] [106 <../contenus/r/code/ABBEYN-densiteGALmu-7.pdf>] [107 <../contenus/r +/code/ABBEYN-densiteGALmu-5.pdf>] [108 + +] +<../contenus/r/code/ABBEYN-callGAL-7.pdf, id=2300, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-callGAL-7.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-callGAL-7.pdf used on input +line 685. +(pdftex.def) Requested size: 433.63878pt x 433.63878pt. + [109] +<../contenus/r/code/ABBEYN-callGAL-5.pdf, id=2306, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-callGAL-5.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-callGAL-5.pdf used on input +line 694. +(pdftex.def) Requested size: 433.63878pt x 433.63878pt. +) [110 <../contenus/r/code/ABBEYN-callGAL-7.pdf>] [111 <../contenus/r/code/ABBE +YN-callGAL-5.pdf>] +\openout2 = `conclusion.aux'. + + +(./conclusion.tex [112 + + + +]) [113] +\openout2 = `annexe1.aux'. + + (./annexe1.tex [114 + + + +] [115] [116] [117] [118] +[119] [120] [121] [122] [123]) [124] +\openout2 = `annexe2.aux'. + + (./annexe2.tex +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [125 + + + +] +[126]) [127] +\openout2 = `annexe3.aux'. + + (./annexe3.tex [128 + + + +]) [129] (./908144032.bbl [130 + + +] +LaTeX Font Info: Try loading font information for T1+cmtt on input line 31. + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1cmtt.fd +File: t1cmtt.fd 1999/05/25 v2.5h Standard LaTeX font definitions +) [131] [132] [133] +[134]) [135] +\openout2 = `deed.aux'. + + (./deed.tex [136 + + +] +Underfull \hbox (badness 10000) in paragraph at lines 17--18 + + [] + + + +File: by.pdf Graphic file (type pdf) + + +Package pdftex.def Info: by.pdf used on input line 32. +(pdftex.def) Requested size: 31.29811pt x 31.29802pt. +File: sa.pdf Graphic file (type pdf) + +Package pdftex.def Info: sa.pdf used on input line 32. +(pdftex.def) Requested size: 31.29811pt x 31.29802pt. + +File: by-sa.pdf Graphic file (type pdf) + + +Package pdftex.def Info: by-sa.pdf used on input line 35. +(pdftex.def) Requested size: 56.04544pt x 19.91692pt. +) [137 <./by.pdf> <./sa.pdf> <./by-sa.pdf>] +\tf@toc=\write6 +\openout6 = `908144032.toc'. + +\tf@lot=\write7 +\openout7 = `908144032.lot'. + +\tf@lof=\write8 +\openout8 = `908144032.lof'. + +Package atveryend Info: Empty hook `BeforeClearDocument' on input line 111. +Package atveryend Info: Empty hook `AfterLastShipout' on input line 111. + (./908144032.aux +(./resume.aux) (./abstract.aux) (./remerciements.aux) (./introduction.aux) +(./chapitre1.aux) (./chapitre2.aux) (./chapitre3.aux) (./chapitre4.aux) +(./chapitre5.aux) (./chapitre6.aux) (./chapitre7.aux) (./chapitre8.aux) +(./chapitre9.aux) (./conclusion.aux) (./annexe1.aux) (./annexe2.aux) +(./annexe3.aux) (./deed.aux)) +Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 111. +Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 111. +Package rerunfilecheck Info: File `908144032.out' has not changed. +(rerunfilecheck) Checksum: 457C8D6A4D148AA50CCB86CA64823596;10766. + + +LaTeX Warning: There were multiply-defined labels. + +Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 111. + ) +Here is how much of TeX's memory you used: + 27340 strings out of 493485 + 465108 string characters out of 3143525 + 660929 words of memory out of 3000000 + 29166 multiletter control sequences out of 15000+200000 + 40296 words of font info for 84 fonts, out of 3000000 for 9000 + 957 hyphenation exceptions out of 8191 + 65i,23n,76p,588b,673s stack positions out of 5000i,500n,10000p,200000b,50000s +{/usr/share/texmf/fonts/enc/dvips/cm-super/cm-super-t1.enc}{/usr/share/texliv +e/texmf-dist/fonts/enc/dvips/base/8r.enc} +Output written on 908144032.pdf (155 pages, 1018622 bytes). +PDF statistics: + 2857 PDF objects out of 2984 (max. 8388607) + 2606 compressed objects within 27 object streams + 828 named destinations out of 1000 (max. 500000) + 1173 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/memoire/908144032.lot b/memoire/908144032.lot new file mode 100644 index 0000000..c8bbde8 --- /dev/null +++ b/memoire/908144032.lot @@ -0,0 +1,38 @@ +\select@language {french} +\select@language {french} +\addvspace {10pt} +\contentsline {table}{\numberline {1.1}{\ignorespaces Domaine et r\IeC {\^o}le des param\IeC {\`e}tres de la distribution L stable de Mandelbrot\relax }}{12}{table.caption.14} +\contentsline {table}{\numberline {1.2}{\ignorespaces Respect des conditions \IeC {\'e}mises par Madan et Seneta pour les diff\IeC {\'e}rents mod\IeC {\`e}les pr\IeC {\'e}sent\IeC {\'e}s\relax }}{17}{table.caption.15} +\addvspace {10pt} +\contentsline {table}{\numberline {2.1}{\ignorespaces Domaine et r\IeC {\^o}le des param\IeC {\`e}tres de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{29}{table.caption.18} +\contentsline {table}{\numberline {2.2}{\ignorespaces Cas sp\IeC {\'e}ciaux de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{35}{table.caption.21} +\contentsline {table}{\numberline {2.3}{\ignorespaces Changements de param\IeC {\'e}trisation\relax }}{38}{table.caption.22} +\addvspace {10pt} +\addvspace {10pt} +\contentsline {table}{\numberline {4.1}{\ignorespaces Tests d'hypoth\IeC {\`e}se param\IeC {\'e}triques pour la m\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{63}{table.caption.25} +\addvspace {10pt} +\contentsline {table}{\numberline {5.1}{\ignorespaces M\IeC {\'e}thodes d'estimation repr\IeC {\'e}sentables par la forme quadratique \textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {eq:generalquad}\unskip \@@italiccorr )}} \relax }}{67}{table.caption.26} +\addvspace {10pt} +\contentsline {table}{\numberline {6.1}{\ignorespaces Contraintes lin\IeC {\'e}aires pour les cas particuliers de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e $GAL(\theta ,\sigma ,\mu ,\tau )$\relax }}{77}{table.caption.27} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\contentsline {table}{\numberline {9.1}{\ignorespaces Statistiques d'ordre de l'\IeC {\'e}chantillon $R_1$\relax }}{97}{table.caption.29} +\contentsline {table}{\numberline {9.2}{\ignorespaces Valeurs relatives aux premiers moments de l'\IeC {\'e}chantillon $R_1$\relax }}{98}{table.caption.30} +\contentsline {table}{\numberline {9.3}{\ignorespaces Test de normalit\IeC {\'e} d'Epps-Pulley pour $R_1$\relax }}{98}{table.caption.32} +\contentsline {table}{\numberline {9.4}{\ignorespaces Param\IeC {\`e}tres $\theta _1$ de la premi\IeC {\`e}re optimisation\relax }}{100}{table.caption.34} +\contentsline {table}{\numberline {9.5}{\ignorespaces Param\IeC {\`e}tres $\theta _1$ de la premi\IeC {\`e}re optimisation\relax }}{102}{table.caption.35} +\contentsline {table}{\numberline {9.6}{\ignorespaces Param\IeC {\`e}tres des donn\IeC {\'e}es $R_1$\relax }}{103}{table.caption.36} +\contentsline {table}{\numberline {9.7}{\ignorespaces Approximation de la densit\IeC {\'e} de $R_1$\relax }}{104}{table.caption.38} +\contentsline {table}{\numberline {9.8}{\ignorespaces Approximation de la fonction de r\IeC {\'e}partition de $R_1$\relax }}{105}{table.caption.39} +\contentsline {table}{\numberline {9.9}{\ignorespaces Courbes de densit\IeC {\'e}\relax }}{105}{table.caption.40} +\contentsline {table}{\numberline {9.10}{\ignorespaces Valeur de l'int\IeC {\'e}grale de l'approximation de la densit\IeC {\'e} par la m\IeC {\'e}thode du point de selle \relax }}{105}{table.caption.41} +\contentsline {table}{\numberline {9.11}{\ignorespaces Test du $\chi ^2$\relax }}{108}{table.caption.44} +\contentsline {table}{\numberline {9.12}{\ignorespaces Test de Kolmogorov-Smirnov\relax }}{108}{table.caption.45} +\contentsline {table}{\numberline {9.13}{\ignorespaces Test de distance minimale bas\IeC {\'e} sur la fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments\relax }}{109}{table.caption.46} +\contentsline {table}{\numberline {9.14}{\ignorespaces Caract\IeC {\'e}ristiques de l'option\relax }}{109}{table.caption.47} +\contentsline {table}{\numberline {9.15}{\ignorespaces Param\IeC {\`e}tres neutres au risque\relax }}{109}{table.caption.48} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\contentsline {table}{\numberline {C.1}{\ignorespaces Prix du titre Abbey National (penny sterling)\relax }}{129}{table.caption.60} diff --git a/memoire/908144032.out b/memoire/908144032.out new file mode 100644 index 0000000..09722e9 --- /dev/null +++ b/memoire/908144032.out @@ -0,0 +1,135 @@ +\BOOKMARK [0][-]{section*.2}{R\351sum\351}{}% 1 +\BOOKMARK [0][-]{section*.4}{Abstract}{}% 2 +\BOOKMARK [0][-]{section*.5}{Table des mati\350res}{}% 3 +\BOOKMARK [0][-]{section*.6}{Liste des tableaux}{}% 4 +\BOOKMARK [0][-]{section*.7}{Liste des figures}{}% 5 +\BOOKMARK [0][-]{section*.9}{Remerciements}{}% 6 +\BOOKMARK [0][-]{section*.11}{Introduction}{}% 7 +\BOOKMARK [0][-]{chapter.1}{Les mod\350les de rendements financiers}{}% 8 +\BOOKMARK [1][-]{section.1.1}{L'utilisation de mod\350les en finance}{chapter.1}% 9 +\BOOKMARK [2][-]{subsection.1.1.1}{Diff\351rents types de mod\350les}{section.1.1}% 10 +\BOOKMARK [2][-]{subsection.1.1.2}{Le risque de mod\351lisation}{section.1.1}% 11 +\BOOKMARK [1][-]{section.1.2}{Les rendements financiers}{chapter.1}% 12 +\BOOKMARK [2][-]{subsection.1.2.1}{D\351finitions et notation}{section.1.2}% 13 +\BOOKMARK [2][-]{subsection.1.2.2}{Rendements cumul\351s}{section.1.2}% 14 +\BOOKMARK [2][-]{subsection.1.2.3}{Donn\351es disponibles}{section.1.2}% 15 +\BOOKMARK [1][-]{section.1.3}{Les premiers mod\350les}{chapter.1}% 16 +\BOOKMARK [2][-]{subsection.1.3.1}{Le mod\350le de Bachelier}{section.1.3}% 17 +\BOOKMARK [2][-]{subsection.1.3.2}{Proposition de Mandelbrot}{section.1.3}% 18 +\BOOKMARK [2][-]{subsection.1.3.3}{Le mod\350le de Press}{section.1.3}% 19 +\BOOKMARK [2][-]{subsection.1.3.4}{Le mod\350le de Praetz}{section.1.3}% 20 +\BOOKMARK [1][-]{section.1.4}{Conditions essentielles de Madan et Seneta}{chapter.1}% 21 +\BOOKMARK [0][-]{chapter.2}{La distribution de Laplace asym\351trique g\351n\351ralis\351e}{}% 22 +\BOOKMARK [1][-]{section.2.1}{Le processus de Laplace}{chapter.2}% 23 +\BOOKMARK [2][-]{subsection.2.1.1}{Le processus gamma}{section.2.1}% 24 +\BOOKMARK [2][-]{subsection.2.1.2}{Le processus de Wiener}{section.2.1}% 25 +\BOOKMARK [2][-]{subsection.2.1.3}{Le processus de Laplace est un processus subordonn\351}{section.2.1}% 26 +\BOOKMARK [1][-]{section.2.2}{Distribution de Laplace asym\351trique g\351n\351ralis\351e}{chapter.2}% 27 +\BOOKMARK [2][-]{subsection.2.2.1}{Fonction caract\351ristique}{section.2.2}% 28 +\BOOKMARK [2][-]{subsection.2.2.2}{Invariance d'\351chelle}{section.2.2}% 29 +\BOOKMARK [2][-]{subsection.2.2.3}{Fonctions g\351n\351ratrices}{section.2.2}% 30 +\BOOKMARK [2][-]{subsection.2.2.4}{Moments et r\364le des param\350tres}{section.2.2}% 31 +\BOOKMARK [2][-]{subsection.2.2.5}{Changement d'\351chelle et de localisation}{section.2.2}% 32 +\BOOKMARK [2][-]{subsection.2.2.6}{Repr\351sentation alternative et simulation}{section.2.2}% 33 +\BOOKMARK [2][-]{subsection.2.2.7}{Fonction de Bessel et densit\351}{section.2.2}% 34 +\BOOKMARK [1][-]{section.2.3}{Cas particuliers}{chapter.2}% 35 +\BOOKMARK [2][-]{subsection.2.3.1}{Distribution de Laplace asym\351trique}{section.2.3}% 36 +\BOOKMARK [1][-]{section.2.4}{Relation avec le mod\350le de madan1990variance}{chapter.2}% 37 +\BOOKMARK [0][-]{chapter.3}{Approximation de la densit\351 et de la fonction de r\351partition}{}% 38 +\BOOKMARK [1][-]{section.3.1}{L'approximation de Laplace}{chapter.3}% 39 +\BOOKMARK [1][-]{section.3.2}{L'approximation de Temme}{chapter.3}% 40 +\BOOKMARK [1][-]{section.3.3}{La m\351thode du point de selle}{chapter.3}% 41 +\BOOKMARK [2][-]{subsection.3.3.1}{Approximation de la densit\351}{section.3.3}% 42 +\BOOKMARK [2][-]{subsection.3.3.2}{Unicit\351 du point de selle}{section.3.3}% 43 +\BOOKMARK [2][-]{subsection.3.3.3}{Approximation de la fonction de r\351partition}{section.3.3}% 44 +\BOOKMARK [2][-]{subsection.3.3.4}{Quelques propri\351t\351s des approximations}{section.3.3}% 45 +\BOOKMARK [1][-]{section.3.4}{Application de la m\351thode du point de selle}{chapter.3}% 46 +\BOOKMARK [2][-]{subsection.3.4.1}{Approximation de la densit\351}{section.3.4}% 47 +\BOOKMARK [2][-]{subsection.3.4.2}{Approximation de la fonction de r\351partition}{section.3.4}% 48 +\BOOKMARK [0][-]{chapter.4}{M\351thode des moments g\351n\351ralis\351e}{}% 49 +\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4}% 50 +\BOOKMARK [2][-]{subsection.4.1.1}{M\351thode classique des moments}{section.4.1}% 51 +\BOOKMARK [1][-]{section.4.2}{M\351thode des moments g\351n\351ralis\351e}{chapter.4}% 52 +\BOOKMARK [2][-]{subsection.4.2.1}{D\351finition}{section.4.2}% 53 +\BOOKMARK [2][-]{subsection.4.2.2}{Convergence}{section.4.2}% 54 +\BOOKMARK [2][-]{subsection.4.2.3}{Matrice de pond\351ration optimale}{section.4.2}% 55 +\BOOKMARK [2][-]{subsection.4.2.4}{M\351thode des moments g\351n\351ralis\351e it\351rative}{section.4.2}% 56 +\BOOKMARK [2][-]{subsection.4.2.5}{Distribution asymptotique des estimateurs}{section.4.2}% 57 +\BOOKMARK [1][-]{section.4.3}{Estimation sous contrainte}{chapter.4}% 58 +\BOOKMARK [2][-]{subsection.4.3.1}{Distribution asymptotique des estimateurs contraints}{section.4.3}% 59 +\BOOKMARK [1][-]{section.4.4}{Tests d'hypoth\350ses param\351triques}{chapter.4}% 60 +\BOOKMARK [2][-]{subsection.4.4.1}{Test de Wald}{section.4.4}% 61 +\BOOKMARK [2][-]{subsection.4.4.2}{Test du multiplicateur de Lagrange}{section.4.4}% 62 +\BOOKMARK [2][-]{subsection.4.4.3}{Test bas\351 sur la statistique de m\351trique de distance}{section.4.4}% 63 +\BOOKMARK [2][-]{subsection.4.4.4}{En r\351sum\351}{section.4.4}% 64 +\BOOKMARK [0][-]{chapter.5}{M\351thode de l'\351quation d'estimation optimale}{}% 65 +\BOOKMARK [1][-]{section.5.1}{\311quations quadratiques optimales}{chapter.5}% 66 +\BOOKMARK [1][-]{section.5.2}{\311quation d'estimation optimale modifi\351e}{chapter.5}% 67 +\BOOKMARK [0][-]{chapter.6}{Estimation des param\350tres de la distribution de Laplace asym\351trique g\351n\351ralis\351e}{}% 68 +\BOOKMARK [1][-]{section.6.1}{Vecteur de param\350tres initiaux}{chapter.6}% 69 +\BOOKMARK [1][-]{section.6.2}{M\351thode des moments g\351n\351ralis\351e}{chapter.6}% 70 +\BOOKMARK [2][-]{subsection.6.2.1}{Matrice de pond\351ration optimale}{section.6.2}% 71 +\BOOKMARK [2][-]{subsection.6.2.2}{Variance-covariance des param\350tres}{section.6.2}% 72 +\BOOKMARK [2][-]{subsection.6.2.3}{Contraintes lin\351aires}{section.6.2}% 73 +\BOOKMARK [1][-]{section.6.3}{M\351thode des \351quations d'estimation optimales}{chapter.6}% 74 +\BOOKMARK [0][-]{chapter.7}{Tests statistiques}{}% 75 +\BOOKMARK [1][-]{section.7.1}{Test de normalit\351}{chapter.7}% 76 +\BOOKMARK [2][-]{subsection.7.1.1}{Test de Shapiro-Wilk}{section.7.1}% 77 +\BOOKMARK [2][-]{subsection.7.1.2}{Test d\220Epps-Pulley}{section.7.1}% 78 +\BOOKMARK [1][-]{section.7.2}{Tests d'ad\351quation}{chapter.7}% 79 +\BOOKMARK [2][-]{subsection.7.2.1}{Test 2 de Pearson}{section.7.2}% 80 +\BOOKMARK [2][-]{subsection.7.2.2}{Test de Kolmogorov-Smirnov}{section.7.2}% 81 +\BOOKMARK [2][-]{subsection.7.2.3}{Test de distance minimale bas\351 sur la fonction g\351n\351ratrice des moments}{section.7.2}% 82 +\BOOKMARK [0][-]{chapter.8}{\311valuation d'options}{}% 83 +\BOOKMARK [1][-]{section.8.1}{D\351finitions}{chapter.8}% 84 +\BOOKMARK [2][-]{subsection.8.1.1}{\311quation martingale}{section.8.1}% 85 +\BOOKMARK [2][-]{subsection.8.1.2}{Param\350tres neutres au risque}{section.8.1}% 86 +\BOOKMARK [1][-]{section.8.2}{Aper\347u du mod\350le de Black-Scholes}{chapter.8}% 87 +\BOOKMARK [1][-]{section.8.3}{M\351thodes d'\351valuation pour options europ\351ennes}{chapter.8}% 88 +\BOOKMARK [2][-]{subsection.8.3.1}{M\351thode de Heston}{section.8.3}% 89 +\BOOKMARK [2][-]{subsection.8.3.2}{M\351thode de Carr et Madan}{section.8.3}% 90 +\BOOKMARK [2][-]{subsection.8.3.3}{Prix d'exercice hors de la monnaie}{section.8.3}% 91 +\BOOKMARK [2][-]{subsection.8.3.4}{Critique de la m\351thode de Carr-Madan}{section.8.3}% 92 +\BOOKMARK [2][-]{subsection.8.3.5}{M\351thode d\220Epps}{section.8.3}% 93 +\BOOKMARK [1][-]{section.8.4}{Particularit\351s}{chapter.8}% 94 +\BOOKMARK [2][-]{subsection.8.4.1}{Option sur actions avec dividendes}{section.8.4}% 95 +\BOOKMARK [2][-]{subsection.8.4.2}{Options sur contrats \340 terme et taux de change}{section.8.4}% 96 +\BOOKMARK [0][-]{chapter.9}{Exemple d'application}{}% 97 +\BOOKMARK [1][-]{section.9.1}{Description des donn\351es}{chapter.9}% 98 +\BOOKMARK [1][-]{section.9.2}{Estimation}{chapter.9}% 99 +\BOOKMARK [1][-]{section.9.3}{Approximation}{chapter.9}% 100 +\BOOKMARK [1][-]{section.9.4}{Graphiques}{chapter.9}% 101 +\BOOKMARK [1][-]{section.9.5}{Tests statistiques}{chapter.9}% 102 +\BOOKMARK [1][-]{section.9.6}{\311valuation d'options}{chapter.9}% 103 +\BOOKMARK [0][-]{section*.52}{Conclusion}{}% 104 +\BOOKMARK [0][-]{appendix.A}{\311l\351ments de th\351orie des probabilit\351s}{}% 105 +\BOOKMARK [1][-]{section.A.1}{D\351finitions de base}{appendix.A}% 106 +\BOOKMARK [1][-]{section.A.2}{Transform\351es d'une variable al\351atoire}{appendix.A}% 107 +\BOOKMARK [2][-]{subsection.A.2.1}{La fonction caract\351ristique}{section.A.2}% 108 +\BOOKMARK [3][-]{section*.53}{Transform\351e de Fourier}{subsection.A.2.1}% 109 +\BOOKMARK [3][-]{section*.54}{D\351finition}{subsection.A.2.1}% 110 +\BOOKMARK [3][-]{section*.55}{Les moments}{subsection.A.2.1}% 111 +\BOOKMARK [2][-]{subsection.A.2.2}{Inversion de la fonction caract\351ristique}{section.A.2}% 112 +\BOOKMARK [3][-]{section*.56}{La densit\351}{subsection.A.2.2}% 113 +\BOOKMARK [3][-]{section*.57}{La fonction de r\351partition}{subsection.A.2.2}% 114 +\BOOKMARK [2][-]{subsection.A.2.3}{La fonction g\351n\351ratrice des moments}{section.A.2}% 115 +\BOOKMARK [2][-]{subsection.A.2.4}{La fonction g\351n\351ratrice des cumulants}{section.A.2}% 116 +\BOOKMARK [2][-]{subsection.A.2.5}{La transform\351e d'Esscher}{section.A.2}% 117 +\BOOKMARK [1][-]{section.A.3}{La transform\351e de Fourier rapide}{appendix.A}% 118 +\BOOKMARK [1][-]{section.A.4}{Processus de L\351vy}{appendix.A}% 119 +\BOOKMARK [2][-]{subsection.A.4.1}{D\351finition et propri\351t\351s}{section.A.4}% 120 +\BOOKMARK [3][-]{section*.58}{Repr\351sentation de L\351vy-Khintchine}{subsection.A.4.1}% 121 +\BOOKMARK [3][-]{section*.59}{Repr\351sentation de L\351vy-It\364}{subsection.A.4.1}% 122 +\BOOKMARK [2][-]{subsection.A.4.2}{Processus subordonn\351}{section.A.4}% 123 +\BOOKMARK [1][-]{section.A.5}{Th\351or\350mes d'int\351gration}{appendix.A}% 124 +\BOOKMARK [2][-]{subsection.A.5.1}{Th\351or\350me de convergence domin\351e de Lebesgue}{section.A.5}% 125 +\BOOKMARK [2][-]{subsection.A.5.2}{Th\351or\350me de Fubini}{section.A.5}% 126 +\BOOKMARK [0][-]{appendix.B}{\311l\351ments de statistique math\351matique}{}% 127 +\BOOKMARK [1][-]{section.B.1}{Loi faible des grands nombres}{appendix.B}% 128 +\BOOKMARK [1][-]{section.B.2}{Th\351or\350me central limite}{appendix.B}% 129 +\BOOKMARK [2][-]{subsection.B.2.1}{Cas univari\351}{section.B.2}% 130 +\BOOKMARK [2][-]{subsection.B.2.2}{Cas multivari\351}{section.B.2}% 131 +\BOOKMARK [1][-]{section.B.3}{M\351thode delta multivari\351e}{appendix.B}% 132 +\BOOKMARK [0][-]{appendix.C}{Donn\351es}{}% 133 +\BOOKMARK [0][-]{section*.62}{Bibliographie}{}% 134 +\BOOKMARK [0][-]{section*.64}{Contrat de partage}{}% 135 diff --git a/memoire/908144032.pdf b/memoire/908144032.pdf new file mode 100644 index 0000000..3f39721 Binary files /dev/null and b/memoire/908144032.pdf differ diff --git a/memoire/908144032.tex b/memoire/908144032.tex new file mode 100644 index 0000000..5c8ac96 --- /dev/null +++ b/memoire/908144032.tex @@ -0,0 +1,111 @@ +%% GABARIT POUR MÉMOIRE STANDARD +%% +%% Consulter la documentation de la classe ulthese pour une +%% description détaillée de la classe, de ce gabarit et des options +%% disponibles. +%% +%% [Ne pas hésiter à supprimer les commentaires après les avoir lus.] +%% +%% Déclaration de la classe avec les langues les plus courantes. Le +%% français sera la langue par défaut du document. +\documentclass[11pt,english,francais]{ulthese} \usepackage{scrtime} +\usepackage{natbib} +\usepackage{tikz} \usepackage{multirow} +%% Encodage utilisé pour les caractères accentués dans les fichiers +%% source du document. Les gabarits sont encodés en UTF-8. +\usepackage[utf8]{inputenc} +%% Charger ici les autres paquetages nécessaires pour le document. +%% Quelques exemples; décommenter au besoin. +\usepackage{amsmath} % recommandé pour les mathématiques +\numberwithin{equation}{section} % pour numéroter les équations selon les sections +\usepackage{amsfonts} \usepackage{amsthm} \usepackage{thmtools} +% \usepackage{mathpazo} % texte en Palatino +% \usepackage{mathptmx} % texte en Times +%% Gestion des hyperliens dans le document. S'assurer que hyperref est +%% le dernier paquetage chargé. +\usepackage{hyperref} \hypersetup{colorlinks,allcolors=ULlinkcolor} +\usepackage{cleveref} +\usepackage[hypcap]{caption} % permet de faire le lien au dessus des figures au lieu d en dessous +\usepackage[off]{auto-pst-pdf} +%% Pour utiliser Computer Modern comme police sans empattements +%% (serif) plutôt que Helvetica, décommenter la ligne ci-dessous. +% \renewcommand{\sfdefault}{cmss} + +%% Options de mise en forme du mode français de babel. Consulter la +%% documentation du paquetage babel pour les options disponibles. +\frenchbsetup{% + CompactItemize=false, % ne pas compacter les listes + ThinSpaceInFrenchNumbers=true % espace fine dans les nombres +} \DeclareMathOperator{\sgn}{sgn} +%% Style de la bibliographie. +\bibliographystyle{plainnatmod} +%% Déclarations de la page titre. Remplacer les éléments entre < >. +%% Supprimer les caractères < >. Couper un long titre ou un long +%% sous-titre manuellement avec \\. Terminer une éventuelle deuxième +%% ligne de titre ou de sous-titre avec \vspace*{-\baselineskip}. +\titre{Modélisation des rendements financiers \\ à l'aide de la + distribution de Laplace asymétrique généralisée + \vspace*{-\baselineskip}} +% \titre{Ceci est un exemple de long titre \\ +% avec saut de ligne manuel \vspace*{-\baselineskip}} +% \soustitre{Sous-titre le cas échéant} +% \soustitre{Ceci est un exemple de long-titre \\ +% avec saut de ligne manuel \vspace*{-\baselineskip}} +\auteur{François Pelletier} \programme{Maîtrise en actuariat} +\annee{2013} +\MSc % ou l'un de \LLM, \MA, \MMus, \MServSoc, \MScGeogr, \MATDR +\setcounter{tocdepth}{4} +\begin{document} +\newtheorem{theo}{Théorème}[subsection] +\newtheorem{hypothese}{Hypothèse}[chapter] + +\frontmatter % pages liminaires + +\pagetitre % production de la page titre + +\include{resume} % résumé français +\include{abstract} % résumé anglais +\cleardoublepage + +\tableofcontents % production de la TdM +\cleardoublepage + +\listoftables % production de la liste des tableaux +\cleardoublepage + +\listoffigures % production de la liste des figures +\cleardoublepage + +%\dedicace{Dédicace si désiré} +%\cleardoublepage + +\epigraphe{Dieu ne se soucie pas de nos difficultés mathématiques. Il intègre empiriquement.}{Albert Einstein} +\cleardoublepage + +\include{remerciements} % remerciements + +\mainmatter % corps du document + +\include{introduction} % introduction +\include{chapitre1} % chapitre 1 +\include{chapitre2} % chapitre 2, etc. +\include{chapitre3} +\include{chapitre4} +\include{chapitre5} +\include{chapitre6} +\include{chapitre7} +\include{chapitre8} +\include{chapitre9} +\include{conclusion} % conclusion + +\appendix % annexes le cas échéant + +\include{annexe1} % Annexe 1 +\include{annexe2} % Annexe 2 +\include{annexe3} % Annexe 3 + +\bibliography{memoire} % production de la bibliographie + +\include{deed} + +\end{document} diff --git a/memoire/908144032.toc b/memoire/908144032.toc new file mode 100644 index 0000000..bd99040 --- /dev/null +++ b/memoire/908144032.toc @@ -0,0 +1,137 @@ +\select@language {french} +\select@language {french} +\contentsline {chapter}{R\IeC {\'e}sum\IeC {\'e}}{iii}{section*.2} +\contentsline {chapter}{Abstract}{v}{section*.4} +\contentsline {chapter}{Table des mati{\`e}res}{vii}{section*.5} +\contentsline {chapter}{Liste des tableaux}{xi}{section*.6} +\contentsline {chapter}{Liste des figures}{xiii}{section*.7} +\contentsline {chapter}{Remerciements}{xvii}{section*.9} +\contentsline {chapter}{Introduction}{1}{section*.11} +\contentsline {chapter}{\chapternumberline {1}Les mod\IeC {\`e}les de rendements financiers}{3}{chapter.1} +\contentsline {section}{\numberline {1.1}L'utilisation de mod\IeC {\`e}les en finance}{3}{section.1.1} +\contentsline {subsection}{\numberline {1.1.1}Diff\IeC {\'e}rents types de mod\IeC {\`e}les}{3}{subsection.1.1.1} +\contentsline {subsection}{\numberline {1.1.2}Le risque de mod\IeC {\'e}lisation}{4}{subsection.1.1.2} +\contentsline {section}{\numberline {1.2}Les rendements financiers}{5}{section.1.2} +\contentsline {subsection}{\numberline {1.2.1}D\IeC {\'e}finitions et notation}{5}{subsection.1.2.1} +\contentsline {subsection}{\numberline {1.2.2}Rendements cumul\IeC {\'e}s}{7}{subsection.1.2.2} +\contentsline {subsection}{\numberline {1.2.3}Donn\IeC {\'e}es disponibles}{8}{subsection.1.2.3} +\contentsline {section}{\numberline {1.3}Les premiers mod\IeC {\`e}les}{8}{section.1.3} +\contentsline {subsection}{\numberline {1.3.1}Le mod\IeC {\`e}le de Bachelier}{8}{subsection.1.3.1} +\contentsline {subsection}{\numberline {1.3.2}Proposition de Mandelbrot}{11}{subsection.1.3.2} +\contentsline {subsection}{\numberline {1.3.3}Le mod\IeC {\`e}le de Press}{12}{subsection.1.3.3} +\contentsline {subsection}{\numberline {1.3.4}Le mod\IeC {\`e}le de Praetz}{15}{subsection.1.3.4} +\contentsline {section}{\numberline {1.4}Conditions essentielles de Madan et Seneta}{16}{section.1.4} +\contentsline {chapter}{\chapternumberline {2}La distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{19}{chapter.2} +\contentsline {section}{\numberline {2.1}Le processus de Laplace}{19}{section.2.1} +\contentsline {subsection}{\numberline {2.1.1}Le processus gamma}{20}{subsection.2.1.1} +\contentsline {subsection}{\numberline {2.1.2}Le processus de Wiener}{21}{subsection.2.1.2} +\contentsline {subsection}{\numberline {2.1.3}Le processus de Laplace est un processus subordonn\IeC {\'e}}{22}{subsection.2.1.3} +\contentsline {section}{\numberline {2.2}Distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{25}{section.2.2} +\contentsline {subsection}{\numberline {2.2.1}Fonction caract\IeC {\'e}ristique}{25}{subsection.2.2.1} +\contentsline {subsection}{\numberline {2.2.2}Invariance d'\IeC {\'e}chelle}{26}{subsection.2.2.2} +\contentsline {subsection}{\numberline {2.2.3}Fonctions g\IeC {\'e}n\IeC {\'e}ratrices}{28}{subsection.2.2.3} +\contentsline {subsection}{\numberline {2.2.4}Moments et r\IeC {\^o}le des param\IeC {\`e}tres}{28}{subsection.2.2.4} +\contentsline {subsection}{\numberline {2.2.5}Changement d'\IeC {\'e}chelle et de localisation}{31}{subsection.2.2.5} +\contentsline {subsection}{\numberline {2.2.6}Repr\IeC {\'e}sentation alternative et simulation}{32}{subsection.2.2.6} +\contentsline {subsection}{\numberline {2.2.7}Fonction de Bessel et densit\IeC {\'e}}{33}{subsection.2.2.7} +\contentsline {section}{\numberline {2.3}Cas particuliers}{35}{section.2.3} +\contentsline {subsection}{\numberline {2.3.1}Distribution de Laplace asym\IeC {\'e}trique}{36}{subsection.2.3.1} +\contentsline {section}{\numberline {2.4}Relation avec le mod\IeC {\`e}le de \cite {madan1990variance}}{37}{section.2.4} +\contentsline {chapter}{\chapternumberline {3}Approximation de la densit\IeC {\'e} et de la fonction de r\IeC {\'e}partition}{39}{chapter.3} +\contentsline {section}{\numberline {3.1}L'approximation de Laplace}{39}{section.3.1} +\contentsline {section}{\numberline {3.2}L'approximation de Temme}{41}{section.3.2} +\contentsline {section}{\numberline {3.3}La m\IeC {\'e}thode du point de selle}{42}{section.3.3} +\contentsline {subsection}{\numberline {3.3.1}Approximation de la densit\IeC {\'e}}{42}{subsection.3.3.1} +\contentsline {subsection}{\numberline {3.3.2}Unicit\IeC {\'e} du point de selle}{44}{subsection.3.3.2} +\contentsline {subsection}{\numberline {3.3.3}Approximation de la fonction de r\IeC {\'e}partition}{44}{subsection.3.3.3} +\contentsline {subsection}{\numberline {3.3.4}Quelques propri\IeC {\'e}t\IeC {\'e}s des approximations}{45}{subsection.3.3.4} +\contentsline {section}{\numberline {3.4}Application de la m\IeC {\'e}thode du point de selle}{46}{section.3.4} +\contentsline {subsection}{\numberline {3.4.1}Approximation de la densit\IeC {\'e}}{46}{subsection.3.4.1} +\contentsline {subsection}{\numberline {3.4.2}Approximation de la fonction de r\IeC {\'e}partition}{47}{subsection.3.4.2} +\contentsline {chapter}{\chapternumberline {4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{49}{chapter.4} +\contentsline {section}{\numberline {4.1}Introduction}{49}{section.4.1} +\contentsline {subsection}{\numberline {4.1.1}M\IeC {\'e}thode classique des moments}{50}{subsection.4.1.1} +\contentsline {section}{\numberline {4.2}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{50}{section.4.2} +\contentsline {subsection}{\numberline {4.2.1}D\IeC {\'e}finition}{51}{subsection.4.2.1} +\contentsline {subsection}{\numberline {4.2.2}Convergence}{52}{subsection.4.2.2} +\contentsline {subsection}{\numberline {4.2.3}Matrice de pond\IeC {\'e}ration optimale}{53}{subsection.4.2.3} +\contentsline {subsection}{\numberline {4.2.4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e it\IeC {\'e}rative}{55}{subsection.4.2.4} +\contentsline {subsection}{\numberline {4.2.5}Distribution asymptotique des estimateurs}{55}{subsection.4.2.5} +\contentsline {section}{\numberline {4.3}Estimation sous contrainte}{58}{section.4.3} +\contentsline {subsection}{\numberline {4.3.1}Distribution asymptotique des estimateurs contraints}{59}{subsection.4.3.1} +\contentsline {section}{\numberline {4.4}Tests d'hypoth\IeC {\`e}ses param\IeC {\'e}triques}{61}{section.4.4} +\contentsline {subsection}{\numberline {4.4.1}Test de Wald}{61}{subsection.4.4.1} +\contentsline {subsection}{\numberline {4.4.2}Test du multiplicateur de Lagrange}{62}{subsection.4.4.2} +\contentsline {subsection}{\numberline {4.4.3}Test bas\IeC {\'e} sur la statistique de m\IeC {\'e}trique de distance}{63}{subsection.4.4.3} +\contentsline {subsection}{\numberline {4.4.4}En r\IeC {\'e}sum\IeC {\'e}}{63}{subsection.4.4.4} +\contentsline {chapter}{\chapternumberline {5}M\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale}{65}{chapter.5} +\contentsline {section}{\numberline {5.1}\IeC {\'E}quations quadratiques optimales}{67}{section.5.1} +\contentsline {section}{\numberline {5.2}\IeC {\'E}quation d'estimation optimale modifi\IeC {\'e}e}{70}{section.5.2} +\contentsline {chapter}{\chapternumberline {6}Estimation des param\IeC {\`e}tres de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{73}{chapter.6} +\contentsline {section}{\numberline {6.1}Vecteur de param\IeC {\`e}tres initiaux}{73}{section.6.1} +\contentsline {section}{\numberline {6.2}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{74}{section.6.2} +\contentsline {subsection}{\numberline {6.2.1}Matrice de pond\IeC {\'e}ration optimale}{75}{subsection.6.2.1} +\contentsline {subsection}{\numberline {6.2.2}Variance-covariance des param\IeC {\`e}tres}{76}{subsection.6.2.2} +\contentsline {subsection}{\numberline {6.2.3}Contraintes lin\IeC {\'e}aires}{77}{subsection.6.2.3} +\contentsline {section}{\numberline {6.3}M\IeC {\'e}thode des \IeC {\'e}quations d'estimation optimales}{78}{section.6.3} +\contentsline {chapter}{\chapternumberline {7}Tests statistiques}{81}{chapter.7} +\contentsline {section}{\numberline {7.1}Test de normalit\IeC {\'e}}{81}{section.7.1} +\contentsline {subsection}{\numberline {7.1.1}Test de Shapiro-Wilk}{81}{subsection.7.1.1} +\contentsline {subsection}{\numberline {7.1.2}Test d\IeC {\textquoteright }Epps-Pulley}{82}{subsection.7.1.2} +\contentsline {section}{\numberline {7.2}Tests d'ad\IeC {\'e}quation}{82}{section.7.2} +\contentsline {subsection}{\numberline {7.2.1}Test $\chi ^2$ de Pearson}{83}{subsection.7.2.1} +\contentsline {subsection}{\numberline {7.2.2}Test de Kolmogorov-Smirnov}{84}{subsection.7.2.2} +\contentsline {subsection}{\numberline {7.2.3}Test de distance minimale bas\IeC {\'e} sur la fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments}{85}{subsection.7.2.3} +\contentsline {chapter}{\chapternumberline {8}\IeC {\'E}valuation d'options}{87}{chapter.8} +\contentsline {section}{\numberline {8.1}D\IeC {\'e}finitions}{87}{section.8.1} +\contentsline {subsection}{\numberline {8.1.1}\IeC {\'E}quation martingale}{88}{subsection.8.1.1} +\contentsline {subsection}{\numberline {8.1.2}Param\IeC {\`e}tres neutres au risque}{89}{subsection.8.1.2} +\contentsline {section}{\numberline {8.2}Aper\IeC {\c c}u du mod\IeC {\`e}le de Black-Scholes}{90}{section.8.2} +\contentsline {section}{\numberline {8.3}M\IeC {\'e}thodes d'\IeC {\'e}valuation pour options europ\IeC {\'e}ennes}{90}{section.8.3} +\contentsline {subsection}{\numberline {8.3.1}M\IeC {\'e}thode de Heston}{91}{subsection.8.3.1} +\contentsline {subsection}{\numberline {8.3.2}M\IeC {\'e}thode de Carr et Madan}{91}{subsection.8.3.2} +\contentsline {subsection}{\numberline {8.3.3}Prix d'exercice hors de la monnaie}{93}{subsection.8.3.3} +\contentsline {subsection}{\numberline {8.3.4}Critique de la m\IeC {\'e}thode de Carr-Madan}{93}{subsection.8.3.4} +\contentsline {subsection}{\numberline {8.3.5}M\IeC {\'e}thode d\IeC {\textquoteright }Epps}{94}{subsection.8.3.5} +\contentsline {section}{\numberline {8.4}Particularit\IeC {\'e}s}{95}{section.8.4} +\contentsline {subsection}{\numberline {8.4.1}Option sur actions avec dividendes}{95}{subsection.8.4.1} +\contentsline {subsection}{\numberline {8.4.2}Options sur contrats \IeC {\`a} terme et taux de change}{95}{subsection.8.4.2} +\contentsline {chapter}{\chapternumberline {9}Exemple d'application}{97}{chapter.9} +\contentsline {section}{\numberline {9.1}Description des donn\IeC {\'e}es}{97}{section.9.1} +\contentsline {section}{\numberline {9.2}Estimation}{99}{section.9.2} +\contentsline {section}{\numberline {9.3}Approximation}{104}{section.9.3} +\contentsline {section}{\numberline {9.4}Graphiques}{105}{section.9.4} +\contentsline {section}{\numberline {9.5}Tests statistiques}{108}{section.9.5} +\contentsline {section}{\numberline {9.6}\IeC {\'E}valuation d'options}{109}{section.9.6} +\contentsline {chapter}{Conclusion}{113}{section*.52} +\contentsline {appendix}{\chapternumberline {A}\IeC {\'E}l\IeC {\'e}ments de th\IeC {\'e}orie des probabilit\IeC {\'e}s}{115}{appendix.A} +\contentsline {section}{\numberline {A.1}D\IeC {\'e}finitions de base}{115}{section.A.1} +\contentsline {section}{\numberline {A.2}Transform\IeC {\'e}es d'une variable al\IeC {\'e}atoire}{116}{section.A.2} +\contentsline {subsection}{\numberline {A.2.1}La fonction caract\IeC {\'e}ristique}{116}{subsection.A.2.1} +\contentsline {subsubsection}{Transform\IeC {\'e}e de Fourier}{116}{section*.53} +\contentsline {subsubsection}{D\IeC {\'e}finition}{116}{section*.54} +\contentsline {subsubsection}{Les moments}{117}{section*.55} +\contentsline {subsection}{\numberline {A.2.2}Inversion de la fonction caract\IeC {\'e}ristique}{117}{subsection.A.2.2} +\contentsline {subsubsection}{La densit\IeC {\'e}}{117}{section*.56} +\contentsline {subsubsection}{La fonction de r\IeC {\'e}partition}{117}{section*.57} +\contentsline {subsection}{\numberline {A.2.3}La fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments}{118}{subsection.A.2.3} +\contentsline {subsection}{\numberline {A.2.4}La fonction g\IeC {\'e}n\IeC {\'e}ratrice des cumulants}{119}{subsection.A.2.4} +\contentsline {subsection}{\numberline {A.2.5}La transform\IeC {\'e}e d'Esscher}{120}{subsection.A.2.5} +\contentsline {section}{\numberline {A.3}La transform\IeC {\'e}e de Fourier rapide}{120}{section.A.3} +\contentsline {section}{\numberline {A.4}Processus de L\IeC {\'e}vy}{121}{section.A.4} +\contentsline {subsection}{\numberline {A.4.1}D\IeC {\'e}finition et propri\IeC {\'e}t\IeC {\'e}s}{121}{subsection.A.4.1} +\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-Khintchine}{122}{section*.58} +\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-It\IeC {\^o}}{122}{section*.59} +\contentsline {subsection}{\numberline {A.4.2}Processus subordonn\IeC {\'e}}{123}{subsection.A.4.2} +\contentsline {section}{\numberline {A.5}Th\IeC {\'e}or\IeC {\`e}mes d'int\IeC {\'e}gration}{123}{section.A.5} +\contentsline {subsection}{\numberline {A.5.1}Th\IeC {\'e}or\IeC {\`e}me de convergence domin\IeC {\'e}e de Lebesgue}{124}{subsection.A.5.1} +\contentsline {subsection}{\numberline {A.5.2}Th\IeC {\'e}or\IeC {\`e}me de Fubini}{124}{subsection.A.5.2} +\contentsline {appendix}{\chapternumberline {B}\IeC {\'E}l\IeC {\'e}ments de statistique math\IeC {\'e}matique}{125}{appendix.B} +\contentsline {section}{\numberline {B.1}Loi faible des grands nombres}{125}{section.B.1} +\contentsline {section}{\numberline {B.2}Th\IeC {\'e}or\IeC {\`e}me central limite}{125}{section.B.2} +\contentsline {subsection}{\numberline {B.2.1}Cas univari\IeC {\'e}}{126}{subsection.B.2.1} +\contentsline {subsection}{\numberline {B.2.2}Cas multivari\IeC {\'e}}{126}{subsection.B.2.2} +\contentsline {section}{\numberline {B.3}M\IeC {\'e}thode delta multivari\IeC {\'e}e}{127}{section.B.3} +\contentsline {appendix}{\chapternumberline {C}Donn\IeC {\'e}es}{129}{appendix.C} +\contentsline {chapter}{Bibliographie}{131}{section*.62} +\contentsline {chapter}{Contrat de partage}{137}{section*.64} diff --git a/memoire/Couverture-PageTitre-908144032.pdf b/memoire/Couverture-PageTitre-908144032.pdf new file mode 100644 index 0000000..00b8e73 Binary files /dev/null and b/memoire/Couverture-PageTitre-908144032.pdf differ diff --git a/memoire/MD5SUM b/memoire/MD5SUM new file mode 100644 index 0000000..402fee4 --- /dev/null +++ b/memoire/MD5SUM @@ -0,0 +1 @@ +cb86f43ece0fa685e93c596c18426965 908144032.pdf diff --git a/memoire/README b/memoire/README new file mode 100644 index 0000000..3512fff --- /dev/null +++ b/memoire/README @@ -0,0 +1,92 @@ +This package contains a LaTeX class and templates for thesis and +memoirs at Université Laval, Québec, Canada. + +LICENCE + +LaTeX Project Public License, version 1.3c or (at your option) any +later version. + +*** The rest of this file is in French *** + +DESCRIPTION + +La classe LaTeX ulthese permet de composer des thèses et mémoires +immédiatement conformes aux règles générales de présentation +matérielle de la Faculté des études supérieures et postdoctorales +(FESP) de l'Université Laval. + +CONTENU DE L'ARCHIVE + +- ulthese.ins: fichier d'installation de la classe; voir ci-dessous; +- ulthese.dtx: fichier source documenté de la classe; +- ulthese.pdf: documentation de la classe; +- ul_p.eps: logo de l'Université Laval en format EPS; +- ul_p.pdf: logo de l'Université Laval en format PDF; +- README: le présent fichier. + +INSTALLATION + +Pour installer la classe ulthese, compiler avec LaTeX le fichier +ulthese.ins, en exécutant + + latex ulthese.ins + +depuis une invite de commande. On peut aussi ouvrir le fichier +ulthese.ins dans son éditeur LaTeX et lancer la compilation avec +LaTeX, pdfTeX ou pdfLaTeX depuis celui-ci. + +La compilation du fichier ulthese.ins générera plusieurs fichiers: + +1. la classe elle-même: + - ulthese.cls + +2. des gabarits pour les documents maîtres de différents types de + thèses et de mémoires: + - gabarit-doctorat.tex + - gabarit-doctorat-mesure.tex + - gabarit-doctorat-multifacultaire.tex + - gabarit-doctorat-cotutelle.tex + - gabarit-doctorat-extension-UdeS.tex + - gabarit-doctorat-extension-UQO.tex + - gabarit-maitrise.tex + - gabarit-maitrise-mesure.tex + - gabarit-maitrise-extension-UQAC.tex + +3. des gabarits pour quelques parties d'un document: + - resume.tex + - abstract.tex + - remerciements.tex + - avantpropos.tex + - introduction.tex + - chapitre1.tex + - chapitre2.tex + - conclusion.tex + - annexe.tex + +Les gabarits inutiles peuvent être supprimés. + +DOCUMENTATION + +Le fichier ulthese.pdf contient la documentation complète de la classe +et des gabarits. Le document peut être recréé à partir du code source +avec les commandes + + pdflatex ulthese.dtx + makeindex -s gind.ist ulthese + makeindex -s gglo.ist -o ulthese.gls ulthese.glo + +AIDE ADDITIONNELLE + +Faire appel aux ressources suivantes, dans l'ordre: + +1. consulter le WikiThèse de l'Université Laval + + http://www.theses.ulaval.ca/wiki/ + +2. consulter les archives de la liste de distribution ulthese-aide + + http://listes.ulaval.ca/listserv/archives/ulthese-aide.html + +3. écrire à la liste de distribution ulthese-aide + + ulthese-aide@listes.ulaval.ca diff --git a/memoire/SHA1SUM b/memoire/SHA1SUM new file mode 100644 index 0000000..ef60b29 --- /dev/null +++ b/memoire/SHA1SUM @@ -0,0 +1 @@ +6f9a85e6a4f4334ba59be60c2a3030db781143bb 908144032.pdf diff --git a/memoire/Thumbs.db b/memoire/Thumbs.db new file mode 100644 index 0000000..acfffe0 Binary files /dev/null and b/memoire/Thumbs.db differ diff --git a/memoire/abstract.aux b/memoire/abstract.aux new file mode 100644 index 0000000..4c6a5ed --- /dev/null +++ b/memoire/abstract.aux @@ -0,0 +1,80 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{toc}{\contentsline {chapter}{Abstract}{v}{section*.4}} +\@setckpt{abstract}{ +\setcounter{page}{6} +\setcounter{equation}{0} +\setcounter{enumi}{0} +\setcounter{enumii}{0} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{0} +\setcounter{section}{0} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{6} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{0} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{0} +\setcounter{Item}{0} +\setcounter{Hfootnote}{0} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{2} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/abstract.log b/memoire/abstract.log new file mode 100644 index 0000000..fa86f31 --- /dev/null +++ b/memoire/abstract.log @@ -0,0 +1,520 @@ +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.7.6) 24 SEP 2013 22:48 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**\input abstract.tex +(./abstract.tex +! Undefined control sequence. +l.1 \chapter + *{Abstract} % ne pas numéroter +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1 \chapter* + {Abstract} % ne pas numéroter +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no * in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.2 \phantomsection + \addcontentsline{toc}{chapter}{Abstract} % inclure dans TdM +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 1--3 +[] + [] + +! Undefined control sequence. +\foreign@language ...agename }\languageshorthands + {none}\csname extras#1\end... +l.4 \begin{otherlanguage*}{english} + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.4 \begin{otherlanguage*}{english} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +) +! Emergency stop. +<*> \input abstract.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 8 strings out of 493485 + 178 string characters out of 3143527 + 47818 words of memory out of 3000000 + 3415 multiletter control sequences out of 15000+200000 + 3640 words of font info for 14 fonts, out of 3000000 for 9000 + 957 hyphenation exceptions out of 8191 + 6i,0n,5p,527b,25s stack positions out of 5000i,500n,10000p,200000b,50000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/memoire/abstract.tex b/memoire/abstract.tex new file mode 100644 index 0000000..b6cc1c9 --- /dev/null +++ b/memoire/abstract.tex @@ -0,0 +1,11 @@ +\chapter*{Abstract} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Abstract} % inclure dans TdM + +\begin{otherlanguage*}{english} + Classical models in finance are based on a set of hypotheses that are not always empirically verifiable. The main objective behind this master's thesis is to show that the generalized asymmetric Laplace distribution is an interesting alternative to those models. To support this idea, we focus on some of its properties to develop parametric estimation, approximation and testing methods, then we work out some principles of derivatives pricing. Finally, we have a numerical example to illustrate these concepts. +\end{otherlanguage*} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/annexe1.aux b/memoire/annexe1.aux new file mode 100644 index 0000000..955b742 --- /dev/null +++ b/memoire/annexe1.aux @@ -0,0 +1,256 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{dodge2004statistique} +\citation{dodge2004statistique} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {appendix}{\chapternumberline {A}\IeC {\'E}l\IeC {\'e}ments de th\IeC {\'e}orie des probabilit\IeC {\'e}s}{115}{appendix.A}} +\@writefile{toc}{\contentsline {section}{\numberline {A.1}D\IeC {\'e}finitions de base}{115}{section.A.1}} +\newlabel{sec:introvariablealeatoire}{{A.1}{115}{Définitions de base}{section.A.1}{}} +\newlabel{sec:introvariablealeatoire@cref}{{[subappendix][1][2147483647,1]A.1}{115}} +\newlabel{eq:condition1probabilite}{{A.1.1}{115}{Définitions de base}{equation.A.1.0.1}{}} +\newlabel{eq:condition1probabilite@cref}{{[equation][1][2147483647,1,1]A.1.1}{115}} +\newlabel{eq:condition2probabilite}{{A.1.2}{115}{Définitions de base}{equation.A.1.0.2}{}} +\newlabel{eq:condition2probabilite@cref}{{[equation][2][2147483647,1,1]A.1.2}{115}} +\newlabel{eq:condition3probabilite}{{A.1.3}{115}{Définitions de base}{equation.A.1.0.3}{}} +\newlabel{eq:condition3probabilite@cref}{{[equation][3][2147483647,1,1]A.1.3}{115}} +\newlabel{eq:densitedefinition}{{A.1.4}{115}{Définitions de base}{equation.A.1.0.4}{}} +\newlabel{eq:densitedefinition@cref}{{[equation][4][2147483647,1,1]A.1.4}{115}} +\citation{stuart1987kendall} +\newlabel{eq:repartitiondefinition}{{A.1.5}{116}{Définitions de base}{equation.A.1.0.5}{}} +\newlabel{eq:repartitiondefinition@cref}{{[equation][5][2147483647,1,1]A.1.5}{116}} +\@writefile{toc}{\contentsline {section}{\numberline {A.2}Transform\IeC {\'e}es d'une variable al\IeC {\'e}atoire}{116}{section.A.2}} +\newlabel{sec:transformees}{{A.2}{116}{Transformées d'une variable aléatoire}{section.A.2}{}} +\newlabel{sec:transformees@cref}{{[subappendix][2][2147483647,1]A.2}{116}} +\@writefile{toc}{\contentsline {subsection}{\numberline {A.2.1}La fonction caract\IeC {\'e}ristique}{116}{subsection.A.2.1}} +\newlabel{sec:fonctioncaract}{{A.2.1}{116}{La fonction caractéristique}{subsection.A.2.1}{}} +\newlabel{sec:fonctioncaract@cref}{{[subsubappendix][1][2147483647,1,2]A.2.1}{116}} +\@writefile{toc}{\contentsline {subsubsection}{Transform\IeC {\'e}e de Fourier}{116}{section*.53}} +\newlabel{sec:transfourier}{{A.2.1}{116}{Transformée de Fourier}{section*.53}{}} +\newlabel{sec:transfourier@cref}{{[subsubappendix][1][2147483647,1,2]A.2.1}{116}} +\newlabel{eq:transfourier}{{A.2.1}{116}{Transformée de Fourier}{equation.A.2.0.1}{}} +\newlabel{eq:transfourier@cref}{{[equation][1][2147483647,1,2]A.2.1}{116}} +\newlabel{eq:transinvfourier}{{A.2.2}{116}{Transformée de Fourier}{equation.A.2.0.2}{}} +\newlabel{eq:transinvfourier@cref}{{[equation][2][2147483647,1,2]A.2.2}{116}} +\@writefile{toc}{\contentsline {subsubsection}{D\IeC {\'e}finition}{116}{section*.54}} +\newlabel{sec:deffncaract}{{A.2.1}{116}{Définition}{section*.54}{}} +\newlabel{sec:deffncaract@cref}{{[subsubappendix][1][2147483647,1,2]A.2.1}{116}} +\newlabel{eq:deffncaract}{{A.2.3}{116}{Définition}{equation.A.2.0.3}{}} +\newlabel{eq:deffncaract@cref}{{[equation][3][2147483647,1,2]A.2.3}{116}} +\citation{lukacs1960characteristic} +\citation{gil1951note} +\citation{epps2007pricing} +\citation{wendel1961non} +\@writefile{toc}{\contentsline {subsubsection}{Les moments}{117}{section*.55}} +\newlabel{sec:momentsfncaract}{{A.2.1}{117}{Les moments}{section*.55}{}} +\newlabel{sec:momentsfncaract@cref}{{[subsubappendix][1][2147483647,1,2]A.2.1}{117}} +\newlabel{eq:defmoments}{{A.2.4}{117}{Les moments}{equation.A.2.0.4}{}} +\newlabel{eq:defmoments@cref}{{[equation][4][2147483647,1,2]A.2.4}{117}} +\newlabel{eq:fncaractmoments}{{A.2.5}{117}{Les moments}{equation.A.2.0.5}{}} +\newlabel{eq:fncaractmoments@cref}{{[equation][5][2147483647,1,2]A.2.5}{117}} +\@writefile{toc}{\contentsline {subsection}{\numberline {A.2.2}Inversion de la fonction caract\IeC {\'e}ristique}{117}{subsection.A.2.2}} +\newlabel{sec:inversioncaract}{{A.2.2}{117}{Inversion de la fonction caractéristique}{subsection.A.2.2}{}} +\newlabel{sec:inversioncaract@cref}{{[subsubappendix][2][2147483647,1,2]A.2.2}{117}} +\@writefile{toc}{\contentsline {subsubsection}{La densit\IeC {\'e}}{117}{section*.56}} +\newlabel{sec:densitefncaract}{{A.2.2}{117}{La densité}{section*.56}{}} +\newlabel{sec:densitefncaract@cref}{{[subsubappendix][2][2147483647,1,2]A.2.2}{117}} +\newlabel{eq:caractdensite}{{A.2.6}{117}{La densité}{equation.A.2.0.6}{}} +\newlabel{eq:caractdensite@cref}{{[equation][6][2147483647,1,2]A.2.6}{117}} +\@writefile{toc}{\contentsline {subsubsection}{La fonction de r\IeC {\'e}partition}{117}{section*.57}} +\newlabel{sec:repartitionfncaract}{{A.2.2}{117}{La fonction de répartition}{section*.57}{}} +\newlabel{sec:repartitionfncaract@cref}{{[subsubappendix][2][2147483647,1,2]A.2.2}{117}} +\newlabel{eq:gilpelaez}{{A.2.7}{117}{La fonction de répartition}{equation.A.2.0.7}{}} +\newlabel{eq:gilpelaez@cref}{{[equation][7][2147483647,1,2]A.2.7}{117}} +\newlabel{eq:gilpelaez2}{{A.2.8}{117}{La fonction de répartition}{equation.A.2.0.8}{}} +\newlabel{eq:gilpelaez2@cref}{{[equation][8][2147483647,1,2]A.2.8}{117}} +\citation{shephard1991characteristic} +\newlabel{eq:inversionfncaract}{{A.2.9}{118}{La fonction de répartition}{equation.A.2.0.9}{}} +\newlabel{eq:inversionfncaract@cref}{{[equation][9][2147483647,1,2]A.2.9}{118}} +\newlabel{eq:shepherd}{{A.2.10}{118}{La fonction de répartition}{equation.A.2.0.10}{}} +\newlabel{eq:shepherd@cref}{{[equation][10][2147483647,1,2]A.2.10}{118}} +\newlabel{eq:approxinvfncaract}{{A.2.11}{118}{La fonction de répartition}{equation.A.2.0.11}{}} +\newlabel{eq:approxinvfncaract@cref}{{[equation][11][2147483647,1,2]A.2.11}{118}} +\newlabel{eq:convocaract}{{A.2.12}{118}{La fonction de répartition}{equation.A.2.0.12}{}} +\newlabel{eq:convocaract@cref}{{[equation][12][2147483647,1,2]A.2.12}{118}} +\newlabel{eq:convocaractIID}{{A.2.13}{118}{La fonction de répartition}{equation.A.2.0.13}{}} +\newlabel{eq:convocaractIID@cref}{{[equation][13][2147483647,1,2]A.2.13}{118}} +\@writefile{toc}{\contentsline {subsection}{\numberline {A.2.3}La fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments}{118}{subsection.A.2.3}} +\newlabel{sec:fgm}{{A.2.3}{118}{La fonction génératrice des moments}{subsection.A.2.3}{}} +\newlabel{sec:fgm@cref}{{[subsubappendix][3][2147483647,1,2]A.2.3}{118}} +\newlabel{eq:deffngenmom}{{A.2.14}{118}{La fonction génératrice des moments}{equation.A.2.0.14}{}} +\newlabel{eq:deffngenmom@cref}{{[equation][14][2147483647,1,2]A.2.14}{118}} +\citation{stuart1987kendall} +\newlabel{eq:fgmmoments}{{A.2.15}{119}{La fonction génératrice des moments}{equation.A.2.0.15}{}} +\newlabel{eq:fgmmoments@cref}{{[equation][15][2147483647,1,2]A.2.15}{119}} +\newlabel{eq:fncaractfgm}{{A.2.16}{119}{La fonction génératrice des moments}{equation.A.2.0.16}{}} +\newlabel{eq:fncaractfgm@cref}{{[equation][16][2147483647,1,2]A.2.16}{119}} +\@writefile{toc}{\contentsline {subsection}{\numberline {A.2.4}La fonction g\IeC {\'e}n\IeC {\'e}ratrice des cumulants}{119}{subsection.A.2.4}} +\newlabel{sec:fgc}{{A.2.4}{119}{La fonction génératrice des cumulants}{subsection.A.2.4}{}} +\newlabel{sec:fgc@cref}{{[subsubappendix][4][2147483647,1,2]A.2.4}{119}} +\newlabel{eq:fgc}{{A.2.17}{119}{La fonction génératrice des cumulants}{equation.A.2.0.17}{}} +\newlabel{eq:fgc@cref}{{[equation][17][2147483647,1,2]A.2.17}{119}} +\newlabel{eq:cumulantsempiriques}{{A.2.18}{119}{La fonction génératrice des cumulants}{equation.A.2.0.18.18}{}} +\newlabel{eq:cumulantsempiriques@cref}{{[equation][18][2147483647,1,2]A.2.18}{119}} +\newlabel{eq:cumulantsempiriques1}{{A.2.18a}{119}{La fonction génératrice des cumulants}{equation.A.2.0.18a}{}} +\newlabel{eq:cumulantsempiriques1@cref}{{[equation][1][2147483647,1,2,18]A.2.18a}{119}} +\newlabel{eq:cumulantsempiriques2}{{A.2.18b}{119}{La fonction génératrice des cumulants}{equation.A.2.0.18b}{}} +\newlabel{eq:cumulantsempiriques2@cref}{{[equation][2][2147483647,1,2,18]A.2.18b}{119}} +\newlabel{eq:cumulantsempiriques3}{{A.2.18c}{119}{La fonction génératrice des cumulants}{equation.A.2.0.18c}{}} +\newlabel{eq:cumulantsempiriques3@cref}{{[equation][3][2147483647,1,2,18]A.2.18c}{119}} +\newlabel{eq:cumulantsempiriques4}{{A.2.18d}{119}{La fonction génératrice des cumulants}{equation.A.2.0.18d}{}} +\newlabel{eq:cumulantsempiriques4@cref}{{[equation][4][2147483647,1,2,18]A.2.18d}{119}} +\newlabel{eq:derivfgc}{{A.2.19}{119}{La fonction génératrice des cumulants}{equation.A.2.0.19}{}} +\newlabel{eq:derivfgc@cref}{{[equation][19][2147483647,1,2]A.2.19}{119}} +\@writefile{toc}{\contentsline {subsection}{\numberline {A.2.5}La transform\IeC {\'e}e d'Esscher}{120}{subsection.A.2.5}} +\newlabel{sec:transesscher}{{A.2.5}{120}{La transformée d'Esscher}{subsection.A.2.5}{}} +\newlabel{sec:transesscher@cref}{{[subsubappendix][5][2147483647,1,2]A.2.5}{120}} +\newlabel{eq:esschertransform}{{A.2.20}{120}{La transformée d'Esscher}{equation.A.2.0.20}{}} +\newlabel{eq:esschertransform@cref}{{[equation][20][2147483647,1,2]A.2.20}{120}} +\newlabel{eq:esscherMx}{{A.2.21}{120}{La transformée d'Esscher}{equation.A.2.0.21}{}} +\newlabel{eq:esscherMx@cref}{{[equation][21][2147483647,1,2]A.2.21}{120}} +\@writefile{toc}{\contentsline {section}{\numberline {A.3}La transform\IeC {\'e}e de Fourier rapide}{120}{section.A.3}} +\newlabel{sec:methodeFFT}{{A.3}{120}{La transformée de Fourier rapide}{section.A.3}{}} +\newlabel{sec:methodeFFT@cref}{{[subappendix][3][2147483647,1]A.3}{120}} +\newlabel{eq:sommefft}{{A.3.1}{120}{La transformée de Fourier rapide}{equation.A.3.0.1}{}} +\newlabel{eq:sommefft@cref}{{[equation][1][2147483647,1,3]A.3.1}{120}} +\newlabel{eq:sommeifft}{{A.3.2}{120}{La transformée de Fourier rapide}{equation.A.3.0.2}{}} +\newlabel{eq:sommeifft@cref}{{[equation][2][2147483647,1,3]A.3.2}{120}} +\citation{barndorff2001levy} +\newlabel{eq:sommefftdensite}{{A.3.3}{121}{La transformée de Fourier rapide}{equation.A.3.0.3.3}{}} +\newlabel{eq:sommefftdensite@cref}{{[equation][3][2147483647,1,3]A.3.3}{121}} +\newlabel{eq:sommefftdensite1}{{A.3.3a}{121}{La transformée de Fourier rapide}{equation.A.3.0.3a}{}} +\newlabel{eq:sommefftdensite1@cref}{{[equation][1][2147483647,1,3,3]A.3.3a}{121}} +\newlabel{eq:sommefftdensite2}{{A.3.3b}{121}{La transformée de Fourier rapide}{equation.A.3.0.3b}{}} +\newlabel{eq:sommefftdensite2@cref}{{[equation][2][2147483647,1,3,3]A.3.3b}{121}} +\newlabel{eq:signalfftfonction}{{A.3.4}{121}{La transformée de Fourier rapide}{equation.A.3.0.4}{}} +\newlabel{eq:signalfftfonction@cref}{{[equation][4][2147483647,1,3]A.3.4}{121}} +\newlabel{eq:fftfonction}{{A.3.5}{121}{La transformée de Fourier rapide}{equation.A.3.0.5}{}} +\newlabel{eq:fftfonction@cref}{{[equation][5][2147483647,1,3]A.3.5}{121}} +\newlabel{eq:densitefftfonction}{{A.3.6}{121}{La transformée de Fourier rapide}{equation.A.3.0.6}{}} +\newlabel{eq:densitefftfonction@cref}{{[equation][6][2147483647,1,3]A.3.6}{121}} +\@writefile{toc}{\contentsline {section}{\numberline {A.4}Processus de L\IeC {\'e}vy}{121}{section.A.4}} +\newlabel{sec:processuslevy}{{A.4}{121}{Processus de Lévy}{section.A.4}{}} +\newlabel{sec:processuslevy@cref}{{[subappendix][4][2147483647,1]A.4}{121}} +\@writefile{toc}{\contentsline {subsection}{\numberline {A.4.1}D\IeC {\'e}finition et propri\IeC {\'e}t\IeC {\'e}s}{121}{subsection.A.4.1}} +\newlabel{sec:defproplevy}{{A.4.1}{121}{Définition et propriétés}{subsection.A.4.1}{}} +\newlabel{sec:defproplevy@cref}{{[subsubappendix][1][2147483647,1,4]A.4.1}{121}} +\citation{applebaum2004levy} +\@writefile{toc}{\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-Khintchine}{122}{section*.58}} +\newlabel{sec:levykhintchine}{{A.4.1}{122}{Représentation de Lévy-Khintchine}{section*.58}{}} +\newlabel{sec:levykhintchine@cref}{{[subsubappendix][1][2147483647,1,4]A.4.1}{122}} +\newlabel{eq:levykhintchine}{{A.4.1}{122}{Représentation de Lévy-Khintchine}{equation.A.4.0.1}{}} +\newlabel{eq:levykhintchine@cref}{{[equation][1][2147483647,1,4]A.4.1}{122}} +\@writefile{toc}{\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-It\IeC {\^o}}{122}{section*.59}} +\newlabel{sec:levyito}{{A.4.1}{122}{Représentation de Lévy-Itô}{section*.59}{}} +\newlabel{sec:levyito@cref}{{[subsubappendix][1][2147483647,1,4]A.4.1}{122}} +\newlabel{eq:levyitodecomposition}{{A.4.3}{122}{Représentation de Lévy-Itô}{equation.A.4.0.3}{}} +\newlabel{eq:levyitodecomposition@cref}{{[equation][3][2147483647,1,4]A.4.3}{122}} +\citation{sato1999levy} +\citation{schoutens2003levy} +\citation{kyprianou2007introductory} +\citation{teschl2004topics} +\@writefile{toc}{\contentsline {subsection}{\numberline {A.4.2}Processus subordonn\IeC {\'e}}{123}{subsection.A.4.2}} +\newlabel{sec:procsubordonne}{{A.4.2}{123}{Processus subordonné}{subsection.A.4.2}{}} +\newlabel{sec:procsubordonne@cref}{{[subsubappendix][2][2147483647,1,4]A.4.2}{123}} +\newlabel{eq:processussubordonne}{{A.4.5}{123}{Processus subordonné}{equation.A.4.0.5}{}} +\newlabel{eq:processussubordonne@cref}{{[equation][5][2147483647,1,4]A.4.5}{123}} +\newlabel{eq:exposantcaractYt}{{A.4.6}{123}{Processus subordonné}{equation.A.4.0.6}{}} +\newlabel{eq:exposantcaractYt@cref}{{[equation][6][2147483647,1,4]A.4.6}{123}} +\newlabel{eq:fncaractYt}{{A.4.7}{123}{Processus subordonné}{equation.A.4.0.7}{}} +\newlabel{eq:fncaractYt@cref}{{[equation][7][2147483647,1,4]A.4.7}{123}} +\newlabel{eq:densiteprocessusyt}{{A.4.8}{123}{Processus subordonné}{equation.A.4.0.8}{}} +\newlabel{eq:densiteprocessusyt@cref}{{[equation][8][2147483647,1,4]A.4.8}{123}} +\@writefile{toc}{\contentsline {section}{\numberline {A.5}Th\IeC {\'e}or\IeC {\`e}mes d'int\IeC {\'e}gration}{123}{section.A.5}} +\newlabel{sec:integration}{{A.5}{123}{Théorèmes d'intégration}{section.A.5}{}} +\newlabel{sec:integration@cref}{{[subappendix][5][2147483647,1]A.5}{123}} +\@writefile{toc}{\contentsline {subsection}{\numberline {A.5.1}Th\IeC {\'e}or\IeC {\`e}me de convergence domin\IeC {\'e}e de Lebesgue}{124}{subsection.A.5.1}} +\newlabel{sec:theor-de-conv}{{A.5.1}{124}{Théorème de convergence dominée de Lebesgue}{subsection.A.5.1}{}} +\newlabel{sec:theor-de-conv@cref}{{[subsubappendix][1][2147483647,1,5]A.5.1}{124}} +\newlabel{eq:sequencetheoremconvdom}{{A.5.1}{124}{Théorème de convergence dominée de Lebesgue}{equation.A.5.0.1}{}} +\newlabel{eq:sequencetheoremconvdom@cref}{{[equation][1][2147483647,1,5]A.5.1}{124}} +\newlabel{eq:theoremeconvdom}{{A.5.2}{124}{Théorème de convergence dominée de Lebesgue}{equation.A.5.0.2}{}} +\newlabel{eq:theoremeconvdom@cref}{{[equation][2][2147483647,1,5]A.5.2}{124}} +\@writefile{toc}{\contentsline {subsection}{\numberline {A.5.2}Th\IeC {\'e}or\IeC {\`e}me de Fubini}{124}{subsection.A.5.2}} +\newlabel{sec:theoreme-de-fubini}{{A.5.2}{124}{Théorème de Fubini}{subsection.A.5.2}{}} +\newlabel{sec:theoreme-de-fubini@cref}{{[subsubappendix][2][2147483647,1,5]A.5.2}{124}} +\newlabel{eq:rectanglefubini}{{A.5.3}{124}{Théorème de Fubini}{equation.A.5.0.3}{}} +\newlabel{eq:rectanglefubini@cref}{{[equation][3][2147483647,1,5]A.5.3}{124}} +\newlabel{eq:thfubini2}{{A.5.4}{124}{Théorème de Fubini}{equation.A.5.0.4}{}} +\newlabel{eq:thfubini2@cref}{{[equation][4][2147483647,1,5]A.5.4}{124}} +\@setckpt{annexe1}{ +\setcounter{page}{125} +\setcounter{equation}{4} +\setcounter{enumi}{2} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{1} +\setcounter{section}{5} +\setcounter{subsection}{2} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{143} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{0} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{35} +\setcounter{Hfootnote}{3} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{126} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/annexe1.tex b/memoire/annexe1.tex new file mode 100644 index 0000000..17efb4c --- /dev/null +++ b/memoire/annexe1.tex @@ -0,0 +1,570 @@ +\chapter{Éléments de théorie des probabilités} + +\section{Définitions de base} +\label{sec:introvariablealeatoire} + +Une \textbf{variable aléatoire} est une variable dont la valeur est +déterminée par une expérience aléatoire. On associe une probabilité à +chacune des valeurs possibles. On appelle \textbf{évènement} tout +ensemble de réalisations possible de l'expérience +aléatoire. L'entièreté de tous les évènements se nomme +l'\textbf{ensemble fondamental} et est notée $\Omega$. + +Une \textbf{probabilité} associée à un évènement $A$ doit satisfaire +un ensemble de trois axiomes \citep{dodge2004statistique}. +\begin{enumerate} +\item Une probabilité est une valeur comprise entre 0 et 1: + \begin{equation} + \label{eq:condition1probabilite} + 0 \leq P(A) \leq 1. + \end{equation} + +\item La probabilité de l'évènement correspondant à l'ensemble + fondamental est 1: + \begin{equation} + \label{eq:condition2probabilite} + P(\Omega) = 1. + \end{equation} + +\item Pour chaque séquence d'évènements mutuellement exclusifs + $A_1,A_2,\ldots$: + \begin{equation} + \label{eq:condition3probabilite} + P \left[ \bigcup_{i=1}^{\infty} A_i \right] = \sum_{i=1}^{\infty} P(A_i). + \end{equation} +\end{enumerate} + +Le respect de ces conditions devra notamment être vérifié lors de +l'utilisation de méthodes numériques. + +Toujours selon \cite{dodge2004statistique}, on définit: +\begin{enumerate} +\item La \textbf{densité}, notée $f_X(x)$, permet de déterminer la + probabilité qu'une variable aléatoire $X$ prenne une valeur dans un + intervalle fixé $\left[ a,b \right]$. En intégrant cette fonction + sur l'intervalle $\left[ a,b \right]$, on obtient cette probabilité: + \begin{equation} + \label{eq:densitedefinition} + P(a \leq X \leq b) = \int_{a}^{b} f_X(x) dx. + \end{equation} +\item La \textbf{fonction de répartition}, notée $F_X(x)$, d’une + variable aléatoire est définie comme la probabilité que celle-ci + prenne une valeur inférieure ou égale à un certain nombre $b \in + \mathbb{R}$: + \begin{equation} + \label{eq:repartitiondefinition} + F_X(b) = P(X \leq b) = \int_{-\infty}^{b} f_X(x) dx. + \end{equation} +\end{enumerate} + + +\section{Transformées d'une variable aléatoire} +\label{sec:transformees} + +Les transformées d'une variable aléatoire résultent de l'application +d'un opérateur intégral, ou à noyau, sur la fonction de densité. + +Certaines d'entre elles permettent de déterminer entièrement leur +distribution. Parmi celles-ci, on retrouve la fonction caractéristique +et les fonctions génératrice des moments et des cumulants, qui sont +les plus couramment utilisées. + +Certaines transformées permettent de modifier la distribution d'une +variable aléatoire. Parmi celles-ci, la transformée d'Esscher, qui est +employée en actuariat et en finance, est aussi étroitement liée au +concept d'utilité en sciences économiques. + +\subsection{La fonction caractéristique} +\label{sec:fonctioncaract} + +\subsubsection{Transformée de Fourier} +\label{sec:transfourier} + +La transformée de Fourier est une opération qui transforme une +fonction $g(\xi)$ en une autre $f(x)$ par l'intégration, sur son +domaine, du produit $\mathrm{e}^{-i x\xi}\,g(\xi)$: +\begin{equation} + \label{eq:transfourier} + \mathcal{F}(g):x\mapsto f(x) = \int_{-\infty}^{\infty} \mathrm{e}^{-ix\xi}\,g(\xi)\,\mathrm{d}\xi. +\end{equation} + +On définit aussi la transformée de Fourier inverse +\index{Transformée!de Fourier!inverse}, qui permet de retrouver la +fonction initiale $g(\xi)$ à partir de la transformée $f(x)$: +\begin{equation} + \label{eq:transinvfourier} + g(\xi)=\mathcal{F}^{-1}(f(x)) = {1 \over 2\pi}\, \int_{-\infty}^{\infty} \mathrm{e}^{ix \xi}\,f(x)\, \mathrm{d}x. +\end{equation} + +\subsubsection{Définition} +\label{sec:deffncaract} + +La fonction caractéristique \index{Fonction!caractéristique} est +définie comme étant la transformée de Fourier inverse de la densité +\index{Fonction!de densité}, dans le cas continu, ou de masse de +probabilité \index{Fonction!de masse de probabilité}, dans le cas +discret. C'est donc une application directe de +\eqref{eq:transinvfourier} qui s'exprime par l'intégrale de +Riemann-Stieltjes: +\begin{equation} + \label{eq:deffncaract} + \phi_X(s) = \int_{-\infty}^{\infty} e^{isx} dF_X(x). +\end{equation} + +Cette intégrale est toujours convergente, comme le démontre +\cite{stuart1987kendall}. + +\subsubsection{Les moments} +\label{sec:momentsfncaract} +On définit le moment d'ordre $r$ d'une distribution de probabilités +comme étant la quantité représentée par l'espérance de la puissance +$r$ d'une variable aléatoire: +\begin{equation} + \label{eq:defmoments} + E[X^r] = \int_{-\infty}^{\infty} x^r dF_X(x). +\end{equation} + +La fonction caractéristique\index{Fonction!caractéristique}, +lorsqu'elle est différenciable, permettra de générer les moments de la +distribution en utilisant la propriété suivante +\citep{lukacs1960characteristic}: +\begin{equation*} + \frac{d^r\phi_X(s)}{ds^r} = i^r \int_{-\infty}^{\infty} e^{isx} x^r dF_X(x). +\end{equation*} + +En posant $s=0$ par la suite, on obtient, pour les différentes +dérivées, l'équation suivante: +\begin{equation*} + \left[ \frac{d^r\phi_X(s)}{ds^r} \right]_{s=0} = i^r\,E[X^r]. +\end{equation*} + +Ce qui nous permet de définir les différents moments de la +distribution à partir de la fonction caractéristique: +\begin{equation} + \label{eq:fncaractmoments} + E[X^r] = (-i)^r\,\left[ \frac{d^r\phi_X(s)}{ds^r} \right]_{s=0}. +\end{equation} + +\subsection{Inversion de la fonction caractéristique} +\label{sec:inversioncaract} + +\subsubsection{La densité} +\label{sec:densitefncaract} + +On obtient la densité de la variable aléatoire $X$ en calculant la +transformée de Fourier \eqref{eq:transfourier} de la fonction +caractéristique suivante: +\begin{equation} + \label{eq:caractdensite} + f_X(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-isx} \phi_X(s) ds. +\end{equation} + +On peut utiliser l'intégration numérique si l'intégrale n'a pas de +solution analytique, la méthode de la transformée de Fourier rapide +(section \ref{sec:methodeFFT}) ou encore la méthode du point de selle. + +\subsubsection{La fonction de répartition} +\label{sec:repartitionfncaract} + +On peut obtenir directement l'expression de la fonction de +répartition, sans passer par l'intégration de la densité de +probabilité, en utilisant le théorème de \cite{gil1951note}: +\begin{equation} + \label{eq:gilpelaez} + F_X(x) = \frac{1}{2} + \frac{1}{2\pi} \int_{0}^{\infty} \frac{e^{isx}\phi_X(-s)-e^{-isx}\phi_X(s)}{is} ds. +\end{equation} + +On peut exprimer cette intégrale sous la forme suivante lorsque la +densité $f(x)$ est strictement continue \citep[p.66]{epps2007pricing}: +\begin{equation} + \label{eq:gilpelaez2} + F_X(x) = \frac{1}{2} - \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{e^{-isx}}{is} \phi(s) ds. +\end{equation} +Cette forme est moins appropriée pour l'intégration numérique, mais +sera utile dans plusieurs calculs. On préfèrera la ramener à la forme +suivante avec le théorème 2 de \cite{wendel1961non}: +\begin{equation} + \label{eq:inversionfncaract} + F_X(x) = \frac{1}{2} - \frac{1}{\pi}\int_{0}^{\infty} \frac{Im\left[e^{-isx}\phi_X(s)\right]}{s} ds. +\end{equation} + +Cette fonction peut être difficile à intégrer de manière efficace +numériquement, surtout lorsqu'on l'évalue en des points situés aux +extrémités de la distribution. On peut alors privilégier l'égalité +donnée par le théorème 4 de \cite{shephard1991characteristic}, qui +utilise le noyau de Fejér afin de réduire l'erreur d'intégration, pour +obtenir le résultat suivant: +\begin{equation} + \label{eq:shepherd} + F_X(x) = \frac{1}{2} - \frac{1}{2\pi} \lim_{n\rightarrow\infty} \int_{0}^{n} \underbrace{\left[ 1-\frac{s}{n} \right]}_{\text{Noyau de Fejér}} \left[ \frac{e^{isx}\phi_X(-s)-e^{-isx}\phi_X(s)}{is} \right] ds. +\end{equation} + +De même, en utilisant le raisonnement qui permet de passer de la +définition de la transformée \eqref{eq:gilpelaez} à celle de son +inverse \eqref{eq:inversionfncaract}, on obtient, pour le résultat +\eqref{eq:shepherd}, l'expression suivante: +\begin{equation} + \label{eq:approxinvfncaract} + F_X(x) = \frac{1}{2} - \frac{1}{\pi} \lim_{n\rightarrow\infty} \int_{0}^{n} \left[ 1-\frac{s}{n} \right] \frac{Im\left[e^{-isx}\phi_X(s)\right]}{s}ds. +\end{equation} + +On peut alors obtenir une approximation en fixant la borne +d'intégration supérieure $n$ qui définit la précision +désirée. Certains logiciels d'intégration numérique prennent en charge +les bornes infinies. + +La fonction caractéristique permet d'identifier la distribution d'une +somme de variables aléatoires indépendantes $Z = X_1, \ldots, X_n$, +appelée produit de convolution et noté $Z = X_1 * \ldots * X_n$: +\begin{equation} + \label{eq:convocaract} + \phi_{Z}(s) = \phi_{X_1+\ldots+X_n}(s) = \prod_{i=1}^n \phi_{X_i}(s). +\end{equation} + +Lorsque les variables aléatoires sommées sont aussi identiquement +distribuées, la fonction caractéristique de $Z$ est la $n^e$ puissance +de celle de $X$: +\begin{equation} + \label{eq:convocaractIID} + \phi_{Z}(s) = \phi_{X_1+\ldots+X_n}(s) = \left[\phi_{X_i}(s)\right]^n. +\end{equation} + +Cette fonction est donc une solution de rechange intéressante à +utiliser lorsque aucune forme analytique pour la densité ou la +fonction de répartition pour une distribution donnée n'existe. + +\subsection{La fonction génératrice des moments} +\label{sec:fgm} +La fonction génératrice des moments $M_X(s)$ +\index{Fonction!génératrice des moments} est définie comme étant la +transformée de Laplace inverse \index{Transformée!de Laplace!inverse} +de la densité: +\begin{equation} + \label{eq:deffngenmom} + M_X(s) = \int_{-\infty}^{\infty} e^{sx} dF_X(x). +\end{equation} + +Cette intégrale, contrairement à \eqref{eq:deffncaract}, ne converge +pas toujours, ce qui signifie que certaines distributions n'ont pas de +fonction génératrice des moments. Comme son nom l'indique, cette +fonction permet de générer les moments de la distribution de la +variable $X$, ce qui se fait essentiellement de la même manière +qu'avec la définition \eqref{eq:fncaractmoments}. Par contre, on n'a +pas à éliminer le terme complexe, ce qui a pour avantage de simplifier +les calculs: +\begin{equation} + \label{eq:fgmmoments} + E[X^r] = \left[ \frac{d^r M_X(s)}{ds^r} \right]_{s=0}. +\end{equation} + +La fonction génératrice des moments peut aussi être obtenue à partir +de la fonction caractéristique : +\begin{equation} + \label{eq:fncaractfgm} + M_X(s) = \phi(-is). +\end{equation} + +Ceci permet d'en déduire qu'elles possèdent des caractéristiques +communes, notamment le produit de convolution. + +\subsection{La fonction génératrice des cumulants} +\label{sec:fgc} +La fonction génératrice des cumulants $K_X(s)$ est définie comme le +logarithme de la fonction génératrice des moments: +\begin{equation} + \label{eq:fgc} + K_X(s) = \ln{M_X(s)}. +\end{equation} + +Cette fonction est utilisée pour générer les cumulants, des quantités +étroitement liées aux moments, qui peuvent à leur tour être utilisés +dans le cadre de méthodes d'estimation paramétrique. Parmi celles-ci, +on retrouve la méthode des cumulants, dont l'objectif est de former un +système d'équations où les valeurs empiriques et théoriques de ces +quantités sont égales. Les valeurs empiriques peuvent être obtenues à +partir des moments échantillonnaux \citep{stuart1987kendall}: +\begin{subequations}\label{eq:cumulantsempiriques} + \begin{align} + \hat{K}_1 &= \hat{m}_1 \label{eq:cumulantsempiriques1}\\ + \hat{K}_2 &= \hat{m}_2' - \hat{m}_1^2 \label{eq:cumulantsempiriques2}\\ + \hat{K}_3 &= \hat{m}_3' - 3\hat{m}_1\hat{m}_2' + 2\hat{m}_1^2 \label{eq:cumulantsempiriques3}\\ + \hat{K}_4 &= \hat{m}_4' - 3\hat{m}_2'^2 - 4\hat{m}_1\hat{m}_3' + 12\hat{m}_1^2\hat{m}_2' - + 6\hat{m}_1^4 \label{eq:cumulantsempiriques4}. + \end{align} +\end{subequations} + +On remarquera aussi la forme de la première dérivée de cette dernière, +qui sera utilisée pour estimer la densité et la fonction de +répartition avec la méthode du point de selle: +\begin{equation} + \label{eq:derivfgc} + \frac{dK_X(s)}{ds} = K^{\prime}_X(s) = \frac{M^{\prime}_X(s)}{M_X(s)}. +\end{equation} + +\subsection{La transformée d'Esscher} +\label{sec:transesscher} + +La transformée d'Esscher transforme la densité en une autre en +utilisant un coefficient exponentiel: +\begin{equation} + \label{eq:esschertransform} + f(x;h)=\frac{e^{hx}f(x)}{\int_{-\infty}^\infty e^{hx} f(x) dx}. +\end{equation} + +Lorsque l'on ne dispose pas d'une forme analytique de la densité, on +peut exprimer la transformée d'Esscher sous la forme de sa fonction +génératrice des moments: +\begin{equation} + \label{eq:esscherMx} + M_X(s;h) = \frac{M_X(s+h)}{M_X(h)}. +\end{equation} + +\section{La transformée de Fourier rapide} +\label{sec:methodeFFT} + +Étant donné l'importance de la transformée de Fourier et de son +inverse en sciences physiques, plusieurs algorithmes ont été +développés afin d'en effectuer l'intégration numérique. Ces +algorithmes se regroupent sous l'appellation de transformée de Fourier +rapide. + +L'existence de ces algorithmes favorise l'utilisation de la fonction +caractéristique et de ses propriétés, notamment pour l'agrégation de +risques en actuariat, ou encore le calcul de probabilités pour des +distributions n'ayant pas de forme explicite pour la fonction de +répartition. + +Ces algorithmes numériques sont utilisés respectivement pour calculer +de manière optimisée des sommes de la forme +\begin{align} + \hat{f}(x_u) &= \sum_{j=1}^{N} \phi(\zeta_j)\,e^{-{2\pi i \over N} (j-1)(u-1) } \qquad u = 1,\dots,n. \label{eq:sommefft} \\ + \hat{\phi}(\zeta_j) &= \sum_{u=1}^{N} f(x_u)\,e^{{2\pi i \over N} + (j-1)(u-1) } \qquad j = 1,\dots,n. \label{eq:sommeifft} +\end{align} + +Par exemple, pour évaluer l'intégrale \eqref{eq:caractdensite} afin +d'obtenir la valeur de la densité sur un certain domaine +$\left[a,b\right]$, on devra discrétiser celle-ci sur nombre $N$ de +points de discrétisation.On définit alors les différents paramètres de +ces deux algorithmes: +\begin{itemize} +\item $\eta = \frac{b-a}{N}$, le pas de discrétisation pour la + variable aléatoire $u$ +\item $\zeta = \eta(j-1)$, la variable de transformation +\item $\lambda = \frac{2\pi}{N\eta}$, le pas de discrétisation pour la + variable de transformation $\zeta$ +\item $c=\frac{N\lambda}{2}$, la borne supérieure d'intégration +\end{itemize} + +On obtient alors une sommation de la forme \eqref{eq:sommefft}: +\begin{subequations}\label{eq:sommefftdensite} + \begin{align} + f_{FFT}(x_u) &\approx \frac{1}{2\pi} \sum_{j=1}^N e^{-i\lambda\zeta_j(u-1)} e^{-ic\zeta_j} \phi(-c+\lambda\,(j-1)) \eta, \label{eq:sommefftdensite1}\\ + &\approx \frac{1}{2\pi} \sum_{j=1}^N + e^{-i\frac{2\pi}{N}(j-1)(u-1)} e^{-ic\zeta_j} + \phi(-c+\lambda\,(j-1)) \eta.\label{eq:sommefftdensite2} + \end{align} +\end{subequations} + +Plusieurs logiciels d'analyse numérique possèdent une implémentation +de la méthode de la transformée de Fourier rapide. Essentiellement, +c'est une fonction $f_{FFT}(X_u)$ qui prend comme argument un vecteur +de valeurs, appelé signal, et qui en retourne un autre, de même +longueur, le spectre de fréquences. En statistique, la fonction +caractéristique empirique constitue le signal et la densité, le +spectre de fréquences. Le signal $X_u$ est défini comme suit, à partir +des constantes définies précédemment: +\begin{align} + \label{eq:signalfftfonction} + X_u &= e^{-i\lambda\zeta_j(u-1)} \phi\left(-c+(j-1)\lambda\right) + \eta. +\end{align} + +La somme \eqref{eq:sommefftdensite1} devient alors +\begin{align} + \label{eq:fftfonction} + f_{FFT}(X_u) &\approx \frac{1}{2\pi} \sum_{j=1}^N X_u e^{-ic\zeta_j}. +\end{align} + +On recouvre la densité aux points $x_u=a+(j-1)\eta, j=1,\ldots,N$ en +appliquant la transformation suivante au vecteur $f_{FFT}(X_u)$: +\begin{align} + \label{eq:densitefftfonction} + f(x_u) = \frac{1}{N} e^{icx_u} f_{FFT}(X_u). +\end{align} + +\section{Processus de Lévy} +\label{sec:processuslevy} + +\subsection{Définition et propriétés} +\label{sec:defproplevy} + +Le processus de Lévy, tel que présenté par \cite{barndorff2001levy}, +est une classe générale de processus stochastiques regroupant entre +autres les processus de Poisson composés et les processus de Wiener. +Il est défini comme continu à droite, limité à gauche (càdlàg), ayant +pour point de départ l'origine et ayant des incréments indépendants et +homogènes. Il est infiniment divisible, tout processus de Lévy peut +ainsi être considéré comme une convolution de plusieurs autres. Cette +propriété est très intéressante dans un contexte de rendements +financiers, car la même distribution pourra être utilisée avec +n'importe quel intervalle d'échantillonnage des prix. + +\subsubsection{Représentation de Lévy-Khintchine} +\label{sec:levykhintchine} + +Le processus de Lévy est généralement représenté par sa fonction +caractéristique sous la forme de Lévy-Khintchine: +\begin{align} + \mathbb{E}\Big[e^{i\theta X(t)} \Big] &= \exp \Bigg( + \underbrace{ait\theta}_{\text{composante de dérive}} \nonumber\\ + &\qquad - + \underbrace{\frac{1}{2}\sigma^2t\theta^2}_{\text{composante de + diffusion}} \nonumber\\ &\qquad + \underbrace{t + \int_{\mathbb{R}\backslash\{0\}} \big( e^{i\theta x}-1 -i\theta x \mathbf{I}_{|x|<1}\big)\,\nu(dx)}_{\text{composante de saut}} \Bigg)\label{eq:levykhintchine}\\ + &= \exp \bigg(-t\Psi(\theta) \bigg). +\end{align} + +$\Psi(\theta)$ est appelé l'exposant caractéristique de l'incrément de +longueur 1 $(X(t+1)-X(t))$. Un processus de Lévy est souvent décrit +par son triplet générateur $(a,\sigma^2,\nu)$. Cette description +permettra de classer certains processus de Lévy selon deux catégories: +\begin{itemize} +\item Si $\sigma^2=0$, $\lbrace X(t) \rbrace$ est un processus de + sauts +\item Si $a=0 \mbox{ et } \sigma^2=0$, alors le processus $\lbrace + X(t) \rbrace$ est un processus de sauts purs +\end{itemize} + +L'élément $\nu(dx)$ de la composante de saut est appelé la mesure de +Lévy. Un exemple de processus de Lévy représenté sous la forme de +Lévy-Khintchine qui ne fait pas partie de ces deux catégories est +défini par le modèle de Press \eqref{eq:fncaractpress}. + +\subsubsection{Représentation de Lévy-Itô} +\label{sec:levyito} +La décomposition de Lévy-Itô est une représentation alternative +décrivant la trajectoire d'une réalisation du processus. Cette +dernière a une interprétation intéressante en finance décrite par +\cite{applebaum2004levy}: +\begin{align} + \label{eq:levyitodecomposition} + X(t) &= \underbrace{at + B_{\sigma^2}(t)}_{\text{mouvement brownien}} \nonumber\\ &\quad+ \underbrace{\int_{|x|<1}x N(t,dx)-t\nu(dx)}_{\text{martingale de sauts purs de carré intégrable}} \nonumber\\ &\quad+ \underbrace{\int_{|x|>1}x N(t,dx)}_{\text{processus de Poisson composé}} \\ + &= X_1(t) + X_2(t) + X_3(t). +\end{align} + +La portion mouvement brownien ($X_1(t)$) décrit le comportement +général du processus, en spécifiant le rendement espéré $a$ et la +volatilité du titre $\sigma^2$. La première intégrale ($X_2(t)$) +décrit le bruit causé par les transactions financières quotidiennes +qui font varier un peu le prix, alors que la seconde ($X_3(t)$) +introduit les sauts occasionnés par des évènements plus rares, comme +les catastrophes naturelles et les crises politiques. Dans les deux +cas, $N(t,dx)$ est une mesure aléatoire de Poisson. + +\subsection{Processus subordonné} +\label{sec:procsubordonne} + +On considère les processus de Lévy $\lbrace X(t) \rbrace$ et $\lbrace +Z(t) \rbrace$. Celui qui suit est défini comme étant un processus +subordonné et aussi un processus de Lévy, comme le démontre +\cite{sato1999levy} et \cite{schoutens2003levy}: +\begin{equation} + \label{eq:processussubordonne} + \lbrace Y(t) \rbrace = \lbrace X\left(Z(t)\right) \rbrace. +\end{equation} + +Les processus de Lévy les plus couramment utilisés en finance sont des +mouvements browniens subordonnés. Ils sont plus faciles à manipuler et +permettent néanmoins de représenter les phénomènes de queues longues +présents dans les distributions de rendements. Une bonne introduction +à ce sujet est faite par \cite{kyprianou2007introductory}. + +Si $\Lambda(\theta)$ est l'exposant caractéristique du processus +$\lbrace X(t)\rbrace$ et $\Xi(\theta)$, celui du processus subordonné +$\lbrace Z(t)\rbrace$, alors celui du processus $\lbrace Y(t)\rbrace$ +prend la forme suivante: +\begin{align} + \label{eq:exposantcaractYt} + \Psi(\theta) &= \Xi(\theta)\circ i\Lambda(\theta) \nonumber\\ + &= \Xi(i\Lambda(\theta)). +\end{align} + +La fonction caractéristique du processus $\lbrace Y(t) \rbrace$ est +alors +\begin{equation} + \label{eq:fncaractYt} + \phi_{Y(t)}(\xi) = e^{-t\Xi(i\Lambda(\xi))}. +\end{equation} + +La densité de la variable aléatoire $Y(t)$ s'obtient à l'aide de la +formule de l'espérance conditionnelle: +\begin{align} + f_{Y(t)}(y) &= E\left[ f_{X(t)}(y|Z(t)=z) \right] \nonumber\\ + &= \int_{-\infty}^{\infty} f_{X(t)}(y|Z(t)=z) \cdot f_{Z(t)}(z) + dz \label{eq:densiteprocessusyt}. +\end{align} + +\section{Théorèmes d'intégration} +\label{sec:integration} + +Ces deux théorèmes sont présentés sans démonstration afin de +complémenter la démonstration de la méthode d'Epps de la section +\ref{sec:epps2007}. La démonstration de ceux-ci nécessite des +connaissances en théorie de l'intégration. On peut retrouver davantage +d'explications sur ceux-ci dans \cite{teschl2004topics}. + +\subsection{Théorème de convergence dominée de Lebesgue} +\label{sec:theor-de-conv} + +Le théorème de convergence dominée de Lebesgue est un des principaux +éléments de la théorie de l'intégration. Il sert à démontrer qu'une +fonction est intégrable, sachant qu'une autre l'est aussi et qu'elle +répond à certaines conditions. + +Soit $(f_n)_{n\in \mathbb{N}}$, une suite de fonctions réelles ou +complexes intégrables dans un intervalle $\mathit{I}$ qui convergent +vers une fonction $f$: +\begin{align} + \label{eq:sequencetheoremconvdom} + f(x) = \lim_{n \to \infty} f_n(x). +\end{align} + +On suppose qu'une fonction $g$ intégrable dans l'intervalle +$\mathit{I}$ existe telle que pour toute valeur de $n$ et pour tout +point $x\in I$ où elle est définie, la valeur absolue de $f_n(x)$ est +inférieure ou égale à $g(x)$: +\begin{align} + \label{eq:theoremeconvdom} + \forall n \in \mathbb{N}, \forall x \in \mathit{I}, |f_n(x)| \leq + g(x). +\end{align} + +Alors, la fonction $f$ est intégrable. + +\subsection{Théorème de Fubini} +\label{sec:theoreme-de-fubini} + +Le théorème de Fubini permet, entre autres, d'inverser l'ordre +d'intégration lorsque certaines conditions sont remplies. + +Soit une fonction $f$ continue sur le rectangle $\mathit{R}$ suivant: +\begin{align} + \label{eq:rectanglefubini} + \mathit{R} &= \left\{(x,y) | a \leq x \leq b, c \leq y \leq d \right\}. +\end{align} + +on peut inverser l'ordre d'intégration: +\begin{align} + \label{eq:thfubini2} + \int_{X\times Y}f(x,y)~ d(\mu\times\nu)(x,y)=\int_X\left[\int_Y + f(x,y)~ d\nu(y)\right]~ d\mu(x)=\int_Y\left[\int_X f(x,y)~ + d\mu(x)\right]~ d\nu(y). +\end{align} + +Il est possible de généraliser cet énoncé en utilisant la théorie de +la mesure. +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/annexe2.aux b/memoire/annexe2.aux new file mode 100644 index 0000000..d915008 --- /dev/null +++ b/memoire/annexe2.aux @@ -0,0 +1,111 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {appendix}{\chapternumberline {B}\IeC {\'E}l\IeC {\'e}ments de statistique math\IeC {\'e}matique}{125}{appendix.B}} +\@writefile{toc}{\contentsline {section}{\numberline {B.1}Loi faible des grands nombres}{125}{section.B.1}} +\newlabel{sec:loifaible}{{B.1}{125}{Loi faible des grands nombres}{section.B.1}{}} +\newlabel{sec:loifaible@cref}{{[subappendix][1][2147483647,2]B.1}{125}} +\newlabel{loifaible}{{B.1.1}{125}{Loi faible des grands nombres}{equation.B.1.0.1}{}} +\newlabel{loifaible@cref}{{[equation][1][2147483647,2,1]B.1.1}{125}} +\@writefile{toc}{\contentsline {section}{\numberline {B.2}Th\IeC {\'e}or\IeC {\`e}me central limite}{125}{section.B.2}} +\newlabel{sec:theor-centr-limite}{{B.2}{125}{Théorème central limite}{section.B.2}{}} +\newlabel{sec:theor-centr-limite@cref}{{[subappendix][2][2147483647,2]B.2}{125}} +\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.1}Cas univari\IeC {\'e}}{126}{subsection.B.2.1}} +\newlabel{sec:cas-univarie}{{B.2.1}{126}{Cas univarié}{subsection.B.2.1}{}} +\newlabel{sec:cas-univarie@cref}{{[subsubappendix][1][2147483647,2,2]B.2.1}{126}} +\newlabel{eq:TCL}{{B.2.1}{126}{Cas univarié}{equation.B.2.0.1}{}} +\newlabel{eq:TCL@cref}{{[equation][1][2147483647,2,2]B.2.1}{126}} +\newlabel{eq:TCL2}{{B.2.2}{126}{Cas univarié}{equation.B.2.0.2}{}} +\newlabel{eq:TCL2@cref}{{[equation][2][2147483647,2,2]B.2.2}{126}} +\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.2}Cas multivari\IeC {\'e}}{126}{subsection.B.2.2}} +\newlabel{sec:cas-multivarie}{{B.2.2}{126}{Cas multivarié}{subsection.B.2.2}{}} +\newlabel{sec:cas-multivarie@cref}{{[subsubappendix][2][2147483647,2,2]B.2.2}{126}} +\newlabel{eq:defmultiX}{{B.2.3}{126}{Cas multivarié}{equation.B.2.0.3}{}} +\newlabel{eq:defmultiX@cref}{{[equation][3][2147483647,2,2]B.2.3}{126}} +\newlabel{eq:TCLmulti1}{{B.2.4}{126}{Cas multivarié}{equation.B.2.0.4}{}} +\newlabel{eq:TCLmulti1@cref}{{[equation][4][2147483647,2,2]B.2.4}{126}} +\newlabel{eq:TCLmulti2}{{B.2.5}{126}{Cas multivarié}{equation.B.2.0.5}{}} +\newlabel{eq:TCLmulti2@cref}{{[equation][5][2147483647,2,2]B.2.5}{126}} +\@writefile{toc}{\contentsline {section}{\numberline {B.3}M\IeC {\'e}thode delta multivari\IeC {\'e}e}{127}{section.B.3}} +\newlabel{sec:deltamethod}{{B.3}{127}{Méthode delta multivariée}{section.B.3}{}} +\newlabel{sec:deltamethod@cref}{{[subappendix][3][2147483647,2]B.3}{127}} +\newlabel{eq:deltamethodmult}{{B.3.4}{127}{Méthode delta multivariée}{equation.B.3.0.4}{}} +\newlabel{eq:deltamethodmult@cref}{{[equation][4][2147483647,2,3]B.3.4}{127}} +\@setckpt{annexe2}{ +\setcounter{page}{128} +\setcounter{equation}{4} +\setcounter{enumi}{2} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{2} +\setcounter{section}{3} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{146} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{0} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{35} +\setcounter{Hfootnote}{3} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{132} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/annexe2.tex b/memoire/annexe2.tex new file mode 100644 index 0000000..2fd1dcd --- /dev/null +++ b/memoire/annexe2.tex @@ -0,0 +1,145 @@ +\chapter{Éléments de statistique mathématique} + +\section{Loi faible des grands nombres} +\label{sec:loifaible} + +La \textbf{loi faible des grands nombres} est un résultat important en +probabilité, car il permet de définir la notion d'estimateur +convergent. Soit une suite de variables aléatoires indépendantes et +identiquement distribuées $\left\{X_T \right\}_{T=1}^{\infty}$ ayant +une espérance $E\left[ X \right]$ et une variance $V\left[ X \right]$ +finies. Selon la loi faible des grands nombres, pour tout nombre réel +strictement positif $\varepsilon$, la probabilité que la différence +entre la moyenne empirique $Y_T=\frac{X_1+X_2+\cdots+X_T}{T}$ et +l'espérance $E\left[ X \right]$ soit supérieure à la valeur +$\varepsilon$ tend vers 0 lorsque $T$ tend vers l'infini. +\begin{align} + \label{loifaible} + \lim_{T \to +\infty} \mathbb{P}\left(\left|Y_T - E\left[ X + \right]\right| \geq \varepsilon\right) = 0 ,\quad + \forall\varepsilon>0 +\end{align} + +On dit alors que la suite d'estimateurs +$\left\{Y_T\right\}_{T=1}^{\infty}$ converge en probabilité vers +l'espérance $E\left[ X \right]$. L'estimateur de l'espérance $Y_T$ est +alors \textbf{convergent}. + +\section{Théorème central limite} +\label{sec:theor-centr-limite} + +Le \textbf{théorème central limite} est un résultat fondamental en +probabilité qui énonce le rôle de la distribution normale. Il démontre +que toute somme de variables aléatoires indépendantes et identiquement +distribuées suit approximativement une loi normale. Ce résultat permet +entre autres d'identifier la distribution limite d'un estimateur +convergent. + +\subsection{Cas univarié} +\label{sec:cas-univarie} + +Soit une suite d'observations $X_1, \ldots, X_T$ d'un échantillon +aléatoire d'une distribution de moyenne $\mu$ et de variance +$\sigma^2$: +\begin{align} + \label{eq:TCL} + Y_T &= \frac{1}{\sqrt{T}\sigma} \left(\sum_{t=1}^T X_t - T\mu\right) \nonumber\\ + &= \frac{\sqrt{T}}{\sigma}\left(\overline{X}_T-\mu\right). +\end{align} +Alors, cette variable aléatoire converge en distribution vers une +variable aléatoire normale centrée réduite: +\begin{align} + \label{eq:TCL2} + Y_T \stackrel{L}{\rightarrow} \mathcal{N}(0,1). +\end{align} + +\subsection{Cas multivarié} +\label{sec:cas-multivarie} + +On peut aussi généraliser ce théorème pour des observations +multivariées. On considère alors une série d'observations +multivariées $\mathbf{X_1}, \ldots, \mathbf{X_T}$ où +\begin{align} + \label{eq:defmultiX} + \mathbf{X_t}=\begin{bmatrix} X_{t(1)} \\ \vdots \\ + X_{t(k)} \end{bmatrix}, \quad t=1,\ldots,T. +\end{align} + +On définit maintenant la variable aléatoire correspondante: +\begin{align} + \label{eq:TCLmulti1} + \mathbf{Y}_T &= \frac{1}{T}\begin{bmatrix} \sum_{t=1}^{T} \left [ + X_{t(1)} \right ] \\ \vdots \\ \sum_{t=1}^{T} \left [ X_{t(k)} + \right ] \end{bmatrix}. +\end{align} + +Dans cette situation, la variable aléatoire converge en distribution +vers une variable aléatoire de distribution normale multivariée +centrée de matrice de variance-covariance $\mathbf{\Sigma}$: +\begin{align} + \label{eq:TCLmulti2} + \sqrt{T}\left(\mathbf{Y}_T - \mu\right)\ \stackrel{L}{\rightarrow}\ + \mathcal{N}_k(0,\mathbf{\Sigma}) +\end{align} + +où +\begin{align*} + \mathbf{\Sigma} &= \begin{bmatrix} + \omega_{(1,1)} &\cdots& \omega_{(1,k)} \\ + \vdots & \ddots & \vdots \\ + \omega_{(k,1)} &\cdots& \omega_{(k,k)} \\ + \end{bmatrix} ,\quad \mbox{avec } \omega_{(j,k)} = \begin{cases} + Var\left[X_{1(j)}\right] , &(j=k) \\ + Cov\left[X_{1(j)},X_{1(k)}\right] , &(j \neq k). + \end{cases} +\end{align*} + +\section{Méthode delta multivariée} +\label{sec:deltamethod} + +Dans le cas univarié, on utilise la méthode delta pour évaluer la +distribution d'une fonction d'un estimateur, en supposant que la +distribution de cet estimateur est asymptotiquement normale de +variance connue. + +Dans le cas multivarié, on estime la distribution d'une fonction d'un +vecteur d'estimateurs, dont la distribution asymptotique est normale +multivariée, avec une matrice de variance-covariance $\Sigma$. + +On a donc, pour un estimateur convergent $\hat\theta_T$, en appliquant +le théorème central limite, le résultat suivant: +\begin{align} + \sqrt{T} (\hat\theta_T - \theta_0) \stackrel{L}{\longrightarrow} + \mathcal{N}(0,\Sigma). +\end{align} +On cherche la distribution d'une fonction $h(\hat\theta_T)$. On +développe cette fonction sous la forme d'une série de Taylor et en +conservant seulement les deux premiers termes: +\begin{align} + h(\hat\theta_T) \approx h(\theta_0) + \nabla + \left[h(\theta_0)\right]'\cdot(\hat\theta_T - \theta_0). +\end{align} +On a donc, après quelques manipulations, que la variance de la +fonction $h(\hat\theta_T)$ est approximativement +\begin{align} + Var\left[h(\hat\theta_T) \right] \approx \nabla + \left[h(\theta_0)\right]' \left(\frac{\Sigma}{T}\right) \nabla + \left[h(\theta_0)\right]. +\end{align} + +La distribution de la fonction $h(\hat\theta_T)$ est alors +asymptotiquement +\begin{align} + \label{eq:deltamethodmult} + h(\hat\theta_T) \stackrel{L}{\longrightarrow} + N\left(h(\theta_0),\nabla \left[h(\theta_0)\right]' + \left(\frac{\Sigma}{T}\right) \nabla + \left[h(\theta_0)\right]\right). +\end{align} + + + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/annexe3.aux b/memoire/annexe3.aux new file mode 100644 index 0000000..3011d44 --- /dev/null +++ b/memoire/annexe3.aux @@ -0,0 +1,87 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {appendix}{\chapternumberline {C}Donn\IeC {\'e}es}{129}{appendix.C}} +\newlabel{chap:donnees}{{C}{129}{Données}{appendix.C}{}} +\newlabel{chap:donnees@cref}{{[appendix][3][2147483647]C}{129}} +\@writefile{lot}{\contentsline {table}{\numberline {C.1}{\ignorespaces Prix du titre Abbey National (penny sterling) du 31 juillet au 8 octobre 1991\relax }}{129}{table.caption.60}} +\newlabel{prixabbeyn}{{C.1}{129}{Prix du titre Abbey National (penny sterling) du 31 juillet au 8 octobre 1991\relax }{table.caption.60}{}} +\newlabel{prixabbeyn@cref}{{[table][1][2147483647,3]C.1}{129}} +\@setckpt{annexe3}{ +\setcounter{page}{130} +\setcounter{equation}{0} +\setcounter{enumi}{2} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{-1} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{3} +\setcounter{section}{0} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{148} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{1} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{35} +\setcounter{Hfootnote}{3} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{133} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/annexe3.tex b/memoire/annexe3.tex new file mode 100644 index 0000000..a29e339 --- /dev/null +++ b/memoire/annexe3.tex @@ -0,0 +1,31 @@ +\chapter{Données} +\label{chap:donnees} + +Ce tableau présente l'échantillon de données utilisées au chapitre 9. + +\begin{table}[htbp] + \caption{Prix du titre Abbey National (penny sterling) du 31 juillet au 8 octobre 1991} + \begin{center} + \begin{tabular}{cccccccccccccc} + \hline + \textbf{Date} & \multicolumn{1}{c}{\textbf{Prix}} & & \textbf{Date} & \multicolumn{1}{c}{\textbf{Prix}} & & \textbf{Date} & \multicolumn{1}{c}{\textbf{Prix}} & & \textbf{Date} & \multicolumn{1}{c}{\textbf{Prix}} & & \textbf{Date} & \multicolumn{1}{c}{\textbf{Prix}} \\ \hline + 31/07 & 296 & & 14/08 & 294 & & 28/08 & 303 & & 11/09 & 296 & & 25/09 & 306 \\ + 01/08 & 296 & & 15/08 & 293 & & 29/08 & 304 & & 12/09 & 301 & & 26/09 & 303 \\ + 02/08 & 300 & & 16/08 & 295 & & 30/08 & 304 & & 13/09 & 298 & & 27/09 & 301 \\ + 05/08 & 302 & & 19/08 & 287 & & 02/09 & 309 & & 16/09 & 295 & & 30/09 & 303 \\ + 06/08 & 300 & & 20/08 & 288 & & 03/09 & 309 & & 17/09 & 295 & & 01/10 & 308 \\ + 07/08 & 304 & & 21/08 & 297 & & 04/09 & 309 & & 18/09 & 293 & & 02/10 & 305 \\ + 08/08 & 303 & & 22/08 & 305 & & 05/09 & 307 & & 19/09 & 292 & & 03/10 & 302 \\ + 09/08 & 299 & & 23/08 & 307 & & 06/09 & 306 & & 20/09 & 297 & & 04/10 & 301 \\ + 12/08 & 293 & & 26/08 & 307 & & 09/09 & 304 & & 23/09 & 294 & & 07/10 & 297 \\ + 13/08 & 294 & & 27/08 & 304 & & 10/09 & 300 & & 24/09 & 293 & & 08/10 & 299 \\ + \hline + \end{tabular} + \end{center} + \label{prixabbeyn} +\end{table} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/avantpropos.aux b/memoire/avantpropos.aux new file mode 100644 index 0000000..ffad164 --- /dev/null +++ b/memoire/avantpropos.aux @@ -0,0 +1,78 @@ +\relax +\@writefile{toc}{\contentsline {chapter}{Avant-propos}{xxi}{section*.11}} +\@setckpt{avantpropos}{ +\setcounter{page}{22} +\setcounter{equation}{0} +\setcounter{enumi}{0} +\setcounter{enumii}{0} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{0} +\setcounter{section}{0} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{22} +\setcounter{lastsheet}{159} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{0} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{0} +\setcounter{Item}{0} +\setcounter{Hfootnote}{0} +\setcounter{memhycontfloat}{0} +\setcounter{bookmark@seq@number}{7} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/avantpropos.tex b/memoire/avantpropos.tex new file mode 100644 index 0000000..bc439c1 --- /dev/null +++ b/memoire/avantpropos.tex @@ -0,0 +1,4 @@ +\chapter*{Avant-propos} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Avant-propos} % inclure dans TdM + +L'avant-propos est surtout nécessaire pour une thèse par article. diff --git a/memoire/by-sa.pdf b/memoire/by-sa.pdf new file mode 100644 index 0000000..575f606 Binary files /dev/null and b/memoire/by-sa.pdf differ diff --git a/memoire/by.pdf b/memoire/by.pdf new file mode 100644 index 0000000..3391984 Binary files /dev/null and b/memoire/by.pdf differ diff --git a/memoire/chapitre1.aux b/memoire/chapitre1.aux new file mode 100644 index 0000000..6d19fec --- /dev/null +++ b/memoire/chapitre1.aux @@ -0,0 +1,204 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{derman1996modelrisk} +\citation{black1973pricing} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {chapter}{\chapternumberline {1}Les mod\IeC {\`e}les de rendements financiers}{3}{chapter.1}} +\@writefile{toc}{\contentsline {section}{\numberline {1.1}L'utilisation de mod\IeC {\`e}les en finance}{3}{section.1.1}} +\newlabel{sec:utilisationmodeles}{{1.1}{3}{L'utilisation de modèles en finance}{section.1.1}{}} +\newlabel{sec:utilisationmodeles@cref}{{[section][1][1]1.1}{3}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.1}Diff\IeC {\'e}rents types de mod\IeC {\`e}les}{3}{subsection.1.1.1}} +\newlabel{sec:differentsmodeles}{{1.1.1}{3}{Différents types de modèles}{subsection.1.1.1}{}} +\newlabel{sec:differentsmodeles@cref}{{[subsection][1][1,1]1.1.1}{3}} +\citation{fama1993common} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.2}Le risque de mod\IeC {\'e}lisation}{4}{subsection.1.1.2}} +\newlabel{sec:modelerisque}{{1.1.2}{4}{Le risque de modélisation}{subsection.1.1.2}{}} +\newlabel{sec:modelerisque@cref}{{[subsection][2][1,1]1.1.2}{4}} +\@writefile{toc}{\contentsline {section}{\numberline {1.2}Les rendements financiers}{5}{section.1.2}} +\newlabel{sec:prixrendements}{{1.2}{5}{Les rendements financiers}{section.1.2}{}} +\newlabel{sec:prixrendements@cref}{{[section][2][1]1.2}{5}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.1}D\IeC {\'e}finitions et notations}{5}{subsection.1.2.1}} +\newlabel{sec:defrendements}{{1.2.1}{5}{Définitions et notations}{subsection.1.2.1}{}} +\newlabel{sec:defrendements@cref}{{[subsection][1][1,2]1.2.1}{5}} +\newlabel{eq:solutiondiffrendement}{{1.2.1}{6}{Définitions et notations}{equation.1.2.1}{}} +\newlabel{eq:solutiondiffrendement@cref}{{[equation][1][1,2]1.2.1}{6}} +\@writefile{loe}{\addvspace {10\p@ }} +\@writefile{loe}{\contentsline {hypothese}{\numberline {1.1}Hypoth\IeC {\`e}se}{6}{hypothese.1.1}} +\newlabel{eq:rendementlogprix}{{1.2.2}{6}{Définitions et notations}{equation.1.2.2}{}} +\newlabel{eq:rendementlogprix@cref}{{[equation][2][1,2]1.2.2}{6}} +\newlabel{eq:rendementcumL}{{1.2.3}{6}{Définitions et notations}{equation.1.2.3}{}} +\newlabel{eq:rendementcumL@cref}{{[equation][3][1,2]1.2.3}{6}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.2}Rendements cumul\IeC {\'e}s}{7}{subsection.1.2.2}} +\newlabel{sec:rendementscum}{{1.2.2}{7}{Rendements cumulés}{subsection.1.2.2}{}} +\newlabel{sec:rendementscum@cref}{{[subsection][2][1,2]1.2.2}{7}} +\@writefile{loe}{\contentsline {hypothese}{\numberline {1.2}Hypoth\IeC {\`e}se}{7}{hypothese.1.2}} +\newlabel{eq:convolutionLR}{{1.2.5}{7}{Rendements cumulés}{equation.1.2.5}{}} +\newlabel{eq:convolutionLR@cref}{{[equation][5][1,2]1.2.5}{7}} +\@writefile{loe}{\contentsline {hypothese}{\numberline {1.3}Hypoth\IeC {\`e}se}{7}{hypothese.1.3}} +\newlabel{eq:rendementsIID}{{1.2.6}{7}{Rendements cumulés}{equation.1.2.6}{}} +\newlabel{eq:rendementsIID@cref}{{[equation][6][1,2]1.2.6}{7}} +\newlabel{eq:convolution2}{{1.2.7}{7}{Rendements cumulés}{equation.1.2.7}{}} +\newlabel{eq:convolution2@cref}{{[equation][7][1,2]1.2.7}{7}} +\citation{bachelier1900theorie} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.3}Donn\IeC {\'e}es disponibles}{8}{subsection.1.2.3}} +\newlabel{sec:donneesdisponibles}{{1.2.3}{8}{Données disponibles}{subsection.1.2.3}{}} +\newlabel{sec:donneesdisponibles@cref}{{[subsection][3][1,2]1.2.3}{8}} +\@writefile{toc}{\contentsline {section}{\numberline {1.3}Les premiers mod\IeC {\`e}les}{8}{section.1.3}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.1}Le mod\IeC {\`e}le de Bachelier}{8}{subsection.1.3.1}} +\newlabel{sec:bachelier}{{1.3.1}{8}{Le modèle de Bachelier}{subsection.1.3.1}{}} +\newlabel{sec:bachelier@cref}{{[subsection][1][1,3]1.3.1}{8}} +\@writefile{lof}{\contentsline {figure}{\numberline {1.1}{\ignorespaces Mod\IeC {\`e}le de Bachelier: probabilit\IeC {\'e} compos\IeC {\'e}e\relax }}{8}{figure.caption.12}} +\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} +\newlabel{fig:bachelier1}{{1.1}{8}{Modèle de Bachelier: probabilité composée\relax }{figure.caption.12}{}} +\newlabel{fig:bachelier1@cref}{{[figure][1][1]1.1}{8}} +\newlabel{eq:probcomposeeB}{{1.3.1}{8}{Le modèle de Bachelier}{equation.1.3.1}{}} +\newlabel{eq:probcomposeeB@cref}{{[equation][1][1,3]1.3.1}{8}} +\citation{mitchell1916critique} +\citation{mitchell1916critique} +\citation{mitchell1916critique} +\newlabel{eq:formeprobB}{{1.3.2}{9}{Le modèle de Bachelier}{equation.1.3.2}{}} +\newlabel{eq:formeprobB@cref}{{[equation][2][1,3]1.3.2}{9}} +\newlabel{eq:bachelier00}{{1.3.5}{9}{Le modèle de Bachelier}{equation.1.3.5}{}} +\newlabel{eq:bachelier00@cref}{{[equation][5][1,3]1.3.5}{9}} +\newlabel{eq:browniengeom}{{1.3.6}{9}{Le modèle de Bachelier}{equation.1.3.6}{}} +\newlabel{eq:browniengeom@cref}{{[equation][6][1,3]1.3.6}{9}} +\newlabel{eq:bachelierwiener2}{{1.3.7}{9}{Le modèle de Bachelier}{equation.1.3.7}{}} +\newlabel{eq:bachelierwiener2@cref}{{[equation][7][1,3]1.3.7}{9}} +\@writefile{lof}{\contentsline {figure}{\numberline {1.2}{\ignorespaces Distribution des rendements annuels de 40 titres boursiers, de 1890 \IeC {\`a} 1915, Table XVIII de \cite {mitchell1916critique}\relax }}{10}{figure.caption.13}} +\newlabel{fig:mitchell1}{{1.2}{10}{Distribution des rendements annuels de 40 titres boursiers, de 1890 à 1915, Table XVIII de \cite {mitchell1916critique}\relax }{figure.caption.13}{}} +\newlabel{fig:mitchell1@cref}{{[figure][2][1]1.2}{10}} +\citation{mandelbrot1963variation} +\citation{walterlevy} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.2}Proposition de Mandelbrot}{11}{subsection.1.3.2}} +\newlabel{sec:mandelbrot}{{1.3.2}{11}{Proposition de Mandelbrot}{subsection.1.3.2}{}} +\newlabel{sec:mandelbrot@cref}{{[subsection][2][1,3]1.3.2}{11}} +\newlabel{enum:mandelbrot}{{1.3.2}{11}{Proposition de Mandelbrot}{subsection.1.3.2}{}} +\newlabel{enum:mandelbrot@cref}{{[subsection][2][1,3]1.3.2}{11}} +\citation{press1967compound} +\@writefile{lot}{\contentsline {table}{\numberline {1.1}{\ignorespaces Domaine et r\IeC {\^o}le des param\IeC {\`e}tres de la distribution L stable de Mandelbrot\relax }}{12}{table.caption.14}} +\newlabel{tab:roleparam}{{1.1}{12}{Domaine et rôle des paramètres de la distribution L stable de Mandelbrot\relax }{table.caption.14}{}} +\newlabel{tab:roleparam@cref}{{[table][1][1]1.1}{12}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.3}Le mod\IeC {\`e}le de Press}{12}{subsection.1.3.3}} +\newlabel{sec:press}{{1.3.3}{12}{Le modèle de Press}{subsection.1.3.3}{}} +\newlabel{sec:press@cref}{{[subsection][3][1,3]1.3.3}{12}} +\newlabel{eq:press67}{{1.3.11}{13}{Le modèle de Press}{equation.1.3.11}{}} +\newlabel{eq:press67@cref}{{[equation][11][1,3]1.3.11}{13}} +\newlabel{eq:fncaractpress}{{1.3.12}{13}{Le modèle de Press}{equation.1.3.12}{}} +\newlabel{eq:fncaractpress@cref}{{[equation][12][1,3]1.3.12}{13}} +\newlabel{eq:fncaractpress2}{{1.3.13}{13}{Le modèle de Press}{equation.1.3.13}{}} +\newlabel{eq:fncaractpress2@cref}{{[equation][13][1,3]1.3.13}{13}} +\citation{merton1976option} +\newlabel{eq:cumulantspress}{{1.3.14}{14}{Le modèle de Press}{equation.1.3.0.14.14}{}} +\newlabel{eq:cumulantspress@cref}{{[equation][14][1,3]1.3.14}{14}} +\newlabel{eq:presscum}{{1.3.15}{14}{Le modèle de Press}{equation.1.3.0.15.15}{}} +\newlabel{eq:presscum@cref}{{[equation][15][1,3]1.3.15}{14}} +\newlabel{eq:presscumtheta}{{1.3.15a}{14}{Le modèle de Press}{equation.1.3.0.15a}{}} +\newlabel{eq:presscumtheta@cref}{{[equation][1][1,3,15]1.3.15a}{14}} +\newlabel{eq:presscumlambda}{{1.3.15b}{14}{Le modèle de Press}{equation.1.3.0.15b}{}} +\newlabel{eq:presscumlambda@cref}{{[equation][2][1,3,15]1.3.15b}{14}} +\newlabel{eq:presscumsigma2}{{1.3.15c}{14}{Le modèle de Press}{equation.1.3.0.15c}{}} +\newlabel{eq:presscumsigma2@cref}{{[equation][3][1,3,15]1.3.15c}{14}} +\newlabel{eq:presscumsigma1}{{1.3.15d}{14}{Le modèle de Press}{equation.1.3.0.15d}{}} +\newlabel{eq:presscumsigma1@cref}{{[equation][4][1,3,15]1.3.15d}{14}} +\newlabel{eq:modelemerton}{{1.3.16}{14}{Le modèle de Press}{equation.1.3.0.16}{}} +\newlabel{eq:modelemerton@cref}{{[equation][16][1,3]1.3.16}{14}} +\citation{praetz1972distribution} +\newlabel{eq:fncaractmerton}{{1.3.18}{15}{Le modèle de Press}{equation.1.3.0.18}{}} +\newlabel{eq:fncaractmerton@cref}{{[equation][18][1,3]1.3.18}{15}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.4}Le mod\IeC {\`e}le de Praetz}{15}{subsection.1.3.4}} +\newlabel{eq:praetz72}{{1.3.20}{15}{Le modèle de Praetz}{equation.1.3.0.20}{}} +\newlabel{eq:praetz72@cref}{{[equation][20][1,3]1.3.20}{15}} +\newlabel{eq:gpraetz}{{1.3.21}{15}{Le modèle de Praetz}{equation.1.3.0.21}{}} +\newlabel{eq:gpraetz@cref}{{[equation][21][1,3]1.3.21}{15}} +\citation{madan1990variance} +\citation{kotz2001laplace} +\newlabel{eq:hpraetz}{{1.3.22}{16}{Le modèle de Praetz}{equation.1.3.0.22}{}} +\newlabel{eq:hpraetz@cref}{{[equation][22][1,3]1.3.22}{16}} +\@writefile{toc}{\contentsline {section}{\numberline {1.4}Conditions essentielles de Madan et Seneta}{16}{section.1.4}} +\newlabel{sec:madanseneta90}{{1.4}{16}{Conditions essentielles de Madan et Seneta}{section.1.4}{}} +\newlabel{sec:madanseneta90@cref}{{[section][4][1]1.4}{16}} +\newlabel{eq:condmadan1}{{1.4.1}{16}{Conditions essentielles de Madan et Seneta}{equation.1.4.0.1}{}} +\newlabel{eq:condmadan1@cref}{{[enumi][1][]1}{16}} +\newlabel{eq:condmadan2}{{1.4.2}{16}{Conditions essentielles de Madan et Seneta}{equation.1.4.0.2}{}} +\newlabel{eq:condmadan2@cref}{{[equation][2][1,4]1.4.2}{16}} +\@writefile{lot}{\contentsline {table}{\numberline {1.2}{\ignorespaces Respect des conditions \IeC {\'e}mises par Madan et Seneta pour les diff\IeC {\'e}rents mod\IeC {\`e}les pr\IeC {\'e}sent\IeC {\'e}s\relax }}{17}{table.caption.15}} +\newlabel{tab:condmadan}{{1.2}{17}{Respect des conditions émises par Madan et Seneta pour les différents modèles présentés\relax }{table.caption.15}{}} +\newlabel{tab:condmadan@cref}{{[table][2][1]1.2}{17}} +\@setckpt{chapitre1}{ +\setcounter{page}{18} +\setcounter{equation}{2} +\setcounter{enumi}{4} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{1} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{1} +\setcounter{section}{4} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{36} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{2} +\setcounter{lofdepth}{1} +\setcounter{table}{2} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{13} +\setcounter{Hfootnote}{1} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{21} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{3} +} diff --git a/memoire/chapitre1.log b/memoire/chapitre1.log new file mode 100644 index 0000000..1d68d49 --- /dev/null +++ b/memoire/chapitre1.log @@ -0,0 +1,29893 @@ +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.7.6) 24 JUL 2013 21:59 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**chapitre1.tex +(./chapitre1.tex +LaTeX2e <2011/06/27> +Babel and hyphenation patterns for english, dumylang, nohyphenation, et +hiopic, farsi, arabic, pinyin, croatian, bulgarian, ukrainian, russian, slovak, + czech, danish, dutch, usenglishmax, ukenglish, finnish, french, basque, ngerma +n, german, swissgerman, ngerman-x-2012-05-30, german-x-2012-05-30, monogreek, g +reek, ibycus, ancientgreek, hungarian, bengali, tamil, hindi, telugu, gujarati, + sanskrit, malayalam, kannada, assamese, marathi, oriya, panjabi, italian, lati +n, latvian, lithuanian, mongolian, mongolianlmc, nynorsk, bokmal, indonesian, e +speranto, coptic, welsh, irish, interlingua, serbian, serbianc, slovenian, friu +lan, romansh, estonian, romanian, armenian, uppersorbian, turkish, afrikaans, i +celandic, kurmanji, polish, portuguese, galician, catalan, spanish, swedish, th +ai, loaded. +! Undefined control sequence. +l.1 \chapter + {Les modèles de rendements financiers} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1 \chapter{L + es modèles de rendements financiers} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 1--2 +[] + [] + +! Undefined control sequence. +l.3 \section + {L'utilisation de modèles en finance} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3 \section{L + 'utilisation de modèles en finance} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 3--5 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.6 I + l est important de considérer les implications de l'utilisation de +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ® in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no  in font nullfont! +Missing character: There is no « in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no R in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no  in font nullfont! +Missing character: There is no » in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{derman1996modelrisk} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.10 publiée par \cite{derman1996modelrisk} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `derman1996modelrisk' on page undefined on input line +10. + +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 6--11 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 6--11 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.12 D + urant les dernières décennies, plusieurs modèles sont apparus afin de +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no D in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 12--21 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.22 P + ar la suite, la confiance développée envers le modèle de tarification +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{black1973pricing} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.23 d'options de \cite{black1973pricing} + et ses extensions a favorisé la +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `black1973pricing' on page undefined on input line 23. + + +Missing character: There is no ] in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 22--31 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 22--31 +[] + [] + +! Undefined control sequence. +l.32 \subsection + {Différents types de modèles} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.32 \subsection{D + ifférents types de modèles} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 32--34 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.35 T + oujours selon Derman, un modèle financier peut être classé parmi au +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no T in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 35--37 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.38 \begin{enumerate} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 38--38 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no ¨ in font cmbx10! +Missing character: There is no , in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (19.16656pt too wide) in paragraph at lines 39--42 +\OT1/cmr/bx/n/10 mo- + [] + + +Overfull \hbox (14.8541pt too wide) in paragraph at lines 39--42 +\OT1/cmr/bx/n/10 dle + [] + + +Overfull \hbox (19.48601pt too wide) in paragraph at lines 39--42 +\OT1/cmr/bx/n/10 fon- + [] + + +Overfull \hbox (15.8124pt too wide) in paragraph at lines 39--42 +\OT1/cmr/bx/n/10 da- + [] + + +Overfull \hbox (25.07626pt too wide) in paragraph at lines 39--42 +\OT1/cmr/bx/n/10 men- + [] + + +Overfull \hbox (13.25685pt too wide) in paragraph at lines 39--42 +\OT1/cmr/bx/n/10 tal + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no ¨ in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (19.16656pt too wide) in paragraph at lines 42--49 +\OT1/cmr/bx/n/10 mo- + [] + + +Overfull \hbox (14.8541pt too wide) in paragraph at lines 42--49 +\OT1/cmr/bx/n/10 dle + [] + + +Overfull \hbox (16.61102pt too wide) in paragraph at lines 42--49 +\OT1/cmr/bx/n/10 ph- + [] + + +Overfull \hbox (25.55542pt too wide) in paragraph at lines 42--49 +\OT1/cmr/bx/n/10 nom- + [] + + +Overfull \hbox (47.7567pt too wide) in paragraph at lines 42--49 +\OT1/cmr/bx/n/10 nologique + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no ¨ in font cmbx10! +Missing character: There is no , in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ´ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (19.16656pt too wide) in paragraph at lines 49--55 +\OT1/cmr/bx/n/10 mo- + [] + + +Overfull \hbox (14.8541pt too wide) in paragraph at lines 49--55 +\OT1/cmr/bx/n/10 dle + [] + + +Overfull \hbox (30.6345pt too wide) in paragraph at lines 49--55 +\OT1/cmr/bx/n/10 statis- + [] + + +Overfull \hbox (25.3957pt too wide) in paragraph at lines 49--55 +\OT1/cmr/bx/n/10 tique + [] + +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 57--67 +[] + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 68--75 +[] + [] + +! Undefined control sequence. +l.76 \subsection + {Le risque de modélisation} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 76--78 +[] + [] + +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 79--80 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 81--84 +[] + [] + +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 85--94 +[] + [] + +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ® in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 95--105 +[] + [] + +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ® in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 106--113 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 114--118 +[] + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 119--127 +[] + [] + +! Undefined control sequence. +l.128 \section + {Les rendements financiers} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 128--130 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 131--140 +[] + [] + + +Overfull \hbox (20.22908pt too wide) in paragraph at lines 131--140 +\OT1/cmr/bx/n/10 ren- + [] + + +Overfull \hbox (15.49298pt too wide) in paragraph at lines 131--140 +\OT1/cmr/bx/n/10 de- + [] + + +Overfull \hbox (25.3957pt too wide) in paragraph at lines 131--140 +\OT1/cmr/bx/n/10 ment + [] + +! Undefined control sequence. +l.141 \subsection + {Définitions et notation} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no D in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 141--143 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 144. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 144. +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 144--152 +[] + [] + + +Overfull \hbox (28.65274pt too wide) in paragraph at lines 144--152 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OML/cmm +/m/it/10 > + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 144--152 +\OT1/cmr/m/n/10 0$ + [] + + +Overfull \hbox (3.61111pt too wide) in paragraph at lines 144--152 +\OML/cmm/m/it/10 t$ + [] + + +Overfull \hbox (19.05907pt too wide) in paragraph at lines 144--152 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (34.44437pt too wide) in paragraph at lines 144--152 +[]$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align* undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.157 \begin{align*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no d in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +! Missing $ inserted. + + $ +l.158 \frac{dS(t)}{dt} + &= R(t) \cdot S(t) +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Extra }, or forgotten $. +\frac #1#2->{\begingroup #1\endgroup \over #2} + +l.158 \frac{dS(t)}{dt} + &= R(t) \cdot S(t) +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Misplaced alignment tab character &. +l.158 \frac{dS(t)}{dt} & + = R(t) \cdot S(t) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.159 \end{align*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.159 \end{align*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.159 \end{align*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 153--160 +[] + [] + + +Overfull \hbox (47.99254pt too wide) in paragraph at lines 153--160 +[]$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.165 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +! Misplaced alignment tab character &. +l.167 S(t) & + = S(0)e^{R(t) \cdot t} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no e in font nullfont! +! Missing $ inserted. + + $ +l.167 S(t) &= S(0)e^ + {R(t) \cdot t} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.168 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.168 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 161--169 +[] + [] + + +Overfull \hbox (23.3021pt too wide) in paragraph at lines 161--169 +\OML/cmm/m/it/10 dS\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (8.81598pt too wide) in paragraph at lines 161--169 +\OML/cmm/m/it/10 dt$ + [] + + +Overfull \hbox (18.09723pt too wide) in paragraph at lines 161--169 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (21.2268pt too wide) in paragraph at lines 161--169 +[]$ + [] + +Missing character: There is no A in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment hypothese undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.172 \begin{hypothese} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{hypothese}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.175 \end{hypothese} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 170--176 +[] + [] + + +Overfull \hbox (19.05907pt too wide) in paragraph at lines 170--176 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (34.44437pt too wide) in paragraph at lines 170--176 +[]$ + [] + + +Overfull \hbox (30.69096pt too wide) in paragraph at lines 170--176 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 ) \OMS/cms +y/m/n/10 6\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (38.74649pt too wide) in paragraph at lines 170--176 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )\OML/cmm/ +m/it/10 ; s \OMS/cmsy/m/n/10 6\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (3.61111pt too wide) in paragraph at lines 170--176 +\OML/cmm/m/it/10 t$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.180 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no R in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +! Misplaced alignment tab character &. +l.181 R(t) & + = \log{(S(t))} - \log{(S(t-1))} \nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +! Missing $ inserted. + + $ +l.181 R(t) &= \log + {(S(t))} - \log{(S(t-1))} \nonumber\\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. + & + +l.182 & + = +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.184 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.184 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 177--185 +[] + [] + + +Overfull \hbox (19.05907pt too wide) in paragraph at lines 177--185 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (3.61111pt too wide) in paragraph at lines 177--185 +\OML/cmm/m/it/10 t$ + [] + + +Overfull \hbox (11.38892pt too wide) in paragraph at lines 177--185 +\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 177--185 +\OT1/cmr/m/n/10 1$ + [] + + +Overfull \hbox (48.23615pt too wide) in paragraph at lines 177--185 +[] [] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (57.6805pt too wide) in paragraph at lines 177--185 +[] [][] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 177--185 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (53.72786pt too wide) in paragraph at lines 177--185 +[] [][]$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.189 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no L in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +! Misplaced alignment tab character &. +l.190 L(t) & + = \sum_{i=1}^{t} R(i) \nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +! Missing $ inserted. + + $ +l.190 L(t) &= \sum + _{i=1}^{t} R(i) \nonumber\\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. + & + +l.191 & + = \sum_{i=1}^{t} \left[\log{(S(i)} - \log{(S(i-1))}\right] \nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.192 & + = \log{(S(t))} - \log{(S(0))} \nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.193 & + = \log{\left(\frac{S(t)}{S(0)}\right)} \label{eq:rendementcumL} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.194 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.194 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 186--195 +[] + [] + + +Overfull \hbox (20.22908pt too wide) in paragraph at lines 186--195 +\OT1/cmr/bx/n/10 ren- + [] + + +Overfull \hbox (15.49298pt too wide) in paragraph at lines 186--195 +\OT1/cmr/bx/n/10 de- + [] + + +Overfull \hbox (25.3957pt too wide) in paragraph at lines 186--195 +\OT1/cmr/bx/n/10 ment + [] + + +Overfull \hbox (15.33325pt too wide) in paragraph at lines 186--195 +\OT1/cmr/bx/n/10 cu- + [] + + +Overfull \hbox (18.84712pt too wide) in paragraph at lines 186--195 +\OT1/cmr/bx/n/10 mul + [] + + +Overfull \hbox (18.19449pt too wide) in paragraph at lines 186--195 +\OML/cmm/m/it/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (18.61111pt too wide) in paragraph at lines 186--195 +[]$ + [] + + +Overfull \hbox (44.56963pt too wide) in paragraph at lines 186--195 +[][] \OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 i\OT1/cmr/m/n/10 )[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 186--195 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (139.03885pt too wide) in paragraph at lines 186--195 +[][] [] [] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 186--195 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (48.23615pt too wide) in paragraph at lines 186--195 +[] [] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (41.84724pt too wide) in paragraph at lines 186--195 +[] [][] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 186--195 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (44.457pt too wide) in paragraph at lines 186--195 +[] [][]$ + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.198 \eqref + {eq:solutiondiffrendement}, mais tenant compte de l'hypothèse +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.200 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no e in font nullfont! +! Missing $ inserted. + + $ +l.201 e^ + {L(t)} &= \frac{S(t)}{S(0)} \nonumber\\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.201 e^{L(t)} & + = \frac{S(t)}{S(0)} \nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.202 S(t) & + = S(0) \cdot e^{L(t)} \nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.203 & + = S(0) \cdot \exp\left(\sum_{i=1}^{t} R(i)\right) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.204 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.204 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 196--205 +[] + [] + + +Overfull \hbox (18.09723pt too wide) in paragraph at lines 196--205 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (19.48613pt too wide) in paragraph at lines 196--205 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (0)$ + [] + + +Overfull \hbox (25.81253pt too wide) in paragraph at lines 196--205 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (17.92923pt too wide) in paragraph at lines 196--205 +[][] + [] + + +Overfull \hbox (28.65274pt too wide) in paragraph at lines 196--205 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (22.26392pt too wide) in paragraph at lines 196--205 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (0) \OMS/cmsy/m/n/10 ^^A + [] + + +Overfull \hbox (19.91328pt too wide) in paragraph at lines 196--205 +\OML/cmm/m/it/10 e[][] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 196--205 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (22.26392pt too wide) in paragraph at lines 196--205 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (0) \OMS/cmsy/m/n/10 ^^A + [] + + +Overfull \hbox (73.45853pt too wide) in paragraph at lines 196--205 +[] []$ + [] + +! Undefined control sequence. +l.206 \subsection + {Rendements cumulés} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no R in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 206--208 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment hypothese undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.210 \begin{hypothese} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{hypothese}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.213 \end{hypothese} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 209--214 +[] + [] + + +Overfull \hbox (36.22702pt too wide) in paragraph at lines 209--214 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 i\OT1/cmr/m/n/10 )\OML/cmm/ +m/it/10 ; i \OMS/cmsy/m/n/10 2 + [] + + +Overfull \hbox (30.83319pt too wide) in paragraph at lines 209--214 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; [] ; t$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.216 utilisant le produit de convolution \eqref + {eq:convocaract}. Soit +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 215--222 +[] + [] + + +Overfull \hbox (18.19449pt too wide) in paragraph at lines 215--222 +\OML/cmm/m/it/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (34.21063pt too wide) in paragraph at lines 215--222 +\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (18.89308pt too wide) in paragraph at lines 215--222 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 i\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (33.82825pt too wide) in paragraph at lines 215--222 +\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (18.19449pt too wide) in paragraph at lines 215--222 +\OML/cmm/m/it/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (18.61111pt too wide) in paragraph at lines 215--222 +[]$ + [] + + +Overfull \hbox (95.99306pt too wide) detected at line 225 +[]\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) += [] \OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 + ) + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ´ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment hypothese undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.228 \begin{hypothese} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{hypothese}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.231 \end{hypothese} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 227--232 +[] + [] + + +Overfull \hbox (36.22702pt too wide) in paragraph at lines 227--232 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 i\OT1/cmr/m/n/10 )\OML/cmm/ +m/it/10 ; i \OMS/cmsy/m/n/10 2 + [] + + +Overfull \hbox (30.83319pt too wide) in paragraph at lines 227--232 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; [] ; t$ + [] + +Missing character: There is no A in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.235 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.237 \phi + _{R}(\xi) = \phi_{R(1)}(\xi) = \ldots &= \phi_{R(t)}(\xi) +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.237 ...hi_{R}(\xi) = \phi_{R(1)}(\xi) = \ldots & + = \phi_{R(t)}(\xi) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.238 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.238 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 233--239 +[] + [] + + +Overfull \hbox (35.68683pt too wide) in paragraph at lines 233--239 +\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (45.92299pt too wide) in paragraph at lines 233--239 +\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (22.22212pt too wide) in paragraph at lines 233--239 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (34.40219pt too wide) in paragraph at lines 233--239 +\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.240 On peut donc simplifier l'expression \eqref + {eq:convolutionLR} pour +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no R in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 240--242 +[] + [] + + +Overfull \hbox (81.3692pt too wide) detected at line 245 +[]\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) += [][] + [] + +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 247--251 +[] + [] + + +Overfull \hbox (18.89308pt too wide) in paragraph at lines 247--251 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 i\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (18.19449pt too wide) in paragraph at lines 247--251 +\OML/cmm/m/it/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (3.61111pt too wide) in paragraph at lines 247--251 +\OML/cmm/m/it/10 t$ + [] + +! Undefined control sequence. +l.252 \subsection + {Données disponibles} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no D in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 252--254 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 255--262 +[] + [] + +! Undefined control sequence. +l.263 \section + {Les premiers modèles} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 263--264 +[] + [] + +! Undefined control sequence. +l.265 \subsection + {Le modèle de Bachelier} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 265--267 +[] + [] + +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{bachelier1900theorie} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.269 ...©té celui de \cite{bachelier1900theorie} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `bachelier1900theorie' on page undefined on input line + 269. + +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 268--270 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 268--270 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +! Undefined control sequence. + \mathbb + +l.274 l'ensemble du domaine réel $\mathbb + {R}$. La probabilité que le titre +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 271--279 +[] + [] + + +Overfull \hbox (45.87503pt too wide) in paragraph at lines 271--279 +[]$ + [] + + +Overfull \hbox (7.67015pt too wide) in paragraph at lines 271--279 +\OML/cmm/m/it/10 R$ + [] + + +Overfull \hbox (18.09723pt too wide) in paragraph at lines 271--279 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...ence `fig:bachelier1' on page \thepage + \space undefined\on@line . +l.287 \ref{fig:bachelier1} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `fig:bachelier1' on page undefined on input line 287. + + +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 280--288 +[] + [] + + +Overfull \hbox (5.71527pt too wide) in paragraph at lines 280--288 +\OML/cmm/m/it/10 x$ + [] + + +Overfull \hbox (12.8681pt too wide) in paragraph at lines 280--288 +\OML/cmm/m/it/10 z \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (5.71527pt too wide) in paragraph at lines 280--288 +\OML/cmm/m/it/10 x$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 280--288 +[] + [] + + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.290 \begin{figure} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! + +! LaTeX Error: Environment tikzpicture undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.292 \begin{tikzpicture} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Undefined control sequence. +l.293 \draw + [->] (0,0) -- (6,0); \foreach \x in {0,2,4} \draw (\x +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no > in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.293 \draw [->] (0,0) -- (6,0); \foreach + \x in {0,2,4} \draw (\x +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.293 \draw [->] (0,0) -- (6,0); \foreach \x + in {0,2,4} \draw (\x +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 4 in font nullfont! +! Undefined control sequence. +l.293 ...0) -- (6,0); \foreach \x in {0,2,4} \draw + (\x +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no ( in font nullfont! +! Undefined control sequence. +l.293 ...- (6,0); \foreach \x in {0,2,4} \draw (\x + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no c in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no ( in font nullfont! +! Undefined control sequence. +l.294 cm,3pt) -- (\x + cm,-3pt); \draw (0,0) node[below=3pt] {$ 0 $} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no c in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.294 cm,3pt) -- (\x cm,-3pt); \draw + (0,0) node[below=3pt] {$ 0 $} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.295 node[above=3pt] {$ S(0) $}; \draw + (2,0) node[below=3pt] {$ t_1 $} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no ( in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.296 node[above=3pt] {$ S(0)+x $}; \draw + (4,0) node[below=3pt] {$ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no ( in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ; in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{tikzpicture}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.298 \end{tikzpicture} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.299 \caption + {Modèle de Bachelier: probabilité composée} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.301 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 290--302 +[] + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 290--302 +\OT1/cmr/m/n/10 0$ + [] + + +Overfull \hbox (19.48613pt too wide) in paragraph at lines 290--302 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (0)$ + [] + + +Overfull \hbox (8.09724pt too wide) in paragraph at lines 290--302 +\OML/cmm/m/it/10 t[]$ + [] + + +Overfull \hbox (27.26393pt too wide) in paragraph at lines 290--302 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (0) + + [] + + +Overfull \hbox (5.71527pt too wide) in paragraph at lines 290--302 +\OML/cmm/m/it/10 x$ + [] + + +Overfull \hbox (15.87505pt too wide) in paragraph at lines 290--302 +\OML/cmm/m/it/10 t[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (8.09724pt too wide) in paragraph at lines 290--302 +\OML/cmm/m/it/10 t[]$ + [] + + +Overfull \hbox (27.26393pt too wide) in paragraph at lines 290--302 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (0) + + [] + + +Overfull \hbox (5.0903pt too wide) in paragraph at lines 290--302 +\OML/cmm/m/it/10 z$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.306 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no f in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +! Missing $ inserted. + + $ +l.308 f(z,t_ + 1+t_2) = \int_{-\infty}^{\infty} f(x,t_1)\cdot f(z-x,t_2) +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.310 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.310 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 303--311 +[] + [] + + +Overfull \hbox (27.52086pt too wide) in paragraph at lines 303--311 +\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x; t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (18.09723pt too wide) in paragraph at lines 303--311 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (19.48613pt too wide) in paragraph at lines 303--311 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (0)$ + [] + + +Overfull \hbox (12.26393pt too wide) in paragraph at lines 303--311 +[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (22.54166pt too wide) in paragraph at lines 303--311 +\OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (55.89589pt too wide) in paragraph at lines 303--311 +[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x; t[]\OT1/cmr/m/n/10 +) \OMS/cmsy/m/n/10 ^^A + [] + + +Overfull \hbox (22.72926pt too wide) in paragraph at lines 303--311 +\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 z \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (24.92361pt too wide) in paragraph at lines 303--311 +\OML/cmm/m/it/10 x; t[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^A + [] + + +Overfull \hbox (10.92014pt too wide) in paragraph at lines 303--311 +\OML/cmm/m/it/10 dx$ + [] + + +! LaTeX Error: The font size command \normalsize is not defined: + there is probably something wrong with the class file. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.311 + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +[1 +! Undefined control sequence. + ...el{sec:utilisationmodeles}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:utilisationmodeles}{{}{}} +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} +! Undefined control sequence. + ...bel{sec:differentsmodeles}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:differentsmodeles}{{}{}} +! Undefined control sequence. + \newlabel{sec:modelerisque}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:modelerisque}{{}{}} +! Undefined control sequence. + \newlabel{sec:prixrendements}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:prixrendements}{{}{}} +! Undefined control sequence. + \newlabel{sec:defrendements}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:defrendements}{{}{}} +! Undefined control sequence. + ...{eq:solutiondiffrendement}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:solutiondiffrendement}{{}{}} +! Undefined control sequence. + ...label{eq:rendementlogprix}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:rendementlogprix}{{}{}} +! Undefined control sequence. + \newlabel{eq:rendementcumL}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:rendementcumL}{{}{}} +! Undefined control sequence. + \newlabel{sec:rendementscum}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:rendementscum}{{}{}} +! Undefined control sequence. + \newlabel{eq:convolutionLR}{{1}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:convolutionLR}{{1}{}} +! Undefined control sequence. + \newlabel{eq:rendementsIID}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:rendementsIID}{{}{}} +! Undefined control sequence. + \newlabel{eq:convolution2}{{2}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:convolution2}{{2}{}} +! Undefined control sequence. + ...el{sec:donneesdisponibles}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:donneesdisponibles}{{}{}} +! Undefined control sequence. + \newlabel{sec:bachelier}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:bachelier}{{}{}} +! Undefined control sequence. + \newlabel{fig:bachelier1}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{fig:bachelier1}{{}{}} +! Undefined control sequence. + \newlabel{eq:probcomposeeB}{{}{\thepage + }} +l.311 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:probcomposeeB}{{}{}} +] +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! + + +! LaTeX Error: Environment align* undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.314 \begin{align*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no f in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no A in font nullfont! +! Missing $ inserted. + + $ +l.316 f(x,t) = A \cdot + \exp \left\{ B^2x^2 \right\} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.317 \end{align*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.317 \end{align*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 312--318 +[] + [] + + +Overfull \hbox (55.82991pt too wide) in paragraph at lines 312--318 +\OMS/cmsy/m/n/10 ^^A [] []$ + [] + +Missing character: There is no A in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.321 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.322 \int + _{-\infty}^{\infty} A \cdot \exp \left\{ B^2x^2 \right\} dx = 1 +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.323 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.323 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 319--324 +[] + [] + + +Overfull \hbox (27.52086pt too wide) in paragraph at lines 319--324 +\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x; t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (31.38892pt too wide) in paragraph at lines 319--324 +[][] \OML/cmm/m/it/10 A \OMS/cmsy/m/n/10 ^^A + [] + + +Overfull \hbox (74.52777pt too wide) in paragraph at lines 319--324 +[] [] \OML/cmm/m/it/10 dx \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 319--324 +\OT1/cmr/m/n/10 1$ + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align* undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.326 \begin{align*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.327 \Leftrightarrow + B=A\sqrt{\pi} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.328 \end{align*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.328 \end{align*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 325--329 +[] + [] + + +Overfull \hbox (10.00002pt too wide) in paragraph at lines 325--329 +\OMS/cmsy/m/n/10 , + [] + + +Overfull \hbox (18.64232pt too wide) in paragraph at lines 325--329 +\OML/cmm/m/it/10 B \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (21.89243pt too wide) in paragraph at lines 325--329 +\OML/cmm/m/it/10 A[]$ + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.331 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no f in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +! Missing $ inserted. + + $ +l.332 f(x,t) = f(0,t) \cdot + \exp \left\{\pi \cdot f(0,t)^2 \cdot x^2 +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.334 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.334 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 330--335 +[] + [] + + +Overfull \hbox (16.27078pt too wide) in paragraph at lines 330--335 +\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 330--335 +\OT1/cmr/m/n/10 0$ + [] + + +Overfull \hbox (18.05553pt too wide) in paragraph at lines 330--335 +\OML/cmm/m/it/10 A \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (26.8056pt too wide) in paragraph at lines 330--335 +\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (95.052pt too wide) in paragraph at lines 330--335 +\OMS/cmsy/m/n/10 ^^A [] []$ + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.336 En reprenant l'intégrale \eqref + {eq:probcomposeeB}, on obtient que la +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.338 \eqref + {eq:formeprobB} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no B in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.339 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no f in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +! Missing $ inserted. + + $ +l.340 f(z,t_ + 1+t_2) = \frac{f(x,1)f(z-x,2)}{\sqrt{f(x,1)^2+f(z-x,2)^2}} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.343 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.343 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 336--344 +[] + [] + + +Overfull \hbox (35.27092pt too wide) in paragraph at lines 336--344 +\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 z; t[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (11.98615pt too wide) in paragraph at lines 336--344 +\OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (12.26393pt too wide) in paragraph at lines 336--344 +[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (22.54166pt too wide) in paragraph at lines 336--344 +\OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (204.41551pt too wide) in paragraph at lines 336--344 +[] [] []$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 345--350 +[] + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 351--354 +[] + [] + + +Overfull \hbox (127.52408pt too wide) detected at line 357 +[]\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 T\OT1/cmr/m/n/10 ) \OMS/c +msy/m/n/10 ^^@ \OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/ +n/10 ) \OMS/cmsy/m/n/10 ^^X \OML/cmm/m/it/10 N [] + [] + +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.365 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +! Misplaced alignment tab character &. +l.367 S(T)-S(t) & + \sim LN \left(0, \sigma (T-t) \right) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Missing $ inserted. + + $ +l.367 S(T)-S(t) &\sim + LN \left(0, \sigma (T-t) \right) +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.368 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.368 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 359--369 +[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 359--369 +\OMS/cmsy/m/n/10 ^^X + [] + + +Overfull \hbox (71.73601pt too wide) in paragraph at lines 359--369 +\OML/cmm/m/it/10 LN []$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.372 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.374 \log + \left(S(T)\right)-\log\left(S(t)\right) &\sim N\left(0,\sigma +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.374 ...\left(S(T)\right)-\log\left(S(t)\right) & + \sim N\left(0,\sigma +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.376 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.376 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 370--377 +[] + [] + + +Overfull \hbox (51.85768pt too wide) in paragraph at lines 370--377 +[] [] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (51.01385pt too wide) in paragraph at lines 370--377 +[] [] \OMS/cmsy/m/n/10 ^^X + [] + + +Overfull \hbox (66.59706pt too wide) in paragraph at lines 370--377 +\OML/cmm/m/it/10 N []$ + [] + +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ...ence `sec:mandelbrot' on page \thepage + \space undefined\on@line . +l.383 ...bordé à la section \ref{sec:mandelbrot} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:mandelbrot' on page undefined on input line 383. + + +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 378--384 +[] + [] + + +Overfull \hbox (18.19449pt too wide) in paragraph at lines 378--384 +\OML/cmm/m/it/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 378--384 +[] + [] + +Missing character: There is no Q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{mitchell1916critique} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.387 référant à \cite{mitchell1916critique} + , on observe que, sur une base +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `mitchell1916critique' on page undefined on input line + 387. + +Missing character: There is no ] in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...rence `fig:mitchell1' on page \thepage + \space undefined\on@line . +l.391 1915 (figure \ref{fig:mitchell1} + ). Il y a donc présence d'une +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `fig:mitchell1' on page undefined on input line 391. + +Missing character: There is no ) in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.393 \begin{figure} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.395 \includegraphics + [scale=0.75]{../varia/mitchell1.pdf} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no f in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.396 \caption + {Distribution des rendements annuels de 40 titres boursiers, +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{mitchell1916critique} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.397 ...able XVIII de \cite{mitchell1916critique} + } +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `mitchell1916critique' on page undefined on input line + 397. + +Missing character: There is no ] in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.399 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 385--400 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 385--400 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 385--400 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 385--400 +[] + [] + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 401--406 +[] + [] + +! Undefined control sequence. +l.407 \subsection + {Proposition de Mandelbrot} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 407--409 +[] + [] + +Missing character: There is no [ in font nullfont! +\citation{mandelbrot1963variation} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.411 \cite{mandelbrot1963variation} + propose un modèle qui vise à combler +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `mandelbrot1963variation' on page undefined on input l +ine 411. + +Missing character: There is no ] in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.412 ... du modèle brownien géométrique \eqref + {eq:browniengeom}. Il +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (25.43053pt too wide) in paragraph at lines 411--416 +[][] + [] + + +Overfull \hbox (35.77754pt too wide) in paragraph at lines 411--416 +\OT1/cmr/m/it/10 pointues + [] + +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 417--419 +[] + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 420--420 +[] + [] + +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.440 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +! Missing $ inserted. + + $ +l.441 s' \cdot + R(t) + s'' \cdot R(t) = s \cdot R(t) +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.442 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.442 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 435--443 +[] + [] + + +Overfull \hbox (19.05907pt too wide) in paragraph at lines 435--443 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (4.6875pt too wide) in paragraph at lines 435--443 +\OML/cmm/m/it/10 s$ + [] + + +Overfull \hbox (7.49306pt too wide) in paragraph at lines 435--443 +\OML/cmm/m/it/10 s[]$ + [] + + +Overfull \hbox (9.79861pt too wide) in paragraph at lines 435--443 +\OML/cmm/m/it/10 s[]$ + [] + + +Overfull \hbox (29.61465pt too wide) in paragraph at lines 435--443 +\OMS/cmsy/m/n/10 ^^A\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/c +mr/m/n/10 ) + + [] + + +Overfull \hbox (12.5764pt too wide) in paragraph at lines 435--443 +\OML/cmm/m/it/10 s[] \OMS/cmsy/m/n/10 ^^A + [] + + +Overfull \hbox (29.61458pt too wide) in paragraph at lines 435--443 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (7.46529pt too wide) in paragraph at lines 435--443 +\OML/cmm/m/it/10 s \OMS/cmsy/m/n/10 ^^A + [] + + +Overfull \hbox (19.05907pt too wide) in paragraph at lines 435--443 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.447 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.448 \log + {(\phi_{R(t)}(\xi))} = i\delta \xi - \gamma |\xi|^{\alpha} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.450 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.450 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 444--451 +[] + [] + + +Overfull \hbox (67.31882pt too wide) in paragraph at lines 444--451 +[] [] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (20.88092pt too wide) in paragraph at lines 444--451 +\OML/cmm/m/it/10 i^^N^^X \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (94.4605pt too wide) in paragraph at lines 444--451 +\OML/cmm/m/it/10 ^^M\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 ^^X\OMS/cmsy/m/n/10 j[] +[]$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 452--454 +[] + [] + + +Overfull \hbox (16.98955pt too wide) in paragraph at lines 452--454 +\OML/cmm/m/it/10 \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 452--454 +\OT1/cmr/m/n/10 2$ + [] + + +Overfull \hbox (16.98955pt too wide) in paragraph at lines 452--454 +\OML/cmm/m/it/10 \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (21.73955pt too wide) in paragraph at lines 452--454 +\OT1/cmr/m/n/10 1[]\OML/cmm/m/it/10 ^^L \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 452--454 +\OT1/cmr/m/n/10 0$ + [] + +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 455--473 +[] + [] + + +Overfull \hbox (15.87843pt too wide) in paragraph at lines 455--473 +\OML/cmm/m/it/10 \OMS/cmsy/m/n/10 2 + [] + + +Overfull \hbox (20.00002pt too wide) in paragraph at lines 455--473 +[]$ + [] + + +Overfull \hbox (16.98955pt too wide) in paragraph at lines 455--473 +\OML/cmm/m/it/10 \OMS/cmsy/m/n/10 6\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 455--473 +\OT1/cmr/m/n/10 1$ + [] + + +Overfull \hbox (15.62842pt too wide) in paragraph at lines 455--473 +\OML/cmm/m/it/10 ^^L \OMS/cmsy/m/n/10 2 + [] + + +Overfull \hbox (27.77782pt too wide) in paragraph at lines 455--473 +[]$ + [] + + +Overfull \hbox (16.28836pt too wide) in paragraph at lines 455--473 +\OML/cmm/m/it/10 ^^M \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (10.38626pt too wide) in paragraph at lines 455--473 +\OML/cmm/m/it/10 s[]$ + [] + + +Overfull \hbox (6.43404pt too wide) in paragraph at lines 455--473 +\OML/cmm/m/it/10 $ + [] + + +Overfull \hbox (4.8229pt too wide) in paragraph at lines 455--473 +\OML/cmm/m/it/10 ^^N$ + [] + +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 474--483 +[] + [] + +! Undefined control sequence. +l.484 \subsection + {Le modèle de Press} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 484--486 +[] + [] + +Missing character: There is no [ in font nullfont! +\citation{press1967compound} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.487 \cite{press1967compound} + propose un modèle statistique basé sur un +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `press1967compound' on page undefined on input line 48 +7. + +Missing character: There is no ] in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (25.43053pt too wide) in paragraph at lines 487--500 +[][] + [] + + +Overfull \hbox (22.22226pt too wide) in paragraph at lines 487--500 +\OML/cmm/m/it/10 W\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 501--510 +[] + [] + + +Overfull \hbox (30.51392pt too wide) in paragraph at lines 501--510 +[]$ + [] + + +Overfull \hbox (9.44447pt too wide) in paragraph at lines 501--510 +\OML/cmm/m/it/10 ^^Ut$ + [] + + +Overfull \hbox (31.24649pt too wide) in paragraph at lines 501--510 +\OML/cmm/m/it/10 Y[]; k \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (47.73596pt too wide) in paragraph at lines 501--510 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; [] ; N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\O +T1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (22.22226pt too wide) in paragraph at lines 501--510 +\OML/cmm/m/it/10 W\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.514 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no s in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +! Misplaced alignment tab character &. +l.516 s(t) & + = s(0) + \sum_{k=1}^{N(t)} Y_k + W(t) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no + in font nullfont! +! Missing $ inserted. + + $ +l.516 s(t) &= s(0) + \sum + _{k=1}^{N(t)} Y_k + W(t) +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.517 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.517 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 511--518 +[] + [] + + +Overfull \hbox (36.63196pt too wide) in paragraph at lines 511--518 +[] \OMS/cmsy/m/n/10 ^^Q + [] + + +Overfull \hbox (50.45837pt too wide) in paragraph at lines 511--518 +[]$ + [] + + +Overfull \hbox (47.58134pt too wide) in paragraph at lines 511--518 +[][] \OML/cmm/m/it/10 Y[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (22.22226pt too wide) in paragraph at lines 511--518 +\OML/cmm/m/it/10 W\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align* undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.521 \begin{align*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no Y in font nullfont! +! Missing $ inserted. + + $ +l.522 Y_ + k &\sim N(\theta,\sigma_2^2) \\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.522 Y_k & + \sim N(\theta,\sigma_2^2) \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.523 W(t) & + \sim N(0,\sigma_1^2 t) \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.524 N(t) & + \sim Poisson(\lambda t) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.525 \end{align*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.525 \end{align*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 519--526 +[] + [] + + +Overfull \hbox (15.47568pt too wide) in paragraph at lines 519--526 +[] \OMS/cmsy/m/n/10 ^^X + [] + + +Overfull \hbox (36.87845pt too wide) in paragraph at lines 519--526 +\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^R; ^^[[]\OT1/cmr/m/n/10 ) +[] + [] + + +Overfull \hbox (32.77777pt too wide) in paragraph at lines 519--526 +\OML/cmm/m/it/10 W\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OMS/cms +y/m/n/10 ^^X + [] + + +Overfull \hbox (40.51735pt too wide) in paragraph at lines 519--526 +\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; ^^[[]t\OT1/cmr/m/n/10 )[ +] + [] + + +Overfull \hbox (31.0694pt too wide) in paragraph at lines 519--526 +\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OMS/cms +y/m/n/10 ^^X + [] + + +Overfull \hbox (53.54825pt too wide) in paragraph at lines 519--526 +\OML/cmm/m/it/10 Poisson\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^Ut\OT1/cmr/m/n/10 ) +$ + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 527--533 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.538 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.540 \phi + _{s(t)}\left(\xi\right) &= E\left[e^{i \xi s(t)} \right] \nonumb... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.540 \phi_{s(t)}\left(\xi\right) & + = E\left[e^{i \xi s(t)} \right] \nonumb... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.541 & + = exp\left\{ i\xi \cdot s(0) \right\} \times exp \left\{ -\frac{t \... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.542 & + = exp\left\{ i\xi \cdot s(0)- \frac{t}{2}\sigma_1^2\xi^2 + \lambda +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.544 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.544 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 534--545 +[] + [] + + +Overfull \hbox (16.07642pt too wide) in paragraph at lines 534--545 +\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (32.11642pt too wide) in paragraph at lines 534--545 +\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (44.33856pt too wide) in paragraph at lines 534--545 +\OML/cmm/m/it/10 ^^^[] [] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (44.5333pt too wide) in paragraph at lines 534--545 +\OML/cmm/m/it/10 E [] [] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 534--545 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (67.8149pt too wide) in paragraph at lines 534--545 +\OML/cmm/m/it/10 exp [] \OMS/cmsy/m/n/10 ^^B + [] + + +Overfull \hbox (67.9476pt too wide) in paragraph at lines 534--545 +\OML/cmm/m/it/10 exp [] \OMS/cmsy/m/n/10 ^^B + [] + + +Overfull \hbox (122.88634pt too wide) in paragraph at lines 534--545 +\OML/cmm/m/it/10 exp [] [] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 534--545 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (204.89801pt too wide) in paragraph at lines 534--545 +\OML/cmm/m/it/10 exp []$ + [] + +Missing character: There is no A in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ´ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.550 processus \eqref + {eq:fncaractpress}. Il s'agit essentiellement de poser +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.552 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.554 \phi + _{\Delta s(t)}\left(\xi\right) &= E\left[e^{i\xi\Delta s(t)} \ri... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.554 \phi_{\Delta s(t)}\left(\xi\right) & + = E\left[e^{i\xi\Delta s(t)} \ri... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.555 & + = exp\left\{-\frac{\sigma_1^2 \xi^2}{2} + \lambda \left[e^{i +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.557 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.557 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 546--558 +[] + [] + + +Overfull \hbox (34.96529pt too wide) in paragraph at lines 546--558 +\OT1/cmr/m/n/10 ^^A\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cm +r/m/n/10 ) = + [] + + +Overfull \hbox (23.85422pt too wide) in paragraph at lines 546--558 +\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OMS/cms +y/m/n/10 ^^@ + [] + + +Overfull \hbox (19.96532pt too wide) in paragraph at lines 546--558 +\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (8.88892pt too wide) in paragraph at lines 546--558 +\OT1/cmr/m/n/10 1)$ + [] + + +Overfull \hbox (38.6859pt too wide) in paragraph at lines 546--558 +\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (28.02083pt too wide) in paragraph at lines 546--558 +\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 (0) = + [] + + +Overfull \hbox (19.16664pt too wide) in paragraph at lines 546--558 +\OT1/cmr/m/n/10 0[]\OML/cmm/m/it/10 t \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 546--558 +\OT1/cmr/m/n/10 1$ + [] + + +Overfull \hbox (50.90804pt too wide) in paragraph at lines 546--558 +\OML/cmm/m/it/10 ^^^[] [] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (51.10277pt too wide) in paragraph at lines 546--558 +\OML/cmm/m/it/10 E [] [] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 546--558 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (156.5769pt too wide) in paragraph at lines 546--558 +\OML/cmm/m/it/10 exp []$ + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment subequations undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.563 \begin{subequations} + \label{eq:cumulantspress} +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.564 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no K in font nullfont! +! Missing $ inserted. + + $ +l.565 K_ + 1 &= \lambda\theta \\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.565 K_1 & + = \lambda\theta \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.566 K_2 & + = \sigma_1^2+\lambda(\theta^2+\sigma_2^2) \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.567 K_3 & + = \lambda\theta(\theta^2+3\sigma_2^2) \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.568 K_4 & + = \lambda(\theta^4 + 6 \theta^2 \sigma_2^2 + 3 \sigma_2^4) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.569 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.569 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{document} ended by \end{subequations}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.570 \end{subequations} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 559--571 +[] + [] + + +Overfull \hbox (24.40977pt too wide) in paragraph at lines 559--571 +\OT1/cmr/m/n/10 ^^A\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cm +r/m/n/10 )$ + [] + + +Overfull \hbox (15.04164pt too wide) in paragraph at lines 559--571 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (10.80559pt too wide) in paragraph at lines 559--571 +\OML/cmm/m/it/10 ^^U^^R[] + [] + + +Overfull \hbox (23.5347pt too wide) in paragraph at lines 559--571 +\OML/cmm/m/it/10 K[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (18.33684pt too wide) in paragraph at lines 559--571 +\OML/cmm/m/it/10 ^^[[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (26.95842pt too wide) in paragraph at lines 559--571 +\OML/cmm/m/it/10 ^^U\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^R[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (14.44794pt too wide) in paragraph at lines 559--571 +\OML/cmm/m/it/10 ^^[[]\OT1/cmr/m/n/10 )[] + [] + + +Overfull \hbox (23.5347pt too wide) in paragraph at lines 559--571 +\OML/cmm/m/it/10 K[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (31.93065pt too wide) in paragraph at lines 559--571 +\OML/cmm/m/it/10 ^^U^^R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^R[] \OT1/cmr/m/n/10 ++ + [] + + +Overfull \hbox (19.44795pt too wide) in paragraph at lines 559--571 +\OT1/cmr/m/n/10 3\OML/cmm/m/it/10 ^^[[]\OT1/cmr/m/n/10 )[] + [] + + +Overfull \hbox (23.5347pt too wide) in paragraph at lines 559--571 +\OML/cmm/m/it/10 K[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (26.95842pt too wide) in paragraph at lines 559--571 +\OML/cmm/m/it/10 ^^U\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^R[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (32.79521pt too wide) in paragraph at lines 559--571 +\OT1/cmr/m/n/10 6\OML/cmm/m/it/10 ^^R[]^^[[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (19.44795pt too wide) in paragraph at lines 559--571 +\OT1/cmr/m/n/10 3\OML/cmm/m/it/10 ^^[[]\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.573 \eqref + {eq:cumulantsempiriques}, on obtient les équations suivantes: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment subequations undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.574 \begin{subequations} + \label{eq:presscum} +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.575 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no 0 in font nullfont! +! Misplaced alignment tab character &. +l.576 0 & + = \hat\theta^4 - \frac{\overline{K}_3}{\overline{K}_1} \hat\the... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +! Missing $ inserted. + + $ +l.576 0 &= \hat + \theta^4 - \frac{\overline{K}_3}{\overline{K}_1} \hat\the... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.577 \hat\lambda & + = \frac{\overline{K}_1}{\hat\theta} \label{eq:presscu... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.578 \hat\sigma_2^2 & + = \frac{\overline{K}_3-\hat\theta^2\overline{K}_1}... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.579 \hat\sigma_1^2 & + = \overline{K}_2 - +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.583 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.583 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{document} ended by \end{subequations}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.584 \end{subequations} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 572--585 +[] + [] + + +Overfull \hbox (17.23616pt too wide) in paragraph at lines 572--585 +[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (30.74875pt too wide) in paragraph at lines 572--585 +[][] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (22.47095pt too wide) in paragraph at lines 572--585 +[][][][] + [] + + +Overfull \hbox (16.38887pt too wide) in paragraph at lines 572--585 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (13.51259pt too wide) in paragraph at lines 572--585 +[][][] + [] + + +Overfull \hbox (21.11455pt too wide) in paragraph at lines 572--585 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (38.8023pt too wide) in paragraph at lines 572--585 +[][][] + [] + + +Overfull \hbox (21.11455pt too wide) in paragraph at lines 572--585 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (21.47226pt too wide) in paragraph at lines 572--585 +[][] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (87.60648pt too wide) in paragraph at lines 572--585 +[] [][]$ + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.586 ...solvant numériquement l'équation \eqref + {eq:presscumtheta} pour le +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.588 équations \eqref + {eq:presscum}, on obtient des estimateurs convergents +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 586--590 +[] + [] + + +Overfull \hbox (4.97223pt too wide) in paragraph at lines 586--590 +[]$ + [] + +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{merton1976option} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.591 ...é présenté par \cite{merton1976option} + , +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `merton1976option' on page undefined on input line 591 +. + +Missing character: There is no ] in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.596 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no d in font nullfont! +Missing character: There is no S in font nullfont! +! Missing $ inserted. + + $ +l.598 \frac{dS}{S} + = (\alpha - \lambda k)dt + \sigma dW + dq +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Extra }, or forgotten $. +\frac #1#2->{\begingroup #1\endgroup \over #2} + +l.598 \frac{dS}{S} + = (\alpha - \lambda k)dt + \sigma dW + dq +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.599 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.599 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.599 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ´ in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.603 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +! Missing $ inserted. + + $ +l.604 S(t) = S(0) \exp + \left\{ (\alpha-\frac{1}{2}\sigma_1^2-\lambda k)t + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.607 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.607 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 591--608 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 591--608 +[] + [] + + +Overfull \hbox (6.43404pt too wide) in paragraph at lines 591--608 +\OML/cmm/m/it/10 $ + [] + + +Overfull \hbox (8.02779pt too wide) in paragraph at lines 591--608 +\OML/cmm/m/it/10 Y$ + [] + + +Overfull \hbox (85.88pt too wide) in paragraph at lines 591--608 +[]$ + [] + + +Overfull \hbox (5.52084pt too wide) in paragraph at lines 591--608 +\OML/cmm/m/it/10 k$ + [] + + +Overfull \hbox (4.8229pt too wide) in paragraph at lines 591--608 +\OML/cmm/m/it/10 q$ + [] + + +Overfull \hbox (249.6343pt too wide) in paragraph at lines 591--608 +[] [] []$ + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.612 ...n caractéristique d'un incrément \eqref + {eq:fncaractpress2} du +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.614 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.616 \phi + _{\Delta s(t)}\left(\xi\right) &= E\left[e^{i\xi\Delta s(t)} \ri... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.616 \phi_{\Delta s(t)}\left(\xi\right) & + = E\left[e^{i\xi\Delta s(t)} \ri... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.617 & + = exp\left\{i\delta \xi -\frac{\sigma_1^2 \xi^2}{2} + \lambda +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.619 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.619 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 609--620 +[] + [] + + +Overfull \hbox (15.37842pt too wide) in paragraph at lines 609--620 +\OML/cmm/m/it/10 ^^N \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (14.21184pt too wide) in paragraph at lines 609--620 +\OML/cmm/m/it/10 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (24.72296pt too wide) in paragraph at lines 609--620 +[]\OML/cmm/m/it/10 ^^[[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (11.3542pt too wide) in paragraph at lines 609--620 +\OML/cmm/m/it/10 ^^Uk$ + [] + + +Overfull \hbox (8.02779pt too wide) in paragraph at lines 609--620 +\OML/cmm/m/it/10 Y$ + [] + + +Overfull \hbox (50.90804pt too wide) in paragraph at lines 609--620 +\OML/cmm/m/it/10 ^^^[] [] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (51.10277pt too wide) in paragraph at lines 609--620 +\OML/cmm/m/it/10 E [] [] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 609--620 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (174.12436pt too wide) in paragraph at lines 609--620 +\OML/cmm/m/it/10 exp []$ + [] + +! Undefined control sequence. +l.621 \subsection + {Le modèle de Praetz} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no z in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 621--622 +[] + [] + +Missing character: There is no [ in font nullfont! +\citation{praetz1972distribution} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.623 \cite{praetz1972distribution} + propose un modèle inspiré par la +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `praetz1972distribution' on page undefined on input li +ne 623. + +Missing character: There is no ] in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (25.43053pt too wide) in paragraph at lines 623--630 +[][] + [] + + +Overfull \hbox (29.0591pt too wide) in paragraph at lines 623--630 +[]$ + [] + + +Overfull \hbox (6.02548pt too wide) in paragraph at lines 623--630 +\OML/cmm/m/it/10 ^^V$ + [] + + +Overfull \hbox (10.55904pt too wide) in paragraph at lines 623--630 +\OML/cmm/m/it/10 ^^[[]$ + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.640 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no f in font nullfont! +! Missing $ inserted. + + $ +l.642 f_ + {R(t)}(r|\sigma^2) &= \frac{1}{\sigma\sqrt{2\pi}}exp\left\{-\frac{... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.642 f_{R(t)}(r|\sigma^2) & + = \frac{1}{\sigma\sqrt{2\pi}}exp\left\{-\frac{... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.643 h_{R(t)}(r) & + = \int_0^{\infty} f_R(r|\sigma^2) g(\sigma^2) d\sigma^2 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.644 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.644 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 631--645 +[] + [] + + +Overfull \hbox (10.55904pt too wide) in paragraph at lines 631--645 +\OML/cmm/m/it/10 ^^[[]$ + [] + + +Overfull \hbox (23.4653pt too wide) in paragraph at lines 631--645 +\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^[[]\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (52.29047pt too wide) in paragraph at lines 631--645 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 ^^[[]\O +T1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (90.20883pt too wide) in paragraph at lines 631--645 +[]\OML/cmm/m/it/10 exp [] [] + [] + + +Overfull \hbox (44.71523pt too wide) in paragraph at lines 631--645 +\OML/cmm/m/it/10 h[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (93.39468pt too wide) in paragraph at lines 631--645 +[][] \OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OMS/cmsy/m/n/10 j\ +OML/cmm/m/it/10 ^^[[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/c +mm/m/it/10 ^^[[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 d^^[[]$ + [] + +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.648 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no g in font nullfont! +Missing character: There is no ( in font nullfont! +! Missing $ inserted. + + $ +l.650 g(\sigma + ^2) &= +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.650 g(\sigma^2) & + = +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.652 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.652 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 646--653 +[] + [] + + +Overfull \hbox (23.4653pt too wide) in paragraph at lines 646--653 +\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^[[]\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (8.78014pt too wide) in paragraph at lines 646--653 +\OML/cmm/m/it/10 m$ + [] + + +Overfull \hbox (9.17363pt too wide) in paragraph at lines 646--653 +\OML/cmm/m/it/10 s[]$ + [] + + +Overfull \hbox (25.00345pt too wide) in paragraph at lines 646--653 +\OML/cmm/m/it/10 ^^[[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (87.08504pt too wide) in paragraph at lines 646--653 +[]$ + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.658 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no h in font nullfont! +! Missing $ inserted. + + $ +l.660 h_ + {R(t)}(r) &= \frac{\Gamma(m)\left[\ 2(m-1)\pi +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.660 h_{R(t)}(r) & + = \frac{\Gamma(m)\left[\ 2(m-1)\pi +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.663 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.663 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 654--664 +[] + [] + + +Overfull \hbox (9.17363pt too wide) in paragraph at lines 654--664 +\OML/cmm/m/it/10 s[]$ + [] + + +Overfull \hbox (19.73225pt too wide) in paragraph at lines 654--664 +[]$ + [] + + +Overfull \hbox (7.67015pt too wide) in paragraph at lines 654--664 +\OML/cmm/m/it/10 R$ + [] + + +Overfull \hbox (13.78015pt too wide) in paragraph at lines 654--664 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 m$ + [] + + +Overfull \hbox (44.2462pt too wide) in paragraph at lines 654--664 +[][]$ + [] + + +Overfull \hbox (38.95364pt too wide) in paragraph at lines 654--664 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (76.55658pt too wide) in paragraph at lines 654--664 +[]$ + [] + +Missing character: There is no D in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 665--675 +[] + [] + + +Overfull \hbox (6.02548pt too wide) in paragraph at lines 665--675 +\OML/cmm/m/it/10 ^^V$ + [] + +! Undefined control sequence. +l.676 \section + {Conditions essentielles de Madan et Seneta} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 676--678 +[] + [] + +Missing character: There is no I in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{madan1990variance} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.680 \cite{madan1990variance} + présentent un ensemble de conditions +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `madan1990variance' on page undefined on input line 68 +0. + +Missing character: There is no ] in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 679--684 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 679--684 +[] + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 685--685 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (7.67015pt too wide) in paragraph at lines 686--692 +\OML/cmm/m/it/10 R$ + [] + + +Overfull \hbox (12.56717pt too wide) in paragraph at lines 686--692 +\OML/cmm/m/it/10 r \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (3.61111pt too wide) in paragraph at lines 686--692 +\OML/cmm/m/it/10 t$ + [] + + +Overfull \hbox (3.61111pt too wide) in paragraph at lines 686--692 +\OML/cmm/m/it/10 t$ + [] + + +Overfull \hbox (4.78937pt too wide) in paragraph at lines 686--692 +\OML/cmm/m/it/10 r$ + [] + + +Overfull \hbox (172.62921pt too wide) in alignment at lines 692--696 + [][][] [] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (6.00235pt too wide) in paragraph at lines 697--701 +\OML/cmm/m/it/10 n$ + [] + + +Overfull \hbox (7.67015pt too wide) in paragraph at lines 697--701 +\OML/cmm/m/it/10 R$ + [] + + +Overfull \hbox (16.55786pt too wide) in paragraph at lines 697--701 +\OML/cmm/m/it/10 n \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 697--701 +\OT1/cmr/m/n/10 4$ + [] + + +Overfull \hbox (119.40256pt too wide) detected at line 704 +[]\OML/cmm/m/it/10 E [] < \OMS/cmsy/m/n/10 1\OML/cmm/m/it/10 ; k \OMS/cmsy/m/n +/10 2 f\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ; +\OT1/cmr/m/n/10 3\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 4\OMS/cmsy/m/n/10 g + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...rence `tab:condmadan' on page \thepage + \space undefined\on@line . +l.720 ...etrouvent à la table \ref{tab:condmadan} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:condmadan' on page undefined on input line 720. + +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.721 \begin{table} + [h] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.736 \caption + {Respect des conditions émises par Madan et Seneta pour les... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no R in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.738 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 719--739 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 719--739 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ... `sec:levykhintchine' on page \thepage + \space undefined\on@line . +l.743 ...ffusion (Section \ref{sec:levykhintchine} + ), ce qui va +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:levykhintchine' on page undefined on input line +743. + +Missing character: There is no ) in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.748 asymétrique généralisée \citep + {kotz2001laplace}, a acquis beaucoup de +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no k in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 740--755 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 740--755 +[] + [] + +! Undefined control sequence. +l.756 \section + {La volatilité} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 756--758 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 759--766 +[] + [] + + +Overfull \hbox (38.81227pt too wide) in paragraph at lines 759--766 +\OT1/cmr/bx/n/10 volatilit + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.775 prévisions \citep + {musiela2005martingale}. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 767--776 +[] + [] + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ´ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +! Undefined control sequence. +l.782 \subsection + {Mesure de la volatilité historique} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no M in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 777--784 +[] + [] + +Missing character: There is no A in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{randal2004non} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.786 ...par fenêtre mobile. \cite{randal2004non} + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `randal2004non' on page undefined on input line 786. + +Missing character: There is no ] in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.789 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.790 \hat + \sigma^2(t) = \frac{1}{2r+1} \sum_{j=-r}^r R(t+j)^2, \qquad +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.792 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.792 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 785--793 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 785--793 +[] + [] + + +Overfull \hbox (21.94795pt too wide) in paragraph at lines 785--793 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (29.61458pt too wide) in paragraph at lines 785--793 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (27.1945pt too wide) in paragraph at lines 785--793 +\OML/cmm/m/it/10 Y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OMS/cms +y/m/n/10 ^^@ + [] + + +Overfull \hbox (34.59721pt too wide) in paragraph at lines 785--793 +\OML/cmm/m/it/10 E []$ + [] + + +Overfull \hbox (32.50346pt too wide) in paragraph at lines 785--793 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (77.80698pt too wide) in paragraph at lines 785--793 +[] [][] \OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OT1/cmr/m/n/10 + + + [] + + +Overfull \hbox (30.56544pt too wide) in paragraph at lines 785--793 +\OML/cmm/m/it/10 j\OT1/cmr/m/n/10 )[]\OML/cmm/m/it/10 ; t \OMS/cmsy/m/n/10 2 + [] + + +Overfull \hbox (55.02536pt too wide) in paragraph at lines 785--793 +[][]$ + [] + +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 794--801 +[] + [] + + +Overfull \hbox (6.00235pt too wide) in paragraph at lines 794--801 +\OML/cmm/m/it/10 n$ + [] + + +Overfull \hbox (13.78015pt too wide) in paragraph at lines 794--801 +\OML/cmm/m/it/10 n \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (9.78938pt too wide) in paragraph at lines 794--801 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 r$ + [] + +! Undefined control sequence. +l.802 \subsection + {Biais de la volatilité implicite} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no B in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 802--804 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ® in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{hull1999options} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.807 ...sous-jacent. Selon \cite{hull1999options} + , ce +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `hull1999options' on page undefined on input line 807. + + +Missing character: There is no ] in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ´ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...ce `fig:volimplicite' on page \thepage + \space undefined\on@line . +l.814 \ref{fig:volimplicite} + pour un exemple). +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `fig:volimplicite' on page undefined on input line 81 +4. + +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.815 \begin{figure} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.817 \includegraphics + []{./bbry-echeance-06-2013-20mai2013.png} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.818 \caption + {Courbe de volatilité implicite, titre BBRY, option d'achat +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no R in font nullfont! +Missing character: There is no Y in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{thevolskew} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.820 source: \cite{thevolskew} + } +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `thevolskew' on page undefined on input line 820. + +Missing character: There is no ] in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.822 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 805--823 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 805--823 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 805--823 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 805--823 +[] + [] + +) +! Emergency stop. +<*> chapitre1.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 51 strings out of 493485 + 739 string characters out of 3143527 + 69818 words of memory out of 3000000 + 3454 multiletter control sequences out of 15000+200000 + 4294 words of font info for 16 fonts, out of 3000000 for 9000 + 957 hyphenation exceptions out of 8191 + 20i,7n,26p,219b,135s stack positions out of 5000i,500n,10000p,200000b,50000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/memoire/chapitre1.tex b/memoire/chapitre1.tex new file mode 100644 index 0000000..0699270 --- /dev/null +++ b/memoire/chapitre1.tex @@ -0,0 +1,834 @@ +\chapter{Les modèles de rendements financiers} + +\section{L'utilisation de modèles en finance} +\label{sec:utilisationmodeles} + +On doit considérer les implications de l'utilisation de modèles en +finance avant d'entreprendre leur étude. On doit aussi prendre +connaissance des différents types ainsi que les risques liés à chacun +d'entre eux. Pour ce faire, on se réfère à la note «Model Risk» +publiée par \cite{derman1996modelrisk}. + +Durant les dernières décennies, plusieurs modèles sont apparus afin de +fournir une approche fondamentale aux concepts de tarification, +d'offre et de demande et d'arbitrage aux intervenants des milieux +financiers. Au cours des années 1970, on se préoccupe particulièrement +des fluctuations des taux d'intérêt, un phénomène qui marque cette +époque. Les notions de duration et de convexité font alors leurs +débuts. Sur les marchés de capitaux propres, on s'intéresse à la +discordance entre le prix négocié des contrats à terme et le prix +raisonnable calculé selon une perspective théorique. + +Puis, la confiance développée envers le modèle de tarification +d'options de \cite{black1973pricing} et ses extensions a favorisé la +croissance du marché des produits dérivés. La puissance de calcul +croissante des ordinateurs a aussi permis l'élaboration et +l'utilisation de modèles de plus en plus sophistiqués. La dépendance +qui peut se développer envers ceux-ci apporte son lot de +considérations. On doit donc se rappeler l'utilisation désirée par les +auteurs de ceux-ci et le risque associé à leur usage à grande échelle. + +\subsection{Différents types de modèles} +\label{sec:differentsmodeles} + +Toujours selon Derman, un modèle financier peut être classé parmi au +moins trois catégories: + +\begin{enumerate} +\item Le \textbf{modèle fondamental}, basé sur un système de postulats + et de données, entre lesquels on peut établir différentes + relations. Le modèle de Black-Scholes en est un exemple. +\item Le \textbf{modèle phénoménologique}, qui présente une + description ou une analogie, afin d'illustrer quelque chose qui ne + peut être directement observé. C'est un modèle moins fondamental, + basé aussi sur des liens de cause à effet. Un modèle qui chercherait + à expliquer l'impact du retrait du porteur de parts majoritaire + d'une entreprise sur la valeur des actions de celle-ci serait + phénoménologique. +\item Le \textbf{modèle statistique}, basé sur une régression ou un + réglage optimal entre différents ensembles de données. On ne cherche + pas ici à expliquer une dynamique, mais à décrire une tendance ou + une corrélation. Le modèle d'évaluation des actifs financiers et + celui des trois facteurs de \cite{fama1993common} en sont des exemples. +\end{enumerate} + +Un modèle financier est en partie basé sur des variables qui +représentent des opinions et des anticipations, et non seulement des +quantités mesurables. Ces variables peuvent être, entre autres, le +rendement et la volatilité future espérés. Cette considération sera +importante notamment lorsque l'on voudra déterminer le prix +raisonnable d'un produit dérivé. En effet, un modèle de tarification +est essentiellement un moyen de refléter l'intuition des acteurs du +marché à propos de ces variables sous la forme d'un prix exprimé dans +une unité monétaire. Un bon modèle doit faciliter l'extrapolation de +ce prix sous certaines conditions de marché. + +Contrairement à la physique classique, un principe fondamental en +finance est l'incertitude. On ne peut anticiper la valeur d'un titre à +un moment donné dans le futur avec la même précision qu'on peut +prévoir la position d'un objet à cet instant. Les outils mathématiques +principalement utilisés seront alors les processus stochastiques, les +statistiques et les distributions de probabilités, en plus du calcul +différentiel et intégral. + +\subsection{Le risque de modélisation} +\label{sec:modelerisque} + +Plusieurs risques inhérents à la modélisation en finance +existent. Quelques-uns d'entre eux seront décrits dans cette section. + +La modélisation peut tout simplement ne pas être applicable à la +situation étudiée. L'exemple le plus probant serait de tenter de +prévoir les mouvements du prix d'un titre financier à court terme. + +Un modèle peut être incorrect pour plusieurs raisons. Entre autres, il +peut ignorer certains facteurs ou poser une hypothèse déterministe +inappropriée sur ceux-ci. Il peut aussi considérer une dynamique +incorrecte pour un des facteurs ou encore une relation inappropriée +entre ceux-ci. Enfin, il peut n'être applicable que sous certaines +conditions bien précises ou encore que son utilisation soit limitée à +court terme, notamment lorsqu'il nécessite un temps de calibration +pour être statistiquement valable. Il peut aussi être inutilisable par +une mauvaise estimation des paramètres. + +Un modèle peut aussi être correct, mais avoir une solution +erronée. Cela se produit notamment lorsqu'on tente de dériver une +solution analytique ou que l'on doit utiliser des méthodes numériques +pour obtenir celle-ci. On se doit, dans ce cas, de connaître l'erreur +maximale possible de la méthode utilisée. Un modèle correct peut aussi +être utilisé dans le mauvais contexte. Par exemple, on pourrait avoir +recours à des paramètres inadéquats de simulation, ou encore réutiliser le +modèle dans une autre situation sans tenir compte des +conditions de validité de celui-ci. + +Son utilisation peut génèrer des prix déraisonnables; on parle alors +d'arbitrage de modèle. Par exemple, si un titre est évalué à l'aide du +modèle d'évaluation des actifs financiers, son prix sera différent de +celui qui serait obtenu avec la régression à trois facteurs de Fama et +French. Un investisseur peut alors faire du profit en achetant le +titre à celui qui demande le prix le plus faible pour le revendre à +celui qui offre le plus élevé. + +L'utilisation de données instables peut produire des résultats +différents selon la période étudiée. La possibilité qu'une estimation +basée sur des données historiques soit erronée doit être considérée. + +Enfin, comme la plupart des modèles financiers sont implémentés sous +forme de logiciels, différents bogues informatiques peuvent se +retrouver dans le code source. On considère entre autres des erreurs +d'arrondissement, de logique et de clarté du code, ainsi que des +particularités du matériel qui n'auraient pas été prises en compte par +le programmeur. Ces erreurs peuvent être difficiles à détecter, c'est +pourquoi un grand nombre de tests devrait être effectués avant de +publier un logiciel de modélisation financière. + +\section{Les rendements financiers} +\label{sec:prixrendements} + +Le \textbf{rendement} est défini comme étant le gain ou la perte de +valeur d'un actif sur une période donnée. Il est constitué des revenus +occasionnés et des gains en capitaux d'un investissement et est +habituellement représenté sous la forme d'un pourcentage. Ces derniers +peuvent prendre la forme de coupons pour les titres à revenus fixes et +de dividendes pour les actions échangées sur les marchés boursiers. On +ne considèrera, dans ce texte, que les titres boursiers sans +dividende, dont le rendement est lié uniquement aux gains en capitaux. + +\subsection{Définitions et notations} +\label{sec:defrendements} + +On définit le prix $S(t)>0$ d'un titre financier observé au temps +$t$. Implicitement, le prix considéré est celui à la fermeture. On +définit aussi le taux de rendement effectif $R(t)$ sur une période +comprise dans l'intervalle de temps $\left[t-1,t\right]$. C'est le +taux composé continument, aussi appelé force d'intérêt, qui aurait +occasionné les mêmes gains ou pertes sur un montant déposé en banque +au cours de la période concernée. Le taux de rendement est la variable +d'intérêt dans le contexte de la modélisation financière. + +On associe le taux de rendement effectif à la différence entre le +logarithme du prix initial et final. Dans la situation où le taux de +rendement est déterministe et non aléatoire, on obtient l'équation +différentielle suivante: +\begin{align*} + \frac{dS(t)}{dt} &= R(t) \cdot S(t). +\end{align*} + +On peut interpréter cette équation en affirmant que la variation du +prix $dS(t)$ sur un intervalle de temps infiniment petit $dt$ est +proportionnelle à la valeur actuelle $S(t)$. Cette équation +différentielle a pour solution générale: +\begin{align} + \label{eq:solutiondiffrendement} + S(t) &= S(0)e^{R(t) \cdot t}. +\end{align} + +Afin de définir les propriétés de l'échantillon sélectionné, on pose +comme hypothèse: +\begin{hypothese} + Le rendement $R(t)$ est constant durant la période définie par + l'intervalle de temps $\left[t-1,t\right]$, mais il est différent + d'une à l'autre: $R(s) \neq R(t), s \neq t$. +\end{hypothese} + +On peut alors représenter le rendement $R(t)$ comme étant la +différence entre les logarithmes des prix observés au temps $t$ et +$t-1$, ou encore le logarithme du quotient de ces mêmes prix: +\begin{align} + R(t) &= \ln{(S(t))} - \ln{(S(t-1))} \nonumber\\ + &= + \ln{\left(\frac{S(t)}{S(t-1)}\right)}. \label{eq:rendementlogprix} +\end{align} + +On définit aussi le \textbf{rendement cumulé} $L(t)$. Il correspond à +la somme des rendements effectifs observés sur l'intervalle +$\left[0,t\right]$: +\begin{align} + L(t) &= \sum_{i=1}^{t} R(i) \nonumber\\ + &= \sum_{i=1}^{t} \left[\ln{(S(i)} - \ln{(S(i-1))}\right] \nonumber\\ + &= \ln{(S(t))} - \ln{(S(0))} \nonumber\\ + &= \ln{\left(\frac{S(t)}{S(0)}\right)}. \label{eq:rendementcumL} +\end{align} + +Cette représentation permet d'exprimer le prix actuel $S(t)$ en +fonction de la valeur initiale $S(0)$ sous une forme similaire à la +solution \eqref{eq:solutiondiffrendement}, mais tenant compte de +l'hypothèse émise précédemment: +\begin{align} + e^{L(t)} &= \frac{S(t)}{S(0)} \nonumber\\ + S(t) &= S(0) \cdot e^{L(t)} \nonumber\\ + &= S(0) \cdot \exp\left(\sum_{i=1}^{t} R(i)\right). +\end{align} + +\subsection{Rendements cumulés} +\label{sec:rendementscum} + +On pose l'hypothèse suivante: +\begin{hypothese} + Les rendements $R(i), i \in 1, \ldots, t$ sont indépendants, mais + pas nécessairement identiquement distribués. +\end{hypothese} + +On peut alors obtenir la distribution du rendement cumulé $L(t)$ en +utilisant le produit de convolution \eqref{eq:convocaract}. +Considérons $\phi_{R(i)}(\xi)$ la fonction caractéristique d'un +rendement $R(i)$ et $\phi_{L(t)}(\xi)$ celle du cumulé $L(t)$. On +obtient alors que cette dernière est égale au produit des fonctions +caractéristiques des rendements effectifs sur chacune des périodes de +l'intervalle $\left[0,t\right]$: +\begin{equation} + \label{eq:convolutionLR} + \phi_{L(t)}(\xi) = \prod_{i=1}^t \phi_{R(i)}(\xi). +\end{equation} + +On considère la situation où l'on posera plutôt l'hypothèse suivante: +\begin{hypothese} + Les rendements $R(i), i \in 1, \ldots, t$ sont à la fois + indépendants et identiquement distribués. +\end{hypothese} + +Alors, la fonction caractéristique des rendements est égale pour +chaque période: +\begin{align} + \label{eq:rendementsIID} + \phi_{R}(\xi) = \phi_{R(1)}(\xi) = \ldots &= \phi_{R(t)}(\xi). +\end{align} + +On peut donc simplifier l'expression \eqref{eq:convolutionLR} pour +obtenir la fonction caractéristique: +\begin{equation} + \label{eq:convolution2} + \phi_{L(t)}(\xi) = \left[\phi_{R}(\xi)\right]^t. +\end{equation} + +Considérer une distribution qui est fermée sous la convolution pour +modéliser les rendements sur une période $R(i)$ peut alors être +intéressant. Le rendement cumulé $L(t)$ pourra aussi être +modélisé à l'aide de la même distribution. Pour ce faire, on modifie +un paramètre d’échelle en fonction de la longueur $t$ de l'intervalle +de temps considéré. + +\subsection{Données disponibles} +\label{sec:donneesdisponibles} + +Les données disponibles auprès des fournisseurs d'informations +financières prennent habituellement la forme de séries chronologiques +discontinues. Celles-ci incluent les prix à l'ouverture, le plus bas +et le plus élevé au courant de la journée ainsi qu'à la fermeture, +pour chaque jour où les marchés financiers sont en activité. Afin de +mesurer le rendement quotidien d'un titre, seuls les prix à la +fermeture seront considérés. + +\section{Les premiers modèles} + +\subsection{Le modèle de Bachelier} +\label{sec:bachelier} + +Un des premiers modèles proposés afin de représenter les rendements +financiers a été celui de \cite{bachelier1900theorie} . + +Le prix d'un titre peut varier, durant une période, de n'importe +quelle valeur comprise dans l'intervalle $\left[ -S(t),\infty +\right]$. Il propose donc que cet intervalle soit remplacé par +l'ensemble du domaine réel $\mathbb{R}$. La probabilité que le titre +atteigne une valeur nulle ou négative ou que celle-ci double devrait +donc être négligeable. Il ajoute aussi que la variation est +indépendante du prix actuel du titre $S(t)$ et que la distribution de +probabilités de celle-ci est symétrique et centrée en ce point. + +Il utilise le principe selon lequel la probabilité que deux évènements +indépendants consécutifs aient lieu est le produit de celles que +chacun d'entre eux se réalise, pour établir la distribution des +variations du prix. Par exemple, la variation du prix sur une première +période prend la valeur $x$ et celle sur une seconde, $z-x$, comme +illustré à la figure \ref{fig:bachelier1}. + +%% ligne du temps +\begin{figure}[!ht] + \centering + \begin{tikzpicture} + \draw [->] (0,0) -- (6,0); \foreach \x in {0,2,4} \draw (\x + cm,3pt) -- (\x cm,-3pt); \draw (0,0) node[below=3pt] {$ 0 $} + node[above=3pt] {$ S(0) $}; \draw (2,0) node[below=3pt] {$ t_1 $} + node[above=3pt] {$ S(0)+x $}; \draw (4,0) node[below=3pt] {$ + t_1+t_2 $} node[above=3pt] {$ S(0)+z $}; + \end{tikzpicture} + \caption{Modèle de Bachelier: probabilité composée} + \label{fig:bachelier1} +\end{figure} + +On définit $f(x,t)$ la fonction de densité de la variation du prix +$S(t)$ par rapport au niveau initial $S(0)$. Alors, selon le principe +précédent, on obtient l'expression +\begin{align} + \label{eq:probcomposeeB} + f(z,t_1+t_2) = \int_{-\infty}^{\infty} f(x,t_1)\cdot f(z-x,t_2) + \cdot dx. +\end{align} + +La solution proposée est que la densité de probabilité soit de la forme +\begin{align*} + \label{eq:formeprobB} + f(x,t) = A \cdot \exp \left\{-B^2x^2 \right\}. +\end{align*} + +Afin que la fonction $f(x,t)$ soit une densité de probabilité, la +condition suivante doit être respectée: +\begin{align} + \int_{-\infty}^{\infty} A \cdot \exp \left\{-B^2x^2 \right\} dx = 1. +\end{align} + +Ceci implique que +\begin{align*} + B&= A\sqrt{\pi}. +\end{align*} + +En posant $x=0$, on a $A=f(0,t)$ et l'on en déduit: +\begin{align} + f(x,t) = f(0,t) \cdot \exp \left\{-\pi \cdot f(0,t)^2 \cdot x^2 + \right\}. +\end{align} + +En reprenant l'intégrale \eqref{eq:probcomposeeB}, on obtient que la +densité de probabilité $f(z,t_1+t_2)$ soit aussi de la forme +\eqref{eq:formeprobB}: +\begin{align} + f(z,t_1+t_2) = \frac{f(x,1)f(z-x,2)}{\sqrt{f(x,1)^2+f(z-x,2)^2}} + \exp \left\{-\pi \frac{f(x,1)f(z-x,2)}{f(x,1)^2+f(z-x,2)^2} z^2 + \right\}. +\end{align} + +On reconnaitra que cette densité est, à un changement de variable +près, une loi normale. La démarche suggère qu'il recherchait une +distribution qui était fermée sous la convolution, une propriété +souhaitable pour un modèle cohérent des rendements financiers. + +Ce modèle implique un processus de Wiener-Bachelier selon lequel les +incréments, ou les changements de prix, suivent une distribution +normale: +\begin{equation} + \label{eq:bachelier00} + S(T)-S(t) \sim N\left(0,\sigma^2 \left(T-t\right)\right). +\end{equation} + +On doit noter que ce modèle implique que la variance des fluctuations +n'est pas proportionnelle au prix initial. Une première correction +sera apportée au modèle afin de considérer le logarithme du prix. Ce +changement permettra d'obtenir un modèle où elle est désormais +proportionnelle au prix initial. Le processus du prix suivra alors un +mouvement brownien géométrique: +\begin{align} + \label{eq:browniengeom} + S(T)-S(t) &\sim LN \left(0, \sigma (T-t) \right). +\end{align} + +Le logarithme du prix suivra alors un processus de Wiener-Bachelier: +\begin{align} + \label{eq:bachelierwiener2} + \ln\left(S(T)\right)-\ln\left(S(t)\right) &\sim N\left(0,\sigma + \left(T-t\right)\right). +\end{align} + +Un des principaux avantages du processus de Bachelier modifié est que +le rendement cumulé $L(t)$ est aussi une variable aléatoire +gaussienne. Cette propriété est appelée L-stabilité ou invariance sous +l'addition. La distribution gaussienne est la seule ayant cette +propriété où le second moment est fini. Le sujet des distributions L +stables sera aussi abordé à la section \ref{sec:mandelbrot}. + +Quelques années après sa publication, ce modèle est l'objet de +critiques de la part d'économistes et de financiers. En se référant à +\cite{mitchell1916critique}, on observe que, sur une base annuelle, +les variations négatives par rapport à la moyenne (149) sont plus +fréquentes que celles qui sont positives (126), pour un ensemble de 40 +titres boursiers, entre 1890 et 1915 (figure \ref{fig:mitchell1}). +Une asymétrie négative des rendements sera alors présente. +\begin{figure}[!ht] + \centering + \includegraphics[scale=0.75]{../varia/mitchell1.pdf} + \caption{Distribution des rendements annuels de 40 titres boursiers, + de 1890 à 1915, Table XVIII de \cite{mitchell1916critique}} + \label{fig:mitchell1} +\end{figure} + +De plus, les variations extrêmes sont plus fréquentes que ne pourrait +le prédire un modèle basé sur un mouvement brownien. La distribution +des rendements aurait donc des queues plus épaisses +\footnote{traduction de l'anglais heavy tailed} que la normale. On +doit trouver un modèle qui permet de tenir compte de ces +particularités. + +\subsection{Proposition de Mandelbrot} +\label{sec:mandelbrot} + + +\cite{mandelbrot1963variation} propose un modèle qui vise à combler +les lacunes du processus brownien géométrique +\eqref{eq:browniengeom}. Il explique que les distributions empiriques +des changements de prix sont habituellement trop \emph{pointues} pour +être considérées comme des échantillons d'une population gaussienne. + +Il identifie différentes caractéristiques qu'un bon modèle des +rendements financiers devrait posséder: + +\begin{enumerate} + \label{enum:mandelbrot} +\item Il doit tenir compte de la fréquence des grands changements de + prix. Il doit donc être basé sur une distribution leptocurtique, + plus pointue au centre que la normale. +\item Il doit permettre des changements instantanés et imprévisibles + de toute amplitude. +\item Il doit admettre une probabilité non nulle que plusieurs + changements consécutifs semblent corrélés. +\item Il doit admettre un processus de prix non stationnaire, car la + variance échantillonnale prend différentes valeurs à travers le + temps. +\end{enumerate} + +La famille de distributions L stable semble être celle qui répond le +mieux à l'ensemble de ces conditions \citep{walterlevy}. L'équation +suivante définit la propriété de L-stabilité de la distribution de la +variable aléatoire des rendements sur une période $R$: +\begin{align} + (a_1 R_1 + b_1) + (a_2 R_2 + b_2) &\stackrel{d}{=} aR + b \\ + \forall a_1,a_2 > 0, \forall b_1, b_2. +\end{align} + +La solution générale de cette équation a été découverte par Lévy en +1925. Le logarithme de la fonction caractéristique de celle-ci prend +la forme suivante: +\begin{align} + \ln{(\phi_{R}(\xi))} = i\delta \xi - \gamma |\xi|^{\alpha} + \left[1+\frac{i\beta \xi}{|\xi|} \tan{\frac{\alpha\pi}{2}} \right]. +\end{align} + +Le domaine et le rôle des paramètres de la distribution L stable est +décrit à la table \ref{tab:roleparam}. La flexibilité apportée par les +quatre paramètres permet de remplir les quatre conditions établies au +début de cette section. De plus, l'absence, dans la majorité des cas, +de moments finis d'ordre supérieur à l'espérance, permet de tenir +compte du mouvement erratique des prix et ainsi produire de larges +discontinuités de son processus. Elle permet aussi d'expliquer +l'apparence de corrélation sérielle, en considérant une probabilité +non négligeable que cette caractéristique soit présente. Cependant, ce +modèle est difficile à appliquer à l'évaluation de produits dérivés +pour cette raison, étant donné que l'on devra être en mesure de +quantifier la volatilité. +\begin{table}[!ht] + \centering + \begin{tabular}{|c|p{1.75cm}|p{2.5cm}|p{6.25cm}|} + \hline + \textbf{Paramètre} & \textbf{Domaine} & \textbf{Rôle} & \textbf{Observations} \\ + \hline + $\alpha$ & $\left]0,2\right]$ & Aplatissement & Plus sa valeur est petite, plus la distribution est leptocurtique. $\alpha=2$ correspond à la distribution normale. \\ + $\beta$ & $\left] -1, 1 \right]$ & Asymétrie & Défini seulement lorsque $\alpha \neq 1$. Lorsque $\alpha=1$ et $\beta=0$, on obtient la distribution de Cauchy. \\ + $\gamma = s^{\alpha}$ & $\mathbb{R}\setminus\{0 \}$ & Échelle & On doit prendre la racine $\alpha$ pour obtenir un paramètre d'échelle $s$ tel que défini par Pearson. \\ + $\delta$ & $\mathbb{R}$ & Localisation & \\ + \hline + \end{tabular} + \caption{Domaine et rôle des paramètres de la distribution L stable de Mandelbrot} + \label{tab:roleparam} +\end{table} + +L'approche classique, selon Mandelbrot, pour expliquer les grands +changements de prix a été de considérer un mélange de deux +distributions normales, dont une pour les fluctuations régulières et +une qui a une variance plus importante, pour les discontinuités. Il +remarque que pour expliquer adéquatement le comportement des données +empiriques, on doit introduire un mélange de plusieurs distributions +normales, ce qui rendrait le modèle plus complexe. Par contre, on +retrouve une approche intéressante avec le modèle présenté à la +section suivante. + +\subsection{Le modèle de Press} +\label{sec:press} + +\cite{press1967compound} propose un modèle statistique basé sur un +processus de Poisson composé auquel on ajoute un mouvement brownien +$W(t)$. C'est donc d'un processus ayant des incréments stationnaires +et indépendants. Il présente donc les caractéristiques d'un processus +de Lévy. Press utilise aussi la transformation logarithmique +\eqref{eq:browniengeom} afin que la variation soit proportionnelle au +prix. Il remarque aussi que le modèle logarithmique de Bachelier est +inadéquat, car il ne tient pas compte des queues de la distribution +empirique des rendements qui sont plus épaisses que celles de la +normale. Il ajoute que le modèle proposé par Mandelbrot est +discutable, car il ne trouve aucune évidence, à partir des données +observées, que la distribution de la population aurait une variance +infinie. + +Le processus de Poisson $\left\{N(t)\right\}$ de paramètre $\lambda t$ +est un processus de comptage qui détermine les occurrences des sauts +$Y_k, k = 1, \ldots, N(t)$. Ces sauts surviennent généralement +lorsqu'une information importante est rendue publique par rapport à un +titre. Ceux-ci sont aussi de distribution normale, mais leur espérance +n'est pas nulle et leur variance est différente de celle du processus +$W(t)$. Cette composante que l'on ajoute au modèle de Bachelier +modifié permet d'expliquer les variations plus importantes et moins +fréquentes observées empiriquement. + +Le processus du logarithme du prix $\left\{s(t)\right\} \equiv +\left\{\ln{(S(t))}\right\}$ est donc représenté par l'équation +suivante: +\begin{align} + \label{eq:press67} + s(t) &= s(0) + \sum_{k=1}^{N(t)} Y_k + W(t). +\end{align} + +On définit les différentes variables aléatoires composant le processus +comme suit: +\begin{align*} + Y_k &\sim N(\theta,\sigma_2^2) \\ + W(t) &\sim N(0,\sigma_1^2 t) \\ + N(t) &\sim Poisson(\lambda t) +\end{align*} + +Comme pour la plupart des processus de Lévy, on ne peut obtenir une +forme explicite pour la fonction de densité, car celle-ci se présente +sous la forme d'une série infinie. On représente alors ces processus +par leur fonction caractéristique, formée par le produit de celles de +leurs différentes composantes. + +La distribution du logarithme du prix $s(t)$ est définie par la +fonction caractéristique $\phi_{s(t)}(\xi)$, qui est le produit de +celle de la constante et celles des processus de Wiener et de Poisson +composé: +\begin{align} + \label{eq:fncaractpress} + \phi_{s(t)}\left(\xi\right) &= E\left[e^{i \xi s(t)} \right] \nonumber \\ + &= exp\left\{ i\xi \cdot s(0) \right\} \times exp \left\{ -\frac{t \sigma_1^2 \xi^2}{2} \right\} \times exp \left\{ \lambda t \left[e^{i \theta \xi-(\sigma_2^2 \xi^2/2)}-1 \right] \right\} \nonumber \\ + &= exp\left\{ i\xi \cdot s(0)- \frac{t}{2}\sigma_1^2\xi^2 + \lambda + t \left[e^{i \theta \xi-(\sigma_2^2 \xi^2/2)}-1 \right] \right\}. +\end{align} + +Afin d'estimer le modèle, on s'intéressera plutôt à la distribution +d'un incrément $\Delta s(t) = s(t)-s(t-1)$ de ce processus. La +fonction caractéristique $\phi_{\Delta s(t)}(\xi)$ de cette variable +aléatoire peut être facilement identifiée à partir de celle du +processus \eqref{eq:fncaractpress}. Essentiellement, on pose $s(0)=0 +\mbox{ et } t=1$, pour obtenir: +\begin{align} + \label{eq:fncaractpress2} + \phi_{\Delta s(t)}\left(\xi\right) &= E\left[e^{i\xi\Delta s(t)} \right] \nonumber \\ + &= exp\left\{-\frac{\sigma_1^2 \xi^2}{2} + \lambda \left[e^{i + \xi\theta -(\sigma_2^2 \xi^2/2)}-1 \right] \right\}. +\end{align} + +Pour estimer les paramètres du modèle, on privilégie la méthode des +cumulants, qui est similaire à la méthode des moments. Considérons les +quatre premiers cumulants de la distribution de l'incrément $\Delta +s(t)$: +\begin{subequations}\label{eq:cumulantspress} + \begin{align} + K_1 &= \lambda\theta \\ + K_2 &= \sigma_1^2+\lambda(\theta^2+\sigma_2^2) \\ + K_3 &= \lambda\theta(\theta^2+3\sigma_2^2) \\ + K_4 &= \lambda(\theta^4 + 6 \theta^2 \sigma_2^2 + 3 \sigma_2^4). + \end{align} +\end{subequations} + +En utilisant les quatre premiers cumulants empiriques +\eqref{eq:cumulantsempiriques}, on obtient les équations suivantes: +\begin{subequations}\label{eq:presscum} + \begin{align} + 0 &= \hat\theta^4 - \frac{\overline{K}_3}{\overline{K}_1} \hat\theta^2 + \frac{3\overline{K}_4}{2\overline{K}_1} \hat\theta - \frac{\overline{K}_3^2}{2\overline{K}_1^2} \label{eq:presscumtheta}\\ + \hat\lambda &= \frac{\overline{K}_1}{\hat\theta} \label{eq:presscumlambda}\\ + \hat\sigma_2^2 &= \frac{\overline{K}_3-\hat\theta^2\overline{K}_1}{3\overline{K}_1} \label{eq:presscumsigma2}\\ + \hat\sigma_1^2 &= \overline{K}_2 - + \frac{\overline{K}_1}{\hat\theta}\left(\hat\theta^2 + + \frac{\overline{K}_3 - \overline{K}_3 \theta^2}{3\overline{K}_1} + \right). \label{eq:presscumsigma1} + \end{align} +\end{subequations} + +En résolvant numériquement l'équation \eqref{eq:presscumtheta} pour le +$\hat\theta$, puis par substitutions successives dans les équations +\eqref{eq:presscum}, on obtient des estimateurs convergents pour les +quatre paramètres du modèle. + +Un modèle similaire a aussi été présenté par \cite{merton1976option}, +cependant, il inclut un paramètre de dérive $\alpha$, et considère que +les sauts $Y$, qui sont des facteurs multiplicatifs, peuvent suivre une autre distribution que la normale. Il +présente le modèle sous la forme d'une équation différentielle +stochastique: +\begin{align} + \label{eq:modelemerton} + \frac{dS}{S} = (\alpha - \lambda k)dt + \sigma dW + dq. +\end{align} +La constante $k$ représente l'espérance de la variation relative si un +saut se produit et $q$, le processus de Poisson composé. La solution +de cette équation est, selon le lemme d'Itô: +\begin{align} + S(t) &= \tilde{S}(0) \exp \left\{ + (\alpha-\frac{1}{2}\sigma^2-\lambda k)t + + \sigma W(t) \right\} + \end{align} + où + \begin{align} + \tilde{S}(0) &= \begin{cases} + S(0) & \text{si } N(t) = 0\\ + S(0) \sum_{k=1}^{N(t)} Y_k & \text{si } N(t) \geq 1. \nonumber + \end{cases} +\end{align} + +En spécifiant un paramètre de dérive $\delta = +\alpha-\frac{1}{2}\sigma^2-\lambda k$ et en considérant que les +sauts $Y$ sont de distribution lognormale, on peut réécrire la +fonction caractéristique d'un incrément \eqref{eq:fncaractpress2} du +modèle de Press: +\begin{align} + \label{eq:fncaractmerton} + \phi_{\Delta s(t)}\left(\xi\right) &= E\left[e^{i\xi\Delta s(t)} \right] \nonumber \\ + &= \exp\left\{i\delta \xi -\frac{\sigma^2 \xi^2}{2} + \lambda + \left[e^{i \xi\theta -(\sigma^2 \xi^2/2)}-1 \right] \right\}. +\end{align} + +L'utilisation de ce modèle présente deux désavantages. L'estimation du +modèle est difficile lorsque la moyenne s'approche de 0, car le +quotient \eqref{eq:presscumlambda} tend alors vers une +indétermination. De plus, contrairement à d'autres modèles, il est +difficile d'identifier le rôle des paramètres par rapport à un moment +en particulier (classification de Pearson), contrairement à ce qu'on +pourra observer avec la distribution de Laplace asymétrique +généralisée. + +\subsection{Le modèle de Praetz} + +\cite{praetz1972distribution} propose un modèle inspiré par la +physique des particules. Il pose comme hypothèse que deux intervalles +qui ne se chevauchent pas forment une marche aléatoire, et que les +éléments qui composent la séquence des rendements financiers $\left\{ + R(t) \right\}$ sont mutuellement indépendants. Il considère qu'un +état stable existe où les rendements suivent une loi normale de +paramètres $\mu$ et $\sigma^2$. + +Cependant, cet état stable n'est jamais réellement atteint, et la +fonction de densité empirique généralement observée suppose une +distribution symétrique concave, pointue au centre et ayant des queues +épaisses. Il fait une analogie entre la température d'un gaz et le +niveau d'activité sur les marchés, où la variance du mouvement +brownien est proportionnelle à ces deux quantités. Il propose que le +paramètre de variance de la normale $\sigma^2$ suive une distribution +$g(\sigma^2)$ ayant un support positif. La distribution conditionnelle +est normale lorsque ce paramètre est connu. +\begin{align} + h_{R(t)}(r) &= \int_0^{\infty} f_{R(t)}(r|\sigma^2) g(\sigma^2) d\sigma^2 \\ + f_{R(t)}(r|\sigma^2) &= \frac{1}{\sigma\sqrt{2\pi}}exp\left\{-\frac{(r-\mu)^2}{2\sigma^2} \right\} \label{eq:praetz72} +\end{align} + +Il propose comme solution acceptable pour la densité $g(\sigma^2)$, la +distribution gamma inverse de paramètres $m$ et $s^2$: +\begin{align} + \label{eq:gpraetz} + g(\sigma^2) &= + \frac{s^{2m}(m-1)^me^{-(m-1)\frac{s^2}{\sigma^2}}}{\sigma^{2(m-1)}\Gamma(m)}. +\end{align} + +Cette distribution a pour moyenne $s^2$ et variance $\frac{s^2}{m-2}$. +La distribution non conditionnelle des rendements $h_{R(t)}$ est +approximativement une Student avec $2m$ degrés de liberté à un facteur +d'échelle de $\left(\frac{m}{m-1}\right)^{1/2}$ près: +\begin{align} + \label{eq:hpraetz} + h_{R(t)}(r) &= \frac{\Gamma(m)\left[\ 2(m-1)\pi + \right]^{1/2}s}{\left[1+\frac{(y-\mu)^2}{s^2(2m-2)} + \right]^{m+1/2}}. +\end{align} + +D'autres distributions pourraient être utilisées au lieu de la gamma +inverse. En utilisant la loi gamma, on obtient la distribution de +Laplace asymétrique généralisée, qui sera l'objet d'une étude +approfondie aux chapitres suivants. Il propose enfin d'utiliser aussi +la distribution a priori gamma inverse pour le paramètre $\mu$. Par +contre, il remarque qu'il obtient aussi une distribution similaire à +celle de Student. Cette généralisation n'est donc pas nécessaire. + +\section{Conditions essentielles de Madan et Seneta} +\label{sec:madanseneta90} + +Inspirés par les travaux de Mandelbrot, Press et Praetz, +\cite{madan1990variance} présentent un ensemble de conditions +considérées essentielles dans l'élaboration d'un modèle de rendements +financiers. Ils se baseront sur celles-ci pour proposer le modèle +Variance Gamma: + +\begin{enumerate} +\item La distribution des rendements $R$ doit avoir une queue + épaisse. Ainsi, la probabilité que cette variable aléatoire ait une + valeur supérieure à $r+t$ avec un $t$ petit, sachant qu'elle est + supérieure à $r$, doit tendre vers 1, ce qui signifie que la + fonction de survie converge lorsque cette quantité est grande. + \begin{eqnarray} + \label{eq:condmadan1} + \lim_{r\rightarrow \infty} P\left[R > r+t | R > r \right] &=& 1 \\ + \bar{F}(r+t) &\sim& \bar{F}(r), \qquad r \rightarrow \infty \nonumber + \end{eqnarray} +\item La distribution doit posséder des moments finis pour les $n$ + premières puissances des rendements $R$. Étant donné que l'on + cherche à modéliser la queue de la distribution, on fixe $n=4$. + \begin{equation} + \label{eq:condmadan2} + E\left[R^k\right] < \infty, \qquad k \in \lbrace 1,2,3,4 \rbrace + \end{equation} +\item + \begin{enumerate} + \item Le modèle doit proposer un processus de temps continu ayant + des accroissements stationnaires et indépendants. + \item Les distributions des accroissements doivent appartenir à la + même famille, quelle que soit leur longueur. Cette condition est + essentielle afin de permettre l'échantillonnage et l'analyse des + séries chronologiques. + \end{enumerate} +\item Le modèle doit permettre une extension multivariée avec une + distribution elliptique afin de conserver la validité du modèle + d'évaluation des actifs financiers. +\end{enumerate} + +Chacun des modèles présentés précédemment respecte la majorité ou +toutes ces conditions. Les résultats se retrouvent à la table +\ref{tab:condmadan}. +\begin{table}[!ht] + \centering + \begin{tabular}{ccccc} + & \multicolumn{4}{c}{\textbf{Conditions}} \\ + \hline + \textbf{Modèles} & 1 & 2 & 3 & 4 \\ + \hline + Mouvement brownien de Bachelier & & $\ast$ & $\ast$ & $\ast$ \\ + Distribution stable symétrique de Mandelbrot & $\ast$ & & & $\ast$ \\ + Processus de Poisson composé de Press & $\ast$ & $\ast$ & $\ast$ & $\ast$ \\ + Mélange gaussien/inverse gamma de Praetz & $\ast$ & $\ast$ & & $\ast$ \\ + Modèle Variance Gamma de Madan et Seneta & $\ast$ & $\ast$ & $\ast$ & $\ast$ \\ + \hline + \end{tabular} + \caption{Respect des conditions émises par Madan et Seneta pour les différents modèles présentés} + \label{tab:condmadan} +\end{table} + +On remarque que le modèle de Press remplit toutes les conditions +émises par Madan et Seneta. Cependant, ils remarqueront que ce n'est +pas un processus de sauts, car il contient aussi une composante de +diffusion (Section \ref{sec:levykhintchine}), ce qui va à l'encontre +de l'intuition derrière la continuité de la trajectoire du prix. C'est +cette dernière observation qui les incitera à proposer le modèle +Variance Gamma, qui est un processus de sauts. Ce modèle, aussi étudié +sous le nom de distribution de Laplace asymétrique généralisée par +\cite{kotz2001laplace}, a acquis beaucoup de notoriété dans le domaine +de la finance mathématique. De plus, avec le développement de +l'informatique et des méthodes numériques, on peut maintenant utiliser +de manière efficace la fonction caractéristique dans le cadre de la +calibration, des tests statistiques et de la tarification +d'options. C'est pourquoi un intérêt particulier est apporté à cette +distribution dans ce texte. + +% \section{La volatilité} +% \label{sec:volatilite} + +% La \textbf{volatilité} est une mesure de l'ampleur des variations du +% prix d'un actif. Elle sert à quantifier le risque lié à un +% investissement, le plus souvent sur un horizon à court terme. Elle se +% calcule le plus souvent à partir des prix des options observés sur les +% marchés, on parle alors de volatilité implicite. Comme elle n'est pas +% mesurable, cette volatilité est le reflet de l'anticipation des +% investisseurs quant aux perspectives du marché. + +% Cependant, on peut toujours mesurer la volatilité historique du prix +% d'un titre à travers les rendements passés. La première étude à ce +% sujet a été faite par Black et Scholes, les auteurs du célèbre modèle +% qui porte désormais leur nom. Ils ont conclu que leur modèle +% surestimait le prix des options pour des actifs sous-jacents ayant une +% volatilité historique élevée, le contraire se produisait lorsqu'elle +% l'était peu. Leur modèle est donc utile à condition que les +% investisseurs puissent faire de bonnes prévisions +% \citep{musiela2005martingale}. + +% De plus, comme il a été expliqué précédemment, les données historiques +% démontrent que la volatilité n'est pas constante avec le temps, mais +% plutôt aléatoire. Dans cette perspective, on pourrait identifier la +% distribution de la volatilité à travers le temps. +% \subsection{Mesure de la volatilité historique} +% \label{sec:mesurevolatilite} + +% Afin d'obtenir un ensemble d'observations de la volatilité historique, +% on utilise une approche par fenêtre mobile. \cite{randal2004non} +% considère un estimateur de variance mobile $\hat\sigma^2(t)$, basé sur +% les rendements centrés $R(t) = Y(t)-E\left[Y(t)\right]$ de la forme +% \begin{align} +% \hat\sigma^2(t) = \frac{1}{2r+1} \sum_{j=-r}^r R(t+j)^2, \qquad +% t\in\left[r+1,n-r\right] \label{mobilevariance} +% \end{align} + +% Étant donné la taille limitée $n$ de l'échantillon des rendements, on +% doit faire un compromis entre la précision des observations de la +% volatilité et le nombre $n-2r$ de celles-ci. L'hypothèse de volatilité +% stochastique fera pencher en faveur d'une fenêtre étroite, ce qui +% procurera un grand nombre d'observations la décrivant à court +% terme. On pourra donc ajuster une distribution de probabilités à +% celles-ci. + +% \subsection{Biais de la volatilité implicite} +% \label{sec:impvolsmile} + +% Le biais de volatilité implicite est un concept qui explique pourquoi +% la volatilité des options croît lorsque le prix d'exercice s'éloigne +% de la valeur actuelle du titre sous-jacent. Selon +% \cite{hull1999options}, ce phénomène a été remarqué sur les marchés +% financiers américains à partir du krach boursier du lundi noir +% \footnote{19 octobre 1987}, et n'est toujours pas entièrement +% expliqué. Généralement, on observe que, pour les options sur indices +% boursiers et taux de change, la courbe de volatilité implicite est +% plutôt symétrique, alors qu'elle est asymétrique pour celles sur +% actions (voir figure \ref{fig:volimplicite} pour un exemple). +% \begin{figure}[!ht] +% \centering +% \includegraphics[]{./bbry-echeance-06-2013-20mai2013.png} +% \caption{Courbe de volatilité implicite, titre BBRY, option d'achat +% avec échéance 06-2013, observée le 20-05-2013, prix de 15.83, +% source: \cite{thevolskew}} +% \label{fig:volimplicite} +% \end{figure} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/chapitre2.aux b/memoire/chapitre2.aux new file mode 100644 index 0000000..43cc892 --- /dev/null +++ b/memoire/chapitre2.aux @@ -0,0 +1,259 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{kozubowski1999class} +\citation{madan1990variance} +\citation{madan1990variance} +\citation{kozubowski1999class} +\citation{kotz2001laplace} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {chapter}{\chapternumberline {2}La distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{19}{chapter.2}} +\@writefile{toc}{\contentsline {section}{\numberline {2.1}Le processus de Laplace}{19}{section.2.1}} +\newlabel{sec:processusGAL}{{2.1}{19}{Le processus de Laplace}{section.2.1}{}} +\newlabel{sec:processusGAL@cref}{{[section][1][2]2.1}{19}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.1.1}Le processus gamma}{20}{subsection.2.1.1}} +\newlabel{sec:processusgamma}{{2.1.1}{20}{Le processus gamma}{subsection.2.1.1}{}} +\newlabel{sec:processusgamma@cref}{{[subsection][1][2,1]2.1.1}{20}} +\newlabel{eq:densitegamma}{{2.1.1}{20}{Le processus gamma}{equation.2.1.0.1}{}} +\newlabel{eq:densitegamma@cref}{{[equation][1][2,1]2.1.1}{20}} +\newlabel{eq:fncaractgamma}{{2.1.2}{20}{Le processus gamma}{equation.2.1.0.2}{}} +\newlabel{eq:fncaractgamma@cref}{{[equation][2][2,1]2.1.2}{20}} +\newlabel{eq:densitegamma1}{{2.1.3}{20}{Le processus gamma}{equation.2.1.0.3}{}} +\newlabel{eq:densitegamma1@cref}{{[equation][3][2,1]2.1.3}{20}} +\citation{spiegel1999schaum} +\citation{spiegel1999schaum} +\newlabel{eq:exposantchargamma}{{2.1.5}{21}{Le processus gamma}{equation.2.1.0.5}{}} +\newlabel{eq:exposantchargamma@cref}{{[equation][5][2,1]2.1.5}{21}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.1.2}Le processus de Wiener}{21}{subsection.2.1.2}} +\newlabel{sec:mouvementbrownien}{{2.1.2}{21}{Le processus de Wiener}{subsection.2.1.2}{}} +\newlabel{sec:mouvementbrownien@cref}{{[subsection][2][2,1]2.1.2}{21}} +\newlabel{eq:incrwiener}{{2.1.7}{21}{Le processus de Wiener}{equation.2.1.0.7}{}} +\newlabel{eq:incrwiener@cref}{{[equation][7][2,1]2.1.7}{21}} +\newlabel{eq:fndensitenormale}{{2.1.8}{21}{Le processus de Wiener}{equation.2.1.0.8}{}} +\newlabel{eq:fndensitenormale@cref}{{[equation][8][2,1]2.1.8}{21}} +\newlabel{eq:fncaractnormale}{{2.1.9}{21}{Le processus de Wiener}{equation.2.1.0.9}{}} +\newlabel{eq:fncaractnormale@cref}{{[equation][9][2,1]2.1.9}{21}} +\newlabel{eq:exposantcaractnormale}{{2.1.12}{22}{Le processus de Wiener}{equation.2.1.0.12}{}} +\newlabel{eq:exposantcaractnormale@cref}{{[equation][12][2,1]2.1.12}{22}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.1.3}Le processus de Laplace est un processus subordonn\IeC {\'e}}{22}{subsection.2.1.3}} +\newlabel{sec:browniensub}{{2.1.3}{22}{Le processus de Laplace est un processus subordonné}{subsection.2.1.3}{}} +\newlabel{sec:browniensub@cref}{{[subsection][3][2,1]2.1.3}{22}} +\newlabel{eq:VGsubordinne}{{2.1.13}{22}{Le processus de Laplace est un processus subordonné}{equation.2.1.0.13}{}} +\newlabel{eq:VGsubordinne@cref}{{[equation][13][2,1]2.1.13}{22}} +\newlabel{eq:exposantcaractLaplace}{{2.1.14}{22}{Le processus de Laplace est un processus subordonné}{equation.2.1.0.14}{}} +\newlabel{eq:exposantcaractLaplace@cref}{{[equation][14][2,1]2.1.14}{22}} +\newlabel{eq:fonctioncaractlaplacesansdrift}{{2.1.15}{23}{Le processus de Laplace est un processus subordonné}{equation.2.1.0.15}{}} +\newlabel{eq:fonctioncaractlaplacesansdrift@cref}{{[equation][15][2,1]2.1.15}{23}} +\@writefile{lof}{\contentsline {figure}{\numberline {2.1}{\ignorespaces Premier incr\IeC {\'e}ment d'un processus subordonn\IeC {\'e}\relax }}{23}{figure.caption.16}} +\newlabel{fig:increment1}{{2.1}{23}{Premier incrément d'un processus subordonné\relax }{figure.caption.16}{}} +\newlabel{fig:increment1@cref}{{[figure][1][2]2.1}{23}} +\@writefile{lof}{\contentsline {figure}{\numberline {2.2}{\ignorespaces Simulation d'un processus de Wiener subordonn\IeC {\'e} par un processus gamma\relax }}{24}{figure.caption.17}} +\newlabel{fig:simgammagauss}{{2.2}{24}{Simulation d'un processus de Wiener subordonné par un processus gamma\relax }{figure.caption.17}{}} +\newlabel{fig:simgammagauss@cref}{{[figure][2][2]2.2}{24}} +\citation{kotz2001laplace} +\citation{madan1990variance} +\newlabel{eq:fncaractprocessuslaplace}{{2.1.16}{25}{Le processus de Laplace est un processus subordonné}{equation.2.1.0.16}{}} +\newlabel{eq:fncaractprocessuslaplace@cref}{{[equation][16][2,1]2.1.16}{25}} +\@writefile{toc}{\contentsline {section}{\numberline {2.2}Distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{25}{section.2.2}} +\newlabel{sec:distributionGAL}{{2.2}{25}{Distribution de Laplace asymétrique généralisée}{section.2.2}{}} +\newlabel{sec:distributionGAL@cref}{{[section][2][2]2.2}{25}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.1}Fonction caract\IeC {\'e}ristique}{25}{subsection.2.2.1}} +\newlabel{sec:fncaractGAL}{{2.2.1}{25}{Fonction caractéristique}{subsection.2.2.1}{}} +\newlabel{sec:fncaractGAL@cref}{{[subsection][1][2,2]2.2.1}{25}} +\newlabel{eq:defvarY-GAL}{{2.2.1}{26}{Fonction caractéristique}{equation.2.2.0.1}{}} +\newlabel{eq:defvarY-GAL@cref}{{[equation][1][2,2]2.2.1}{26}} +\newlabel{eq:Ynormalconditionnel}{{2.2.2}{26}{Fonction caractéristique}{equation.2.2.0.2}{}} +\newlabel{eq:Ynormalconditionnel@cref}{{[equation][2][2,2]2.2.2}{26}} +\newlabel{eq:fncaractGALmu}{{2.2.3}{26}{Fonction caractéristique}{equation.2.2.0.3}{}} +\newlabel{eq:fncaractGALmu@cref}{{[equation][3][2,2]2.2.3}{26}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.2}Invariance d'\IeC {\'e}chelle}{26}{subsection.2.2.2}} +\newlabel{sec:invariance-dechelle}{{2.2.2}{26}{Invariance d'échelle}{subsection.2.2.2}{}} +\newlabel{sec:invariance-dechelle@cref}{{[subsection][2][2,2]2.2.2}{26}} +\newlabel{eq:mukappa}{{2.2.4}{27}{Invariance d'échelle}{equation.2.2.0.4}{}} +\newlabel{eq:mukappa@cref}{{[equation][4][2,2]2.2.4}{27}} +\newlabel{eq:kappamu}{{2.2.5}{27}{Invariance d'échelle}{equation.2.2.0.5}{}} +\newlabel{eq:kappamu@cref}{{[equation][5][2,2]2.2.5}{27}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.3}Fonctions g\IeC {\'e}n\IeC {\'e}ratrices}{28}{subsection.2.2.3}} +\newlabel{eq:fgmGAL}{{2.2.10}{28}{Fonctions génératrices}{equation.2.2.0.10}{}} +\newlabel{eq:fgmGAL@cref}{{[equation][10][2,2]2.2.10}{28}} +\newlabel{eq:fgmGALcond}{{2.2.11}{28}{Fonctions génératrices}{equation.2.2.0.11}{}} +\newlabel{eq:fgmGALcond@cref}{{[equation][11][2,2]2.2.11}{28}} +\newlabel{eq:fgmmomentsGAL}{{2.2.12}{28}{Fonctions génératrices}{equation.2.2.0.12}{}} +\newlabel{eq:fgmmomentsGAL@cref}{{[equation][12][2,2]2.2.12}{28}} +\newlabel{eq:fgcGAL}{{2.2.13}{28}{Fonctions génératrices}{equation.2.2.0.13}{}} +\newlabel{eq:fgcGAL@cref}{{[equation][13][2,2]2.2.13}{28}} +\newlabel{eq:fgccumGAL}{{2.2.14}{28}{Fonctions génératrices}{equation.2.2.0.14}{}} +\newlabel{eq:fgccumGAL@cref}{{[equation][14][2,2]2.2.14}{28}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.4}Moments et r\IeC {\^o}le des param\IeC {\`e}tres}{28}{subsection.2.2.4}} +\newlabel{sec:momentsGAL}{{2.2.4}{28}{Moments et rôle des paramètres}{subsection.2.2.4}{}} +\newlabel{sec:momentsGAL@cref}{{[subsection][4][2,2]2.2.4}{28}} +\newlabel{eq:momentsGAL}{{2.2.15}{29}{Moments et rôle des paramètres}{equation.2.2.0.15.15}{}} +\newlabel{eq:momentsGAL@cref}{{[equation][15][2,2]2.2.15}{29}} +\newlabel{eq:moments1GAL}{{2.2.15a}{29}{Moments et rôle des paramètres}{equation.2.2.0.15a}{}} +\newlabel{eq:moments1GAL@cref}{{[equation][1][2,2,15]2.2.15a}{29}} +\newlabel{eq:moments2GAL}{{2.2.15b}{29}{Moments et rôle des paramètres}{equation.2.2.0.15b}{}} +\newlabel{eq:moments2GAL@cref}{{[equation][2][2,2,15]2.2.15b}{29}} +\newlabel{eq:moments3GAL}{{2.2.15c}{29}{Moments et rôle des paramètres}{equation.2.2.0.15c}{}} +\newlabel{eq:moments3GAL@cref}{{[equation][3][2,2,15]2.2.15c}{29}} +\newlabel{eq:moments4GAL}{{2.2.15d}{29}{Moments et rôle des paramètres}{equation.2.2.0.15d}{}} +\newlabel{eq:moments4GAL@cref}{{[equation][4][2,2,15]2.2.15d}{29}} +\newlabel{eq:moments56GAL}{{2.2.16}{29}{Moments et rôle des paramètres}{equation.2.2.0.16.16}{}} +\newlabel{eq:moments56GAL@cref}{{[equation][16][2,2]2.2.16}{29}} +\newlabel{eq:moments5GAL}{{2.2.16a}{29}{Moments et rôle des paramètres}{equation.2.2.0.16a}{}} +\newlabel{eq:moments5GAL@cref}{{[equation][1][2,2,16]2.2.16a}{29}} +\newlabel{eq:moments6GAL}{{2.2.16b}{29}{Moments et rôle des paramètres}{equation.2.2.0.16b}{}} +\newlabel{eq:moments6GAL@cref}{{[equation][2][2,2,16]2.2.16b}{29}} +\citation{madan1990variance} +\@writefile{lot}{\contentsline {table}{\numberline {2.1}{\ignorespaces Domaine et r\IeC {\^o}le des param\IeC {\`e}tres de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{30}{table.caption.18}} +\newlabel{tab:roleparamGAL}{{2.1}{30}{Domaine et rôle des paramètres de la distribution de Laplace asymétrique généralisée\relax }{table.caption.18}{}} +\newlabel{tab:roleparamGAL@cref}{{[table][1][2]2.1}{30}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.5}Changement d'\IeC {\'e}chelle et de localisation}{30}{subsection.2.2.5}} +\newlabel{sec:transGAL}{{2.2.5}{30}{Changement d'échelle et de localisation}{subsection.2.2.5}{}} +\newlabel{sec:transGAL@cref}{{[subsection][5][2,2]2.2.5}{30}} +\citation{kotz2001laplace} +\@writefile{lof}{\contentsline {figure}{\numberline {2.3}{\ignorespaces Fonction de densit\IeC {\'e} de la distribution Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e avec diff\IeC {\'e}rents param\IeC {\`e}tres: $GAL(y;\theta ,\sigma ,\kappa ,\tau )$\relax }}{31}{figure.caption.19}} +\newlabel{fig:densiteGAL}{{2.3}{31}{Fonction de densité de la distribution Laplace asymétrique généralisée avec différents paramètres: $GAL(y;\theta ,\sigma ,\kappa ,\tau )$\relax }{figure.caption.19}{}} +\newlabel{fig:densiteGAL@cref}{{[figure][3][2]2.3}{31}} +\newlabel{eq:GALscaletrans}{{2.2.17}{32}{Changement d'échelle et de localisation}{equation.2.2.0.17}{}} +\newlabel{eq:GALscaletrans@cref}{{[equation][17][2,2]2.2.17}{32}} +\newlabel{eq:transparamGALNS}{{2.2.18}{32}{Changement d'échelle et de localisation}{equation.2.2.0.18}{}} +\newlabel{eq:transparamGALNS@cref}{{[equation][18][2,2]2.2.18}{32}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.6}Repr\IeC {\'e}sentation alternative et simulation}{32}{subsection.2.2.6}} +\newlabel{sec:simulationGAL}{{2.2.6}{32}{Représentation alternative et simulation}{subsection.2.2.6}{}} +\newlabel{sec:simulationGAL@cref}{{[subsection][6][2,2]2.2.6}{32}} +\newlabel{eq:distributiongammaformealt}{{2.2.19}{32}{Représentation alternative et simulation}{equation.2.2.0.19}{}} +\newlabel{eq:distributiongammaformealt@cref}{{[equation][19][2,2]2.2.19}{32}} +\newlabel{eq:processuslaplace2gamma}{{2.2.20}{32}{Représentation alternative et simulation}{equation.2.2.0.20}{}} +\newlabel{eq:processuslaplace2gamma@cref}{{[equation][20][2,2]2.2.20}{32}} +\newlabel{eq:differencegamma}{{2.2.21}{32}{Représentation alternative et simulation}{equation.2.2.0.21}{}} +\newlabel{eq:differencegamma@cref}{{[equation][21][2,2]2.2.21}{32}} +\newlabel{eq:differencegamma2}{{2.2.22}{32}{Représentation alternative et simulation}{equation.2.2.0.22}{}} +\newlabel{eq:differencegamma2@cref}{{[equation][22][2,2]2.2.22}{32}} +\citation{abramowitz1965handbook} +\citation{kozubowski1999class} +\citation{kotz2001laplace} +\newlabel{eq:equivalence2gammaFC}{{2.2.23}{33}{Représentation alternative et simulation}{equation.2.2.0.23}{}} +\newlabel{eq:equivalence2gammaFC@cref}{{[equation][23][2,2]2.2.23}{33}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.7}Fonction de Bessel et densit\IeC {\'e}}{33}{subsection.2.2.7}} +\newlabel{sec:besseldensiteGAL}{{2.2.7}{33}{Fonction de Bessel et densité}{subsection.2.2.7}{}} +\newlabel{sec:besseldensiteGAL@cref}{{[subsection][7][2,2]2.2.7}{33}} +\newlabel{eq:BesselK}{{2.2.24}{33}{Fonction de Bessel et densité}{equation.2.2.0.24}{}} +\newlabel{eq:BesselK@cref}{{[equation][24][2,2]2.2.24}{33}} +\newlabel{eq:densitekotz}{{2.2.25}{33}{Fonction de Bessel et densité}{equation.2.2.0.25}{}} +\newlabel{eq:densitekotz@cref}{{[equation][25][2,2]2.2.25}{33}} +\citation{RpackageVarianceGamma} +\@writefile{lof}{\contentsline {figure}{\numberline {2.4}{\ignorespaces Histogramme et estimateur de densit\IeC {\'e} par noyau de 2500 r\IeC {\'e}alisations de la variable al\IeC {\'e}atoire $Y\sim GAL(\theta =0,\sigma =1,\kappa =2,\tau =1)$\relax }}{34}{figure.caption.20}} +\newlabel{fig:simulGAL}{{2.4}{34}{Histogramme et estimateur de densité par noyau de 2500 réalisations de la variable aléatoire $Y\sim GAL(\theta =0,\sigma =1,\kappa =2,\tau =1)$\relax }{figure.caption.20}{}} +\newlabel{fig:simulGAL@cref}{{[figure][4][2]2.4}{34}} +\newlabel{eq:densitekotz2001}{{2.2.26}{34}{Fonction de Bessel et densité}{equation.2.2.0.26}{}} +\newlabel{eq:densitekotz2001@cref}{{[equation][26][2,2]2.2.26}{34}} +\citation{hinkley1977estimation} +\citation{kozubowski1999class} +\citation{kozubowski2001asymmetric} +\@writefile{toc}{\contentsline {section}{\numberline {2.3}Cas particuliers}{35}{section.2.3}} +\newlabel{sec:cas-particuliers}{{2.3}{35}{Cas particuliers}{section.2.3}{}} +\newlabel{sec:cas-particuliers@cref}{{[section][3][2]2.3}{35}} +\@writefile{lot}{\contentsline {table}{\numberline {2.2}{\ignorespaces Cas sp\IeC {\'e}ciaux de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{35}{table.caption.21}} +\newlabel{tab:casspeciauxGAL}{{2.2}{35}{Cas spéciaux de la distribution de Laplace asymétrique généralisée\relax }{table.caption.21}{}} +\newlabel{tab:casspeciauxGAL@cref}{{[table][2][2]2.2}{35}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.3.1}Distribution de Laplace asym\IeC {\'e}trique}{35}{subsection.2.3.1}} +\newlabel{sec:distributionAL}{{2.3.1}{35}{Distribution de Laplace asymétrique}{subsection.2.3.1}{}} +\newlabel{sec:distributionAL@cref}{{[subsection][1][2,3]2.3.1}{35}} +\newlabel{eq:densiteAL}{{2.3.1}{36}{Distribution de Laplace asymétrique}{equation.2.3.0.1}{}} +\newlabel{eq:densiteAL@cref}{{[equation][1][2,3]2.3.1}{36}} +\newlabel{eq:caractALkappa}{{2.3.2}{36}{Distribution de Laplace asymétrique}{equation.2.3.0.2}{}} +\newlabel{eq:caractALkappa@cref}{{[equation][2][2,3]2.3.2}{36}} +\newlabel{eq:caractALmu}{{2.3.3}{36}{Distribution de Laplace asymétrique}{equation.2.3.0.3}{}} +\newlabel{eq:caractALmu@cref}{{[equation][3][2,3]2.3.3}{36}} +\newlabel{eq:ALrepsimul}{{2.3.4}{36}{Distribution de Laplace asymétrique}{equation.2.3.0.4}{}} +\newlabel{eq:ALrepsimul@cref}{{[equation][4][2,3]2.3.4}{36}} +\newlabel{eq:simulALunif}{{2.3.5}{36}{Distribution de Laplace asymétrique}{equation.2.3.0.5}{}} +\newlabel{eq:simulALunif@cref}{{[equation][5][2,3]2.3.5}{36}} +\citation{madan1990variance} +\citation{madan1990variance} +\citation{kotz2001laplace} +\citation{madan1990variance} +\citation{kotz2001laplace} +\@writefile{toc}{\contentsline {section}{\numberline {2.4}Relation avec le mod\IeC {\`e}le de \cite {madan1990variance}}{37}{section.2.4}} +\newlabel{sec:relation-avec-le}{{2.4}{37}{Relation avec le modèle de \cite {madan1990variance}}{section.2.4}{}} +\newlabel{sec:relation-avec-le@cref}{{[section][4][2]2.4}{37}} +\@writefile{lot}{\contentsline {table}{\numberline {2.3}{\ignorespaces Changements de param\IeC {\'e}trisation\relax }}{37}{table.caption.22}} +\newlabel{tab:chparam}{{2.3}{37}{Changements de paramétrisation\relax }{table.caption.22}{}} +\newlabel{tab:chparam@cref}{{[table][3][2]2.3}{37}} +\@setckpt{chapitre2}{ +\setcounter{page}{38} +\setcounter{equation}{0} +\setcounter{enumi}{3} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{2} +\setcounter{section}{4} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{56} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{4} +\setcounter{lofdepth}{1} +\setcounter{table}{3} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{16} +\setcounter{Hfootnote}{1} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{37} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/chapitre2.tex b/memoire/chapitre2.tex new file mode 100644 index 0000000..6ddf8e2 --- /dev/null +++ b/memoire/chapitre2.tex @@ -0,0 +1,979 @@ +\chapter{La distribution de Laplace asymétrique + généralisée} % numéroté + +Dans ce chapitre, on présente, en premier lieu, le processus de +Laplace ainsi que les deux processus sous-jacents à sa construction, +le processus gamma et le processus de Wiener. Ensuite, on présente la +distribution de Laplace asymétrique généralisée et ses principales +propriétés qui seront utilisées pour modéliser les rendements de +titres financiers. Puis, on présente quelques cas particuliers. Dans +le chapitre suivant, on présentera différentes méthodes pour obtenir +une approximation de la fonction de densité et la fonction de +répartition. + +La distribution de Laplace asymétrique généralisée a été +principalement étudiée par \cite{kozubowski1999class}. Cependant, elle +a été introduite près d'une décennie auparavant par +\cite{madan1990variance}, sous le nom de distribution Variance +Gamma. La différence entre les approches des deux auteurs est +majeure. \cite{madan1990variance} développent un modèle financier à +partir du mouvement brownien géométrique, qu'ils généralisent en +proposant que la variance suive une distribution +gamma. \cite{kozubowski1999class} généralisent la distribution de +Laplace asymétrique. Leur approche est plus générale, car ils ne +cherchent pas à développer un modèle financier, mais une nouvelle +classe de distributions utilisable dans divers domaines +scientifiques. Étant donné leur approche plus détaillée et plus +intuitive, c'est leur formulation du modèle qui sera développée. On +rappellera enfin que les deux modèles sont équivalents même si leurs +paramétrisations sont différentes. + +\section{Le processus de Laplace} +\label{sec:processusGAL} + +Le processus de Laplace est défini comme étant un processus de Wiener +subordonné par un processus gamma. En d'autres termes, c'est un +processus de Wiener évalué à des temps aléatoires déterminés par un +processus gamma. Selon \cite{kotz2001laplace}, ce dernier est à la +distribution de Laplace ce que le mouvement brownien est à la loi +normale. Il est aussi un cas particulier des processus de Lévy, et en +conserve donc la principale propriété, celle d'être infiniment +divisible. + +Il a certains points en commun avec le mouvement brownien dont des +moments finis pour tout ordre et des incréments indépendants et +stationnaires. Cependant, la plupart des caractéristiques diffèrent: +\begin{itemize} +\item Discontinuité des trajectoires (processus de sauts); +\item Distribution asymétrique des accroissements; +\item Paramètre d'échelle et de temps entièrement dissociés. +\end{itemize} + +Enfin, il possède une représentation alternative qui n'implique aucun +processus de Wiener. Il peut en fait être représenté comme la +différence de deux processus gamma indépendants. On peut le +représenter en utilisant la forme générale des processus de Lévy. + +\subsection{Le processus gamma} +\label{sec:processusgamma} + +Le processus gamma, noté $\left\{G(t;\tau,\beta)\right\}$, est un +processus de sauts purs (donc aucune composante de dérive ni de +diffusion) dont les incréments $G(t+1;\tau,\beta) - G(t;\tau,\beta)$ +suivent une distribution gamma de paramètres de forme $\tau$ et +d'échelle $\beta$, définie par les fonctions de densité +$f_{\tau,\beta}(x)$ et caractéristique $\phi_{\tau,\beta}(\xi)$: +\begin{align} + f_{\tau,\beta}(x) &= \frac{\beta^\tau}{\Gamma(\tau)} x^{\tau \,-\, 1} e^{- \beta x } 1_{\lbrace x\geq\,0 \rbrace} \label{eq:densitegamma} \\ + \phi_{\tau,\beta}(\xi) &= E\left[e^{i\xi\,X} \right] \nonumber\\ + &= \int_{0}^{\infty} e^{i\xi\,x} f_{\tau,\beta}(x) dx \nonumber\\ + &= + \frac{1}{\left(1-\frac{i\xi}{\beta}\right)^{\tau}} \label{eq:fncaractgamma}. +\end{align} + +On s'intéresse à la situation où le paramètre d'échelle est de valeur +unitaire ($\beta=1$). Le processus gamma agit alors à titre de +compteur et sa valeur $G(t;\tau,\beta=1)$ au temps $t$ correspondra au +nombre de sauts depuis $t=0$. La fonction de densité +$G(t+1;\tau,\beta=1) - G(t;\tau,\beta=1)$ sera alors: +\begin{align} + f_{\tau,\beta=1}(x) &= \frac{1}{\Gamma(\tau)} x^{\tau \,-\, 1} e^{- x} 1_{\lbrace x\geq\,0 \rbrace}. \label{eq:densitegamma1} +\end{align} + +Le paramètre $\tau$, qui définit la forme de la distribution, +déterminera la fréquence moyenne des sauts du processus gamma +$\Gamma(t;\tau,\beta=1)$, étant donné l'espérance $E[G(t)] = \tau\cdot +t$. La fonction caractéristique de ce processus sera donc +$\phi(\xi,t;\tau,\beta=1)$, en utilisant la propriété de convolution +\eqref{eq:convocaractIID} (même si le temps $t$ n'est pas entier, car +la distribution est infiniment divisible): +\begin{align} + \phi(\xi,t;\tau,\beta=1) &= \left[\frac{1}{\left(1-\frac{i\xi}{1}\right)^{\tau}}\right]^t \nonumber\\ + &= \frac{1}{\left(1-i\xi\right)^{\tau\cdot t}}. +\end{align} + +On peut réécrire la fonction caractéristique d'un incrément de ce +processus $\phi(\xi;t=1;\tau,\beta=1)$ sous la représentation de +Lévy-Khintchine \eqref{eq:levykhintchine}, avec l'exposant +caractéristique $\Xi(\zeta)$: +\begin{align} + \label{eq:exposantchargamma} + \Xi(\zeta;t=1;\tau,\beta=1) &= \tau\ln{\left(1-i\zeta\right)} \\ + &= \tau \left(e^{0} - e^{-\infty}\right) \ln{\left(1-i\zeta \right)} \nonumber\\ + &= \tau \int_{0}^{\infty} \frac{e^{-x} - e^{-(1-i\zeta)x}}{x} dx \nonumber\\ + &\qquad\mbox{(intégrale de Frullani \citep{spiegel1999schaum}, p.115)} \nonumber\\ + &= \tau \int_{0}^{\infty} + \left(1-e^{i\zeta\,x}\right)\frac{1}{x}e^{-x}dx. +\end{align} + +On a donc, par cette représentation, la démonstration que le processus +gamma est un processus de sauts purs. Il pourra donc être utilisé +comme subordonnant dans la construction d'un processus subordonné +\eqref{eq:processussubordonne}. + +\subsection{Le processus de Wiener} +\label{sec:mouvementbrownien} + +Le processus de Wiener $\left\{W(t;\mu,\sigma^2)\right\}$ est un +processus de diffusion avec dérive. Il n'a donc pas de composante de +saut. Ses incréments suivent une distribution normale: +\begin{align} + \label{eq:incrwiener} + W(t+1;\mu,\sigma^2) - W(t;\mu,\sigma^2) \sim N(\mu,\sigma^2). +\end{align} +Cette distribution est définie par la fonction de densité +$f_{\mu,\sigma}(x)$ et la fonction caractéristique +$\phi_{\mu,\sigma}(\xi)$: +\begin{align} + f_{\mu,\sigma}(x) &= \frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left\{\frac{1}{2} \left(\frac{x-\mu}{\sigma} \right)^2\right\}} \label{eq:fndensitenormale} \\ + \phi_{\mu,\sigma}(\xi) &= \exp\left\{ + i\mu\xi-\frac{\sigma^2\xi^2}{2} + \right\} \label{eq:fncaractnormale}. +\end{align} + +Notons que la variance d'un incrément est proportionnelle à la +longueur de celui-ci. Soit deux incréments indépendants d'un même +processus: $I_1 = W(t+q;\mu,\sigma^2) - W(t;\mu,\sigma^2) \sim +N(q\mu,q\sigma^2) \mbox{ et } I_2 = W(t+q+s;\mu,\sigma^2) - +W(t+q;\mu,\sigma^2) \sim N(s\mu,s\sigma^2)$. La somme de ces +incréments suit une distribution normale dont la moyenne et la +variance sont respectivement la somme de celles des deux incréments: +\begin{align} + I_1+I_2 \sim N((q+s)\mu, (q+s)\sigma^2). +\end{align} + +Comme la distribution normale est aussi infiniment divisible, on peut +obtenir la fonction caractéristique du processus +$\phi(\xi;t;\mu,\sigma^2)$ en utilisant la propriété de convolution +\eqref{eq:convocaractIID}: +\begin{align} + \phi(\xi;t;\mu,\sigma^2) = \exp\left\{ i\mu t + \xi-\frac{\sigma^2t\xi^2}{2} \right\}. +\end{align} + +On déduit donc facilement l'exposant caractéristique +$\Lambda(\xi;t=1;\mu,\sigma^2)$ d'un incrément de ce processus, sous +la représentation de Lévy-Khintchine: +\begin{align} + \label{eq:exposantcaractnormale} + \Lambda(\xi;t=1;\mu,\sigma^2) = -(i\mu \xi-\frac{\sigma^2\xi^2}{2}). +\end{align} + +Ceci démontre que le processus de Wiener est un processus avec dérive +et diffusion, mais sans composante de saut. Il pourra donc être +utilisé pour construire un processus subordonné +\eqref{eq:processussubordonne}. + +\subsection{Le processus de Laplace est un processus subordonné} +\label{sec:browniensub} + +On considère un processus gamma $G(t;\tau,\beta=1)$ et un processus de +Wiener $W(t;\mu,\sigma^2)$. On se rappelle que la variance d'un +incrément \eqref{eq:incrwiener} de ce dernier est proportionnelle à la +longueur de l'intervalle de temps. En utilisant une propriété appelée +la subordination, on peut modifier l'échelle de temps du processus de +Wiener de sorte que la variance soit aléatoire pour tout intervalle +. Tout processus de Lévy peut être utilisé comme subordonnant pour +définir cette échelle de temps. Si on utilise le processus gamma, on +obtiendra le processus de Laplace sans dérive +$\left\{Y(t;\sigma,\mu,\tau)\right\}$ défini comme suit: +\begin{align} + \label{eq:VGsubordinne} + \lbrace Y(t;\sigma,\mu,\tau)\rbrace &\equiv \lbrace + W(G(t;\tau,\beta=1);\mu,\sigma^2)\rbrace. +\end{align} + +On obtient l'exposant caractéristique $\Psi(\xi,t=1;\sigma,\mu,\tau)$ +d'un incrément $Y(t+1;\sigma,\mu,\tau)-Y(t;\sigma,\mu,\tau)$ en +utilisant la propriété de subordination définie par l'équation +\eqref{eq:exposantcaractYt}, où $\Xi(\zeta,t=1;\tau,\beta=1)$ est +l'exposant caractéristique du processus gamma et +$\Lambda(\xi,t=1;\mu,\sigma^2)$ celui du processus de Wiener: +\begin{align} + \label{eq:exposantcaractLaplace} + \Psi(\xi,t=1;\sigma,\mu,\tau) &= \Xi(i\Lambda(\xi,t=1;\mu,\sigma^2),t=1;\tau,\beta=1) \nonumber\\ + &= \tau \ln{\left(1-i(i\Lambda(\xi)) \right)} \nonumber\\ + &= \tau \ln{\left(1+(\frac{\sigma^2 \xi^2}{2} - i \mu \xi) \right)}. +\end{align} + +Le processus de Laplace sans dérive est donc, par définition, un +processus de Lévy et par conséquent infiniment divisible. En utilisant +l'exposant caractéristique \eqref{eq:exposantcaractLaplace} et la +définition \eqref{eq:fncaractYt}, on obtient sa fonction +caractéristique: +\begin{align} + \label{eq:fonctioncaractlaplacesansdrift} + \phi_{Y(t;\sigma,\mu,\tau)}(\xi) &= \exp{\left\{-t \cdot \Psi(\xi,t=1;\sigma,\mu,\tau)\right\}} \nonumber\\ + &= \exp{\left\{-t \cdot \left(\tau \ln{\left(1+(\frac{\sigma^2 + \xi^2}{2} - i \mu + \xi) \right)} \right)\right\}} \nonumber\\ + &= \left(1+\frac{\sigma^2 \xi^2}{2} - i \mu \xi\right)^{-\tau \cdot + t}. +\end{align} + +Une manière simple pour expliquer le mécanisme derrière le processus +subordonné est d'en construire une trajectoire à l'aide de la +simulation. + +On simule un temps d'arrivée $T_1$, de distribution gamma, puis une +hauteur de saut $X_1$, de distribution normale. On obtient ainsi le +premier incrément de la trajectoire, tel qu'illustré à la figure +\ref{fig:increment1}. +\begin{figure}[!ht] + \centering \input{"../graphiques/increment1.tex"} + \caption{Premier incrément d'un processus subordonné} + \label{fig:increment1} +\end{figure} + +Une réalisation d'une trajectoire de ce processus par simulation se +trouve à la figure \ref{fig:simgammagauss}. + +\begin{figure}[!ht] + \centering + \includegraphics[scale=.8]{"../graphiques/CH3-SIMGAMMAGAUSS"} + \caption{Simulation d'un processus de Wiener subordonné par un + processus gamma} + \label{fig:simgammagauss} +\end{figure} + +Le processus gamma $G(t;\tau,\beta=1)$, en tant que subordonnant dans +ce cas-ci, définit une échelle de temps économique , selon laquelle +on situe l'arrivée d'évènements pouvant influencer le prix d'un titre +financier. Cette dernière ne peut être mesurée, elle est donc +abstraite. L'échelle de temps où sont effectuées les observations du +processus correspond au temps calendrier. C'est la seule qui +puisse être mesurée. Étant donné que ces deux échelles sont +indépendantes, plusieurs sauts entre deux observations sont possibles. +L'échelle de temps économique est donc soit étirée, soit compressée, +en comparaison au temps calendrier. Autrement dit, si l'on définit +une journée économique comme étant l'intervalle de temps entre deux +sauts, on peut en avoir plusieurs au cours d'une seule journée de +calendrier $(G(t+1;\tau,\beta=1)-G(t;\tau,\beta=1) > 1)$. À l'opposé, +une d'entre elles peut chevaucher plusieurs journées calendrier +$(G(t+1;\tau,\beta=1)-G(t;\tau,\beta=1) \leq 1)$. + +Un processus stochastique qui représente le comportement du prix d'un +titre financier doit inclure une composante de dérive. Celle-ci +exprime le rendement moyen réalisé et est indépendante du processus de +sauts. Pour cette raison, on ajoute un coefficient de dérive $\theta$ +au processus de Laplace sans dérive, pour obtenir sa forme +générale. Comme ce coefficient est constant, on peut multiplier la +fonction caractéristique \eqref{eq:fonctioncaractlaplacesansdrift} par +la transformée de Fourier inverse du produit de celui-ci et de la +longueur de l'intervalle de temps $t$, $\mathcal{F}^{-1}(\theta \cdot +t) = e^{i\xi\theta \cdot t}$, pour obtenir celle du processus de +Laplace: +\begin{align} + \phi_{Y(t;\theta,\sigma,\mu,\tau)}(\xi) &= \frac{e^{i\xi\theta\cdot t}}{\left(1+\frac{\sigma^2\xi^2}{2}- i\mu \xi \right)^{\tau \cdot t}} \nonumber\\ + &= \left(\frac{e^{i\xi\theta}}{\left(1+\frac{\sigma^2\xi^2}{2}- i\mu + \xi + \right)^{\tau}}\right)^{t} \label{eq:fncaractprocessuslaplace}. +\end{align} + +Le processus $\left\{Y(t;\theta,\sigma,\mu,\tau)\right\}$ définit, +dans le contexte financier, l'évolution du logarithme du prix, tel que +présenté par \cite{kotz2001laplace}. Pour des fins de simplification, +on fixe le prix initial à 1: $Y(0;\theta,\sigma,\mu,\tau) = 0$ + +La fonction caractéristique \eqref{eq:fncaractprocessuslaplace} +constituera la principale représentation du processus de Laplace pour +la suite de ce texte. La construction du modèle «Variance Gamma» de +\cite{madan1990variance} est similaire, à l'exception que la +paramétrisation et le processus gamma utilisés sont différents, ce qui +rend leur approche moins intuitive, bien que le résultat soit +équivalent. + +\section{Distribution de Laplace asymétrique généralisée} +\label{sec:distributionGAL} + +La distribution de Laplace asymétrique généralisée caractérise un +intervalle du processus de Laplace avec dérive. Aussi appelée +distribution de Bessel, elle a été introduite par Karl Pearson en +1929, en lien avec la covariance d'un échantillon tiré d'une +population normale à deux variables. C'est aussi une généralisation de +la distribution de Laplace asymétrique qui sera présentée à la section +\ref{sec:distributionAL}. + +\subsection{Fonction caractéristique} +\label{sec:fncaractGAL} + +On définit cette distribution principalement par sa fonction +caractéristique. Celle-ci s'obtient facilement à partir de la fonction +caractéristique du processus de Laplace avec dérive +\eqref{eq:fncaractprocessuslaplace}, en considérant un incrément de +longueur $t=1$. À partir de la définition de ce dernier +\eqref{eq:VGsubordinne}, on déduit qu'elle est en fait un mélange de +la loi normale dont le paramètre de variance suit une distribution +gamma. + +$Y$ est une variable aléatoire définie comme étant la somme: +\begin{itemize} +\item d'un paramètre de translation $\theta$, +\item du produit: + \begin{itemize} + \item d'une variable aléatoire $W$ issue d'une distribution gamma + \eqref{eq:densitegamma1} + \item et d'un paramètre $\mu$ + \end{itemize} +\item et du produit: + \begin{itemize} + \item d'un paramètre $\sigma$, + \item de la racine carrée de la variable aléatoire $W$ + \item et d'une variable aléatoire $Z$ issue d'une distribution + normale centrée réduite: + \begin{align} + \label{eq:defvarY-GAL} + &Y = \theta + \mu W + \sigma \sqrt{W} Z + \end{align} + où + \begin{align} + Z \sim N(0,1) \mbox{ et } W \sim \Gamma(\tau,\beta=1). \nonumber + \end{align} + \end{itemize} +\end{itemize} + +Alors, la variable aléatoire $Y$, sachant que $W=w$, suit une +distribution normale de moyenne $w\mu$ et de variance $w\sigma^2$: +\begin{align} + \label{eq:Ynormalconditionnel} + (Y|W=w) \sim N(w\mu,w\sigma^2). +\end{align} + +La fonction caractéristique de la variable aléatoire $Y$ peut donc +être obtenue en utilisant celle de la loi normale $\phi^{N}_{\mu + w,w\sigma^2}(t)$ et la formule de l'espérance conditionnelle: +\begin{align*} + \phi_Y(t;\theta,\sigma,\mu,\tau) &= E\left[E\left[e^{itY} | W \right] \right] \\ + &= \int_0^{\infty} E \left[ e^{it(\theta + \mu w+\sigma\sqrt{w}Z)} \right] g(w) dw \quad \mbox{(en utilisant \eqref{eq:defvarY-GAL})} \\ + &= e^{i\theta t}\int_0^{\infty} \phi^{N}_{\mu w,w\sigma^2}(t)g(w)dw \\ + &= e^{i\theta t}\int_0^{\infty} e^{ iw\mu t-\frac{w\sigma^2t^2}{2}} \times \frac{1}{\Gamma (\tau)} w^{\tau-1}e^{-w} dw \\ + &= e^{i\theta t}\int_0^{\infty} \frac{1}{\Gamma (\tau)} w^{\tau-1} e^{-w(1+\frac{1}{2} \sigma^2 t^2 - i\mu t)} dw. +\end{align*} + +En complétant l'intérieur de l'intégrale de façon à retrouver la +densité de la loi gamma de paramètres $\alpha=\tau$ et +$\beta=\left(1+\frac{1}{2} \sigma^2 t^2 - i\mu t \right)^{\tau}$, on +obtient la fonction caractéristique de la variable aléatoire $Y$ de +distribution Laplace asymétrique généralisée: +\begin{align} + \label{eq:fncaractGALmu} + \phi_Y(t;\theta,\sigma,\mu,\tau) &= \frac{e^{i\theta t}}{{\left(1+\frac{1}{2} \sigma^2 t^2 - i\mu t \right)^{\tau}}}\int_0^{\infty} \frac{\left(1+\frac{1}{2} \sigma^2 t^2 - i\mu t \right)^{\tau}}{\Gamma (\tau)} w^{\tau-1} e^{-w(1+\frac{1}{2} \sigma^2 t^2 - i\mu t)} dw \nonumber\\ + &= \frac{e^{i\theta t}}{\left(1+\frac{1}{2} \sigma^2 t^2 - i\mu t + \right)^{\tau}}. +\end{align} + +\subsection{Invariance d'échelle} +\label{sec:invariance-dechelle} + +Une seconde paramétrisation pour la famille de distributions de +Laplace introduit la propriété d'invariance d'échelle. Cette propriété +permet d'appliquer un changement d'échelle à une variable aléatoire en +modifiant un seul paramètre sans que la valeur des autres ne soit +affectée. Si l'on revient à la définition de la variable aléatoire +conditionnelle $Y|W$ \eqref{eq:Ynormalconditionnel}, on remarque que +les paramètres $\mu \text{ et } \sigma^2$ sont influencés par la valeur +de $W$. Ils sont donc corrélés. En introduisant un paramètre +d'invariance d'échelle $\kappa$ en remplacement de $\mu$, on élimine +cette corrélation. Le paramètre $\kappa$ est obtenu à l'aide de la +transformation suivante: +\begin{equation} + \label{eq:mukappa} + \kappa = \frac{\sqrt{2\sigma^2+\mu^2}-\mu}{\sqrt{2}\sigma}. +\end{equation} + +À l'inverse, on peut retrouver le paramètre $\mu$ en l'isolant dans +l'équation précédente. On obtient donc: +\begin{equation} + \label{eq:kappamu} + \mu = \frac{\sigma}{\sqrt{2}}\left(\frac{1}{\kappa}-\kappa \right). +\end{equation} + +On définit la fonction vectorielle $T_{\mu\rightarrow\kappa}$ comme +étant la transformation qui permet le passage de la forme en $\mu$ à +celle en $\kappa$: +\begin{align} + T_{\mu\rightarrow\kappa}(\theta, \sigma, \mu, \tau) &= + \left[\begin{array}{c} \theta \\ \sigma \\ + \frac{\sqrt{2\sigma^2+\mu^2}-\mu}{\sqrt{2}\sigma} \\ \tau + \end{array}\right]. +\end{align} + +On définit aussi la transformation inverse $T_{\kappa\rightarrow\mu}$: +\begin{align} + T_{\kappa\rightarrow\mu}(\theta, \sigma, \kappa, \tau) &= + \left[\begin{array}{c} \theta \\ \sigma \\ + \frac{\sigma}{\sqrt{2}}\left(\frac{1}{\kappa}-\kappa \right) \\ + \tau + \end{array}\right]. +\end{align} + +Cette notation permet d'utiliser un vecteur de paramètres. On peut +obtenir la matrice de variance-covariance d'une forme paramétrique en +connaissant celle de l'autre et en utilisant le gradient de ces +transformations. + +Pour la première transformation, on a: +\begin{align} + \nabla T_{\mu\rightarrow\kappa} &= \left[ + \begin{array}[]{cccc} + 1&0&0&0 \\ + 0&1&0&0 \\ + 0&\frac{\mu\sqrt{4\sigma^2+\mu^2}-\mu^2}{2\sigma^2\sqrt{4\sigma^2+mu^2}} & -\frac{\sqrt{4\sigma^2+\mu^2}-\mu}{2\sigma\sqrt{4\sigma^2+\mu^2}} & 0 \\ + 0&0&0&1 + \end{array} + \right]. +\end{align} + +Pour la seconde transformation, on a: +\begin{align} + \nabla T_{\kappa\rightarrow\mu} &= \left[ + \begin{array}[]{cccc} + 1&0&0&0 \\ + 0&1&0&0 \\ + 0&-\frac{\kappa^2-1}{\sqrt{2}\kappa} & -\frac{\left(\kappa^2+1\right)\sigma}{\sqrt{2}\kappa^2} & 0 \\ + 0&0&0&1 + \end{array} + \right]. +\end{align} + +La forme en $\mu$ sera privilégiée pour l'estimation, car elle est +plus compacte. Cependant, certaines propriétés de la distribution font +appel à la forme utilisant le paramètre $\kappa$. + +\subsection{Fonctions génératrices} + +En utilisant la relation \eqref{eq:fncaractfgm}, on obtient la +fonction génératrice des moments à partir de la fonction +caractéristique: +\begin{align} + M_{Y}(\xi) &= \phi_{Y}(-i\xi) \nonumber\\ + &=\frac{e^{\theta \xi}}{\left(1-\frac{1}{2} \sigma^2 \xi^2 - \mu \xi + \right)^{\tau}}, \label{eq:fgmGAL}\\ + &\quad \mbox{où } 1-\frac{1}{2} \sigma^2 \xi^2 - \mu \xi > 0. + \label{eq:fgmGALcond} +\end{align} + +La condition \ref{eq:fgmGALcond} permet de s'assurer que le +dénominateur prend une valeur réelle strictement positive. Cette +condition impose aussi une restriction à l'espace des paramètres +$\Omega$. La fonction génératrice des moments permet d'obtenir tous +les moments $E[Y^r]$ d'une variable aléatoire en la dérivant +successivement par rapport à la variable de transformation et en +égalant cette dernière à 0: +\begin{align} + \label{eq:fgmmomentsGAL} + E[Y^r] = \left[ \frac{d^r M_Y(\xi)}{d\xi^r} \right]_{\xi=0}. +\end{align} + +La fonction génératrice des cumulants $K_{Y}(\xi)$ est aussi +intéressante à utiliser. On l'obtient à partir du logarithme de la +fonction génératrice des moments \eqref{eq:fgmmoments}: +\begin{align} + \label{eq:fgcGAL} + K_Y(\xi) &= \ln(M_Y(\xi)) \nonumber\\ + &= \ln\left(\frac{e^{\theta \xi}}{\left(1-\frac{1}{2} \sigma^2 + \xi^2 - \mu \xi \right)^{\tau}}\right),\qquad 1-\frac{1}{2} + \sigma^2 \xi^2 - \mu \xi > 0. +\end{align} + +Elle a une utilité similaire à la fonction génératrice des moments, +sauf qu'elle permet d'obtenir les cumulants $K_r$ de la distribution +en la dérivant successivement par rapport à la variable de +transformation et en égalant celle-ci à 0: +\begin{align} + \label{eq:fgccumGAL} + K_r = \left[ \frac{d^r \ln{(M_Y(\xi))}}{d\xi^r} \right]_{\xi=0}. +\end{align} + +\subsection{Moments et rôle des paramètres} +\label{sec:momentsGAL} + +En mathématiques, les moments sont des quantités décrivant la forme +d'un ensemble de points. En statistique, les moments décrivent +certaines caractéristiques d'une population ou d'un échantillon. Ces +caractéristiques sont utilisées dans la sélection d'une distribution +de probabilité appropriée pour représenter la population à partir de +l'échantillon, ce qu'on appelle l'inférence statistique. Les moments +bruts et centraux sont évalués par rapport à 0 et à la moyenne +respectivement. + +On obtient les premiers moments bruts de cette distribution à l'aide +de la relation décrite précédemment \eqref{eq:fgmmomentsGAL}. +\begin{align*} + E[Y] &= \theta+\tau\,\mu \\ + E[Y^2] &= {\theta}^{2}+2\,\mu\,\tau\,\theta+\tau\,{\sigma}^{2}+{\mu}^{2}\,{\tau}^{2}+{\mu}^{2}\,\tau \\ + E[Y^3] &= {\theta}^{3}+3\,\mu\,\tau\,{\theta}^{2}+\left( 3\,\tau\,{\sigma}^{2}+3\,{\mu}^{2}\,{\tau}^{2} +3\,{\mu}^{2}\,\tau\right) \,\theta \\ + &\quad + \left( 3\,\mu\,{\tau}^{2}+3\,\mu\,\tau\right) \,{\sigma}^{2}+{\mu}^{3}\,{\tau}^{3}+3\,{\mu}^{3}\,{\tau}^{2}+2\,{\mu}^{3}\,\tau \\ + E[Y^4] &= {\theta}^{4}+4\,\mu\,\tau\,{\theta}^{3}+\left( 6\,\tau\,{\sigma}^{2}+6\,{\mu}^{2}\,{\tau}^{2}+6\,{\mu}^{2}\,\tau\right) \,{\theta}^{2}\\ + &\quad+\left( \left( 12\,\mu\,{\tau}^{2}+12\,\mu\,\tau\right) \,{\sigma}^{2}+4\,{\mu}^{3}\,{\tau}^{3}+12\,{\mu}^{3}\,{\tau}^{2}+8\,{\mu}^{3}\,\tau\right) \,\theta \\ + &\quad+\left( 3\,{\tau}^{2}+3\,\tau\right) \,{\sigma}^{4}+\left( + 6\,{\mu}^{2}\,{\tau}^{3}+18\,{\mu}^{2}\,{\tau}^{2}+12\,{\mu}^{2}\,\tau\right) + \,{\sigma}^{2} \\ + &\quad+{\mu}^{4}\,{\tau}^{4}+6\,{\mu}^{4}\,{\tau}^{3}+11\,{\mu}^{4}\,{\tau}^{2}+6\,{\mu}^{4}\,\tau. +\end{align*} + +À partir de ceux-ci, on obtient aussi les quatre premiers moments +centraux: +\begin{subequations}\label{eq:momentsGAL} + \begin{align} + m_1 &= E[Y] = \theta+\tau\,\mu \label{eq:moments1GAL}\\ + m_2 &= E[(Y-m_1)^2] = \tau\,\sigma^2+\tau\,\mu^2\label{eq:moments2GAL}\\ + m_3 &= E[(Y-m_1)^3] = 3\,\tau\,{\sigma}^{2}\mu+2\,\tau\,{\mu}^{3}\label{eq:moments3GAL}\\ + m_4 &= E[(Y-m_1)^4] = + 3\,{\tau}^{2}{\sigma}^{4}+3\,{\tau}^{2}{\mu}^{4}+3\,\tau\,{\sigma}^{4}+6\,\tau\,{\mu}^{4}+6\,{\tau}^{2}{\mu}^{2}{\sigma}^{2}+12\,\tau\,{\sigma}^{2}{\mu}^{2}.\label{eq:moments4GAL} + \end{align} +\end{subequations} + +On résume le domaine et le rôle des paramètres à la table +\ref{tab:roleparamGAL}. +\begin{table}[!ht] + \centering + \begin{tabular}{cp{1.75cm}p{2.5cm}p{6.25cm}} + \hline + \textbf{Paramètre} & \textbf{Domaine} & \textbf{Rôle} & \textbf{Observations} \\ + \hline + $\theta$ & $\mathbb{R}$ & Localisation & N'influence que la moyenne. Équivaut au mode lorsque $\mu=0$. \\ + $\sigma$ & $\mathbb{R}^{+} \setminus \lbrace 0 \rbrace$ & Échelle & Vrai paramètre d'échelle lorsque $\kappa$ est utilisé. \\ + $\mu$ & $\mathbb{R}$ & Asymétrie & Distribution asymétrique à gauche lorsque négatif et à droite lorsque positif. Déplace la moyenne dans la même direction. Corrélation positive avec la variance et le coefficient d'aplatissement.\\ + $\kappa$ & $\mathbb{R}^{+} \setminus \lbrace 0 \rbrace$ & Asymétrie & Valeur dans l'intervalle $\left[0,1\right[$ lorsque $\mu<0$, dans $\left[1,\infty \right]$ lorsque $\mu \geq0$. \\ + $\tau$ & $\mathbb{R}^{+} \setminus \lbrace 0 \rbrace$ & Aplatissement & Négativement corrélé avec le coefficient d'aplatissement. Une petite valeur donne une distribution pointue. \\ + \hline + \end{tabular} + \caption{Domaine et rôle des paramètres de la distribution de Laplace asymétrique généralisée} + \label{tab:roleparamGAL} +\end{table} + +On retrouve quelques exemples de courbes de la fonction de densité +avec différents paramètres d'asymétrie et d'aplatissement à la figure +\ref{fig:densiteGAL}. +\begin{figure}[!ht] + \centering + \includegraphics[scale=0.8]{../contenus/r/code/dGAL-exemples.pdf} + \caption{Fonction de densité de la distribution Laplace asymétrique + généralisée avec différents paramètres: + $GAL(y;\theta,\sigma,\kappa,\tau)$} + \label{fig:densiteGAL} +\end{figure} + +Afin de comparer la distribution de Laplace asymétrique généralisée +avec la normale, que l'on cherche à remplacer dans le contexte des +rendements financiers, le comportement des coefficients d'asymétrie +$\gamma_1$ et d'aplatissement $\gamma_2$ de celle-ci peut être +intéressant à observer. On obtient ces derniers à partir des moments +centraux \eqref{eq:momentsGAL} : +\begin{subequations}\label{eq:moments56GAL} + \begin{align} + \gamma_1(Y) &= \frac{m_3}{(m_2)^{3/2}} = + \frac{2\,{\mu}^{3}+3\,{\sigma}^{2}\,\mu}{{\left( {\mu}^{2}+{\sigma}^{2}\right) }^{3/2}\,\sqrt{\tau}}\label{eq:moments5GAL}\\ + \gamma_2(Y) &= \frac{m_4}{(m_2)^{2}} - 3 = \frac{\left( + 3\,{\mu}^{4}+6\,{\sigma}^{2}\,{\mu}^{2}+3\,{\sigma}^{4}\right) + \,{\tau}^{2}+\left( + 6\,{\mu}^{4}+12\,{\sigma}^{2}\,{\mu}^{2}+3\,{\sigma}^{4}\right) + \,\tau}{{\left( {\mu}^{2}+{\sigma}^{2}\right) }^{2}\,{\tau}^{2}} + - 3.\label{eq:moments6GAL} + \end{align} +\end{subequations} + +Le coefficient d'asymétrie de la normale vaut $\gamma_1^N(Y)=0$ et +celui d'excès d'aplatissement, $\gamma_2^N(Y)=0$, puisqu'il a été +défini à partir de cette distribution. Si l'on fixe tous les +paramètres sauf $\mu$, la valeur minimale du coefficient d'excès +d'aplatissement $\gamma_2(Y)$ est atteinte lorsque $\mu$ est de 0: +\begin{align*} + \min_{\mu} \gamma_2(Y) &= \left[\gamma_2(Y)\right]_{\mu=0} \\ + &= \frac{3}{\tau}. +\end{align*} +Étant donné que le paramètre $\tau$ est strictement positif, la valeur +minimale que peut prendre le coefficient d'excès d'aplatissement +$\gamma_2(Y)$ est plus grande que 0. Par conséquent, l'aplatissement de la +distribution de Laplace asymétrique généralisée sera toujours +supérieur à celui de la normale; c'est donc une distribution +leptocurtique, ce qui est une propriété désirable selon les +observations de \cite{madan1990variance}. + +\subsection{Changement d'échelle et de localisation} +\label{sec:transGAL} + +Parfois, on doit modifier un ensemble de données afin d'effectuer des +comparaisons ou pour bénéficier des avantages d'une méthode +numérique. Les principales transformations utilisées sont: +\begin{enumerate} +\item un changement de localisation, où l'on additionne une constante + à l'ensemble des données +\item un changement d'échelle, où l'on multiplie l'ensemble des + données par un facteur +\item une combinaison des deux transformations précédentes. +\end{enumerate} + +Soit deux constantes $a$, $b\neq0$ et une variable aléatoire $X$ qui +suit une distribution Laplace asymétrique généralisée: $X \sim +GAL(\theta,\sigma,\kappa,\tau)$. $Y$ correspond à la somme: +\begin{itemize} +\item de la constante $b$ et +\item du produit: + \begin{itemize} + \item de la constante $a$ et + \item de la variable aléatoire $X$. + \end{itemize} +\end{itemize} + +Selon \cite{kotz2001laplace}, en utilisant le paramètre $\kappa$, on +peut effectuer un changement d'échelle et de localisation et obtenir +une variable aléatoire qui suit encore cette distribution: +\begin{itemize} +\item le paramètre de localisation $\theta$ subit la même + transformation que la variable aléatoire $X$; +\item le paramètre d'échelle $\sigma$ est multiplié par la constante + $a$; +\item le paramètre $\kappa$ est inversé si cette constante est + négative. Ce dernier cas est en fait une réflexion de la + distribution autour du mode. +\end{itemize} + +On a donc: +\begin{align} + \label{eq:GALscaletrans} + Y = aX + b \sim GAL(a \theta + b,a\sigma,\kappa^{\sgn{a}\cdot + 1},\tau). +\end{align} + +On estime les paramètres $\hat\theta, \hat\sigma, \hat\mu$ et +$\hat\tau$ sur un ensemble de données centrées et réduites $X_t$ à +partir d'un échantillon original $Y_t$. $\hat{m} \text{ et } +\hat{s}>0$ sont respectivement la moyenne et l'écart-type de +l'échantillon $Y_t$. En utilisant l'équation de transformation +\eqref{eq:GALscaletrans} avec les paramètres estimés précédemment, on +peut alors obtenir ceux correspondants pour l'échantillon: +\begin{align} + \label{eq:transparamGALNS} + Y_t = \hat{s} X_t + \hat{m} \sim GAL(\hat{s} \theta + + \hat{m},\hat{s}\sigma,\kappa,\tau) . +\end{align} + +Le paramètre $\kappa$ n'est pas modifié puisque l'écart-type $\hat{s}$ +est strictement positif. Cette propriété permettra de pratiquer +l'estimation sur des données centrées et réduites, ce qui diminue le +risque d'erreurs numériques sans nuire à sa précision, puisqu'aucune +information contenue dans l'échantillon n'est perdue. + +\subsection{Représentation alternative et simulation} +\label{sec:simulationGAL} + +Le processus de Laplace $Y(t;\theta,\sigma,\kappa,\tau)$ peut être +représenté sous la forme d'une différence de deux processus gamma +$G_1(t;\tau) - G_2(t;\tau)$ à laquelle on additionne une composante de +dérive. Le premier processus compte les gains, alors que le second, +les pertes. Les deux ont des incréments qui suivent une distribution +gamma $\Gamma(\alpha=\tau, \beta=1)$, c'est-à-dire la même +distribution que le processus +subordonnant utilisé à la section \ref{sec:browniensub}: +\begin{align} + \label{eq:distributiongammaformealt} + G_i(\tau) \sim \Gamma\left(\tau,\beta=1 \right), \qquad i\in\left\{ + 1,2 \right\}. +\end{align} + +Il peut être représenté sous la forme du processus composé suivant: +\begin{align} + \label{eq:processuslaplace2gamma} + Y(t) \stackrel{d}{=} \theta + + \frac{\sigma\sqrt{2}}{2}\left(\frac{1}{\kappa} G_1(t) - \kappa + G_2(t)\right). +\end{align} + +La distribution de Laplace asymétrique généralisée peut alors être +représentée sous la forme d'un incrément de ce processus. Les +variables aléatoires $G_1$ et $G_2$ sont respectivement des +réalisations indépendantes de distribution gamma avec densité +\eqref{eq:densitegamma1}: +\begin{align} + \label{eq:differencegamma} + Y \stackrel{d}{=} \theta + \frac{\sigma\sqrt{2}}{2} \left( + \frac{1}{\kappa} G_1 - \kappa G_2 \right). +\end{align} + +Simuler une réalisation de chacune d'entre elles suffira pour obtenir +une réalisation de la distribution gamma. On peut aussi réécrire la +définition précédente de la variable aléatoire $Y$ +\eqref{eq:differencegamma} sous la forme simplifiée suivante: +\begin{align} + \label{eq:differencegamma2} + Y \stackrel{d}{=} \theta + \left(G_3-G_4 \right). +\end{align} + +Les deux variables aléatoires sont alors définies comme suit: +\begin{align*} + G_3 \sim \Gamma\left(\tau,\beta=\frac{\sqrt{2}}{\kappa\sigma} \right)\\ + G_4 \sim \Gamma\left(\tau,\beta=\frac{\kappa\sigma}{\sqrt{2}} + \right). +\end{align*} + +On peut facilement démontrer l'équivalence en distribution +\eqref{eq:differencegamma2}. En effectuant le produit des fonctions +caractéristiques respectives $\phi_{Y}(\xi)$, $\phi_{G_3}(\xi)$ et +$\phi_{G_4}(\xi)$ des variables aléatoires $Y$, $G_3$ et $G_4$, on +retrouve celle de la forme en $\kappa$: +\begin{align} + \label{eq:equivalence2gammaFC} + \phi_{Y}(\xi) &= e^{i\xi\theta} \cdot \phi_{G_3}(\xi) \cdot \phi_{G_4}(\xi) \nonumber\\ + &= e^{i\xi\theta} \cdot \left(\frac{1}{1+i\xi\cdot\frac{\kappa\sigma}{\sqrt{2}}}\right)^{\tau} \cdot \left(\frac{1}{1-i\xi\cdot\frac{\kappa\sigma}{\sqrt{2}}}\right)^{\tau} \nonumber\\ + &= + \frac{e^{i\xi\theta}}{\left(1-i\xi\frac{\sigma}{\sqrt{2}}\left(\frac{1}{\kappa}-\kappa + \right) + \frac{\sigma^2\xi^2}{2}\right)^{\tau}}. +\end{align} + +La figure \ref{fig:simulGAL} présente un histogramme et un estimateur +de densité par noyau d'un échantillon de 2500 réalisations de la +variable aléatoire $Y$, suivant une distribution de Laplace +asymétrique généralisée de paramètres $\theta=0$, $\sigma=1$, +$\kappa=2$ et $\tau=1$. On remarquera que le paramètre $\kappa$ +produit une distribution asymétrique à gauche puisqu'il prend une +valeur supérieure à 1. + +\begin{figure}[!ht] + \centering + \includegraphics[scale=0.8]{"../graphiques/CH3-SIMULGAL0121"} + \caption{Histogramme et estimateur de densité par noyau de 2500 + réalisations de la variable aléatoire $Y\sim + GAL(\theta=0,\sigma=1,\kappa=2,\tau=1)$} + \label{fig:simulGAL} +\end{figure} + +\subsection{Fonction de Bessel et densité} +\label{sec:besseldensiteGAL} + +La densité de la distribution Laplace asymétrique généralisée peut +être exprimée à l'aide de la fonction de Bessel modifiée du troisième +type \citep{abramowitz1965handbook}: +\begin{align} + \label{eq:BesselK} + K_{\lambda}(u) = + \frac{(u/2)^{\lambda}\Gamma(1/2)}{\Gamma(\lambda+1/2)} + \int_1^{\infty} e^{-ut} (t^2-1)^{\lambda-1/2}dt,\qquad \lambda \geq + -1/2. +\end{align} + +On définit les paramètres $\delta = \mu\sigma^{-1}$ et $\eta = +\sqrt{1+\delta^2}$ afin de simplifier la notation. La démonstration +qui permet d'obtenir la fonction de densité est fondée sur la +définition de la distribution utilisant la différence de deux +processus gamma \eqref{eq:differencegamma}. Elle est développée par +\cite{kozubowski1999class}: +\begin{align} + \label{eq:densitekotz} + f(x) &= \frac{e^{-\kappa^{\sgn(x)}|x|}}{\Gamma(\tau)} \int_{0}^{\infty} y^{\tau-1}(|x|+y)^{\tau-1}e^{-\eta y} dy \nonumber\\ + &= \frac{1}{\Gamma(\tau)\sqrt{\pi}} \left(\frac{|x|}{\eta} + \right)^{\tau-1/2} e^{\delta x/2} K_{\tau-1/2}\left(\frac{\eta + |x|}{2}\right). +\end{align} + +Afin d'incorporer le paramètre de localisation $\theta$, on remplace +$x$ par la valeur centrée $x-\theta$, et les paramètres $\eta$ et +$\delta$ par ceux d'origine, tels que définis précédemment, pour +obtenir la forme suivante \citep{kotz2001laplace}: +\begin{align} + \label{eq:densitekotz2001} + f(x) = + \frac{\sqrt{2}e^{\frac{\sqrt{2}}{2\sigma}(1/\kappa-\kappa)(x-\theta)}}{\sqrt{\pi}\sigma^{\tau+1/2}\Gamma(\tau)} + \bigg(\frac{\sqrt{2}|x-\theta|}{\kappa+1/\kappa} \bigg)^{\tau-1/2} + K_{\tau-1/2}\left(\frac{\sqrt{2}}{2\sigma}\left(\frac{1}{k}+\kappa + \right)|x-\theta| \right). +\end{align} + +Cette fonction a été implémentée dans un algorithme de maximum de +vraisemblance pour le logiciel GNU R par +\cite{RpackageVarianceGamma}. Elle reste cependant sensible +numériquement en plus de demander un temps de calcul important pour de +grands échantillons. De plus, comme la fonction de densité n'est pas +différentiable, on ne peut pas obtenir les estimateurs du maximum de +vraisemblance et leurs matrices de variance-covariance. Donc, +il est impossible d'effectuer de tests d'adéquation ou d'hypothèse avec ces +résultats. C'est une des principales raisons qui ont motivé la +recherche d'autres méthodes d'estimation plus efficaces pour cette +distribution. + +\section{Cas particuliers} +\label{sec:cas-particuliers} + +La distribution de Laplace asymétrique généralisée peut être +considérée comme une extension de plusieurs cas particuliers. Ces +distributions ainsi que leur notation et leur fonction de densité sont +présentées dans la table \ref{tab:casspeciauxGAL}. +\begin{table}[!ht] + \centering + \begin{tabular}{cccc} + \hline + \textbf{Cas} & \textbf{Distribution} & \textbf{Notation} & \textbf{Densité} \\ + \hline + $\theta=0$ & \multirow{4}{*}{Exponentielle de moyenne $\mu>0$} & \multirow{4}{*}{$GAL(0,0,\mu,1)$} & \multirow{4}{*}{$\frac{1}{\mu}e^{-x/\mu}\,(x > 0)$} \\ + $\sigma=0$ & & & \\ + $\mu>0$ & & & \\ + $\tau=1$ & & & \\ \hline + $\theta=0$ & \multirow{3}{*}{Gamma de paramètres $\alpha=\tau$, $\beta=\mu>0$} & + \multirow{3}{*}{$GAL(0,0,\mu,\tau)$} & \multirow{3}{*}{$\frac{x^{\tau-1}e^{-x/\mu}}{\mu^\tau\Gamma(\tau)},(x > 0)$} \\ + $\sigma=0$ & & & \\ + $\mu>0$ & & & \\ \hline + $\sigma>0$ & \multirow{3}{*}{Laplace symétrique $(\sigma>0)$} & + \multirow{3}{*}{$GAL(\theta,\sigma,0,1)$} & \multirow{3}{*}{$\frac{1}{\sqrt{2}\sigma}e^{-\sqrt{2}|x-\theta|/\sigma},(x \in \mathbb{R})$} \\ + $\mu=0$ & & & \\ + $\tau=1$ & & & \\ \hline + $\sigma>0$ & \multirow{3}{*}{Laplace asymétrique $(\sigma>0)$} & + \multirow{3}{*}{$GAL(\theta,\sigma,\mu,1)$} & \multirow{3}{*}{$\frac{\sqrt{2}\kappa}{\sigma(1+\kappa^2)}\begin{cases}e^{\frac{\sqrt{2}\kappa}{\sigma}(\theta-x)},\quad & x\geq\theta \\ e^{\frac{\sqrt{2}}{\sigma\kappa}(x-\theta)},\quad & x < \theta \end{cases}$} \\ + $\mu \neq 0$ & & & \\ + $\tau=1$ & & & \\ \hline + $\sigma=0$ & \multirow{3}{*}{Dégénérée à $\theta$} & \multirow{3}{*}{$GAL(\theta,0,0,0)$} & \multirow{3}{*}{$1_{x=\theta}$} \\ + $\mu=0$ & & & \\ + $\tau=0$ & & & \\ \hline + \end{tabular} + \caption{Cas spéciaux de la distribution de Laplace asymétrique généralisée} + \label{tab:casspeciauxGAL} +\end{table} + +Les deux premiers cas spéciaux sont des distributions de probabilité +classiques. L’attention sera portée sur la distribution de Laplace +asymétrique et son cas spécial, celle de Laplace symétrique. La +dégénérée n'a aucune application pratique, mais en mentionner +l'existence est pertinent, puisque ce ne sont pas toutes les +distributions qui admettent ce cas. + +\subsection{Distribution de Laplace asymétrique} +\label{sec:distributionAL} + +La distribution de Laplace asymétrique a été introduite +par \cite{hinkley1977estimation} et étudiée en profondeur par +\cite{kozubowski1999class}. Cette distribution a elle aussi plusieurs +caractéristiques intéressantes pour remplacer la normale, dans le +contexte de la modélisation financière, comme le présentent +\cite{kozubowski2001asymmetric}. Elle est notée +$AL(\theta,\sigma,\kappa)$. + +Cette distribution, en tant que cas particulier de la distribution de +Laplace asymétrique généralisée, conserve la majorité de ses +propriétés. Entre autres, elle est infiniment divisible et possède des moments +finis. Les paramètres $\theta$, $\sigma$ et $\kappa$ conservent leurs +rôles et propriétés définis à la section \ref{sec:momentsGAL}. On peut +donc contrôler la localisation, l'échelle et l'asymétrie de la +distribution. Cependant, aucun paramètre ne permet de contrôler le +coefficient d'aplatissement. + +La densité de probabilité de cette distribution +$f(x;\theta,\sigma,\kappa)$ présente une forme analytique simple qui +ne requiert pas le recours à la fonction de Bessel modifiée de +troisième type \eqref{eq:BesselK}. +\begin{align} + \label{eq:densiteAL} + f(x;\theta,\sigma,\kappa) = + \frac{\sqrt{2}}{\sigma}\frac{\kappa}{1+\kappa^2} \begin{cases} + \exp\left(\frac{-\sqrt{2}\kappa}{\sigma}\left( x-\theta \right)\right),\quad & x\geq\theta \\ + \exp\left( \frac{\sqrt{2}}{\theta\kappa}\left(\theta - x \right) + \right),\quad & x<\theta.\end{cases} +\end{align} + +La fonction caractéristique de cette distribution prend la forme +suivante: +\begin{align} + \label{eq:caractALkappa} + \phi(t;\theta,\sigma,\kappa) = \frac{e^{i \theta + t}}{1+\frac{\sigma^2 t^2}{2}- + i\frac{\sigma}{\sqrt{2}}\left(\frac{1}{\kappa}-\kappa \right) + t}\,. +\end{align} + +Cette dernière est utilisée principalement lorsque l'application d'une +propriété nécessite une distribution avec un seul paramètre d'échelle. +La seconde paramétrisation de cette distribution utilise le paramètre +$\mu$ qui n'est pas invariant d'échelle. Cette paramétrisation donne +une forme beaucoup plus simple de la fonction caractéristique +\eqref{eq:caractALkappa}: +\begin{align} + \label{eq:caractALmu} + \phi(t\theta,\sigma,\mu) = \frac{e^{i \theta t}}{1+\frac{\sigma^2 + t^2}{2}- i\mu t}. +\end{align} + +On reconnait la fonction caractéristique d'un incrément du processus +de Laplace \eqref{eq:fncaractprocessuslaplace} avec les paramètres +($\tau=1$, $t=1$). Une forme standardisée de cette distribution existe +aussi avec le paramètre d'échelle $\sigma=1$ et de dérive $\theta=0$. +Enfin, on peut avoir une représentation alternative de cette +distribution, utilisée principalement pour la simulation, basée sur +l'exponentielle: +\begin{align} + \label{eq:ALrepsimul} + Y &= \theta + \frac{\sigma}{\sqrt{2}} \left(\frac{1}{\kappa}W_1 - k W_2 \right) +\end{align} +où +\begin{align*} + W_1 \sim \mbox{Exponentielle}\left(\frac{\sigma}{\sqrt{2}\kappa}\right) \nonumber \mbox{ et } W_2 \sim \mbox{Exponentielle}\left(\frac{\sigma\kappa}{\sqrt{2}}\right) +\end{align*} + +Cette méthode n'est cependant pas la plus efficace pour simuler une +réalisation $Y$ de la distribution de Laplace asymétrique, car on peut +la faire à partir de deux réalisations indépendantes $U_1$ et $U_2 $ +d'une variable aléatoire uniforme définie sur l'intervalle +$\left[0,1\right]$: +\begin{align} + \label{eq:simulALunif} + Y = \theta + \frac{\sigma}{\sqrt{2}} + \ln{\frac{U_1^\kappa}{U_2^{1/\kappa}}}. +\end{align} + +Lorsque le paramètre $\tau$ de la distribution de Laplace asymétrique +généralisée est un entier, on peut générer une réalisation $X$ de +celle-ci en sommant un total de $\tau$ réalisations indépendantes de +la variable aléatoire $Y$: +\begin{align*} + X &= \sum_{i=1}^{\tau} Y_{i} \\ + Y_{i} &\sim AL(\theta,\sigma,\kappa). +\end{align*} + +\section{Relation avec le modèle de \cite{madan1990variance}} +\label{sec:relation-avec-le} + +Le modèle Variance Gamma de \cite{madan1990variance} utilise une +paramétrisation différente de celle de \cite{kotz2001laplace}. On +doit, afin de conserver une compatibilité des développements des +prochains chapitres, détailler les changements de paramètres qui +permettent de passer d'un modèle à l'autre. Le modèle Variance Gamma +utilise les paramètres $C,\sigma,\theta \mbox{ et } \nu$. Afin +d'éviter la confusion, les paramètres seront indicés de 1 à 3 selon le +modèle d'où ils proviennent. Le modèle 1 sera celui de +\cite{madan1990variance}, les modèles 2 et 3 seront respectivement les +paramétrisations en $\mu$ et en $\kappa$ de +\cite{kotz2001laplace}. Les différentes équivalences se trouvent à la +table \ref{tab:chparam}. +\begin{table}[!ht] + \centering + \begin{tabular}{cccccc} + \hline + \textbf{Vers le modèle} & & & \textbf{À partir du modèle} & & \textbf{À partir du modèle} \\ \hline + \textbf{1} & & & \textbf{2} & & \textbf{3} \\ \hline + + & $C$ & $=$ & $\theta_2$ & $=$ & $\theta_3$ \\ + \multicolumn{ 1}{l}{} & $\sigma_1$ & $=$ & $\sqrt{\tau_2}\sigma_2$ & $=$ & $\sqrt{\tau_3}\sigma_3$ \\ + \multicolumn{ 1}{l}{} & $\theta_1$ & $=$ & $\mu\tau_2$ & $=$ & $\tau_3\sigma_3(1/\kappa - \kappa)/\sqrt{2}$ \\ + \multicolumn{ 1}{l}{} & $\nu$ & $=$ & $1/\tau_2$ & $=$ & $1/\tau_3$ \\ + \hline + \textbf{2} & & & \textbf{1} & & \textbf{3} \\ \hline + & $\theta_2$ & $=$ & $C$ & $=$ & $\theta_3$ \\ + \multicolumn{ 1}{l}{} & $\sigma_2$ & $=$ & $\sigma_1\sqrt{\nu}$ & $=$ & $\sigma_3$ \\ + \multicolumn{ 1}{l}{} & $\mu$ & $=$ & $\theta_1*\nu$ & $=$ & $\sigma_3(1/\kappa - \kappa)/\sqrt{2}$ \\ + \multicolumn{ 1}{l}{} & $\tau_2$ & $=$ & $1/\nu$ & $=$ & $\tau_3$ \\ + \hline + \textbf{3} & & & \textbf{1} & & \textbf{2} \\ \hline + & $\theta_3$ & $=$ & $C$ & $=$ & $\theta_2$ \\ + \multicolumn{ 1}{l}{} & $\sigma_3$ & $=$ & $\sigma_1\sqrt{\nu}$ & $=$ & $\sigma_2$ \\ + \multicolumn{ 1}{l}{} & $\kappa$ & $=$ & $\frac{\sqrt{2(\sigma_1\sqrt{\nu})^2 + (\theta_1*\nu)^2} – \theta_1*\nu}{\sigma_1\sqrt{\nu}\sqrt{2}}$ & $=$ & $\frac{\sqrt{2(\sigma_2)^2 + \mu^2} - \mu}{\sigma_2\sqrt{2}}$ \\ + \multicolumn{ 1}{l}{} & $\tau_3$ & $=$ & $1/\nu$ & $=$ & $\tau_2$ \\ \hline + \end{tabular} + \caption{Changements de paramétrisation} + \label{tab:chparam} +\end{table} + + + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: + diff --git a/memoire/chapitre3.aux b/memoire/chapitre3.aux new file mode 100644 index 0000000..2cc35d2 --- /dev/null +++ b/memoire/chapitre3.aux @@ -0,0 +1,182 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{lugannani1980saddle} +\citation{butler2007saddlepoint} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {chapter}{\chapternumberline {3}Approximation de la densit\IeC {\'e} et de la fonction de r\IeC {\'e}partition}{39}{chapter.3}} +\@writefile{toc}{\contentsline {section}{\numberline {3.1}L'approximation de Laplace}{39}{section.3.1}} +\newlabel{sec:appr-de-lapl}{{3.1}{39}{L'approximation de Laplace}{section.3.1}{}} +\newlabel{sec:appr-de-lapl@cref}{{[section][1][3]3.1}{39}} +\@writefile{lof}{\contentsline {figure}{\numberline {3.1}{\ignorespaces Probabilit\IeC {\'e} \IeC {\`a} l'extr\IeC {\'e}mit\IeC {\'e} du support\relax }}{40}{figure.caption.23}} +\newlabel{fig:probdroite}{{3.1}{40}{Probabilité à l'extrémité du support\relax }{figure.caption.23}{}} +\newlabel{fig:probdroite@cref}{{[figure][1][3]3.1}{40}} +\newlabel{eq:approxlaplace1}{{3.1.1}{40}{L'approximation de Laplace}{equation.3.1.0.1}{}} +\newlabel{eq:approxlaplace1@cref}{{[equation][1][3,1]3.1.1}{40}} +\newlabel{eq:taylorgx}{{3.1.2}{40}{L'approximation de Laplace}{equation.3.1.0.2}{}} +\newlabel{eq:taylorgx@cref}{{[equation][2][3,1]3.1.2}{40}} +\newlabel{eq:integraleapproxlaplace1}{{3.1.3}{40}{L'approximation de Laplace}{equation.3.1.0.3}{}} +\newlabel{eq:integraleapproxlaplace1@cref}{{[equation][3][3,1]3.1.3}{40}} +\newlabel{eq:approxlaplace2}{{3.1.4}{40}{L'approximation de Laplace}{equation.3.1.0.4}{}} +\newlabel{eq:approxlaplace2@cref}{{[equation][4][3,1]3.1.4}{40}} +\newlabel{eq:kappailaplace}{{3.1.5}{40}{L'approximation de Laplace}{equation.3.1.0.5}{}} +\newlabel{eq:kappailaplace@cref}{{[equation][5][3,1]3.1.5}{40}} +\citation{daniels1954saddlepoint} +\@writefile{toc}{\contentsline {section}{\numberline {3.2}L'approximation de Temme}{41}{section.3.2}} +\newlabel{sec:lappr-de-temme}{{3.2}{41}{L'approximation de Temme}{section.3.2}{}} +\newlabel{sec:lappr-de-temme@cref}{{[section][2][3]3.2}{41}} +\@writefile{lof}{\contentsline {figure}{\numberline {3.2}{\ignorespaces Distribution normale tronqu\IeC {\'e}e \IeC {\`a} $w_0$\relax }}{41}{figure.caption.24}} +\newlabel{fig:normaletronque}{{3.2}{41}{Distribution normale tronquée à $w_0$\relax }{figure.caption.24}{}} +\newlabel{fig:normaletronque@cref}{{[figure][2][3]3.2}{41}} +\newlabel{eq:temmeintegrale}{{3.2.2}{41}{L'approximation de Temme}{equation.3.2.0.2}{}} +\newlabel{eq:temmeintegrale@cref}{{[equation][2][3,2]3.2.2}{41}} +\@writefile{toc}{\contentsline {section}{\numberline {3.3}La m\IeC {\'e}thode du point de selle}{42}{section.3.3}} +\newlabel{sec:pointcolGAL}{{3.3}{42}{La méthode du point de selle}{section.3.3}{}} +\newlabel{sec:pointcolGAL@cref}{{[section][3][3]3.3}{42}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.1}Approximation de la densit\IeC {\'e}}{42}{subsection.3.3.1}} +\newlabel{sec:appr-de-prem}{{3.3.1}{42}{Approximation de la densité}{subsection.3.3.1}{}} +\newlabel{sec:appr-de-prem@cref}{{[subsection][1][3,3]3.3.1}{42}} +\newlabel{eq:FGMdefinitionsaddle}{{3.3.1}{42}{Approximation de la densité}{equation.3.3.0.1}{}} +\newlabel{eq:FGMdefinitionsaddle@cref}{{[equation][1][3,3]3.3.1}{42}} +\newlabel{eq:FGMdaniels}{{3.3.2}{42}{Approximation de la densité}{equation.3.3.0.2}{}} +\newlabel{eq:FGMdaniels@cref}{{[equation][2][3,3]3.3.2}{42}} +\newlabel{eq:laplacesaddle1}{{3.3.4}{42}{Approximation de la densité}{equation.3.3.0.4}{}} +\newlabel{eq:laplacesaddle1@cref}{{[equation][4][3,3]3.3.4}{42}} +\newlabel{eq:premierordresaddle}{{3.3.5}{42}{Approximation de la densité}{equation.3.3.0.5}{}} +\newlabel{eq:premierordresaddle@cref}{{[equation][5][3,3]3.3.5}{42}} +\newlabel{eq:secondordresaddle}{{3.3.6}{43}{Approximation de la densité}{equation.3.3.0.6}{}} +\newlabel{eq:secondordresaddle@cref}{{[equation][6][3,3]3.3.6}{43}} +\newlabel{eq:lnfysaddlelaplace}{{3.3.7}{43}{Approximation de la densité}{equation.3.3.0.7}{}} +\newlabel{eq:lnfysaddlelaplace@cref}{{[equation][7][3,3]3.3.7}{43}} +\newlabel{eq:dlnfysaddlelaplace}{{3.3.8}{43}{Approximation de la densité}{equation.3.3.0.8}{}} +\newlabel{eq:dlnfysaddlelaplace@cref}{{[equation][8][3,3]3.3.8}{43}} +\newlabel{eq:premierodreapprox}{{3.3.9}{43}{Approximation de la densité}{equation.3.3.0.9}{}} +\newlabel{eq:premierodreapprox@cref}{{[equation][9][3,3]3.3.9}{43}} +\newlabel{eq:saddlepoint}{{3.3.10}{43}{Approximation de la densité}{equation.3.3.0.10}{}} +\newlabel{eq:saddlepoint@cref}{{[equation][10][3,3]3.3.10}{43}} +\newlabel{eq:deriveesecondesaddlepoint}{{3.3.11}{43}{Approximation de la densité}{equation.3.3.0.11}{}} +\newlabel{eq:deriveesecondesaddlepoint@cref}{{[equation][11][3,3]3.3.11}{43}} +\newlabel{eq:approximationsaddlepointordre1}{{3.3.12}{43}{Approximation de la densité}{equation.3.3.0.12}{}} +\newlabel{eq:approximationsaddlepointordre1@cref}{{[equation][12][3,3]3.3.12}{43}} +\citation{daniels1954saddlepoint} +\newlabel{eq:normalisationsaddle}{{3.3.13}{44}{Approximation de la densité}{equation.3.3.0.13.13}{}} +\newlabel{eq:normalisationsaddle@cref}{{[equation][13][3,3]3.3.13}{44}} +\newlabel{eq:normalisationsaddle1}{{3.3.13a}{44}{Approximation de la densité}{equation.3.3.0.13a}{}} +\newlabel{eq:normalisationsaddle1@cref}{{[equation][1][3,3,13]3.3.13a}{44}} +\newlabel{eq:normalisationsaddle2}{{3.3.13b}{44}{Approximation de la densité}{equation.3.3.0.13b}{}} +\newlabel{eq:normalisationsaddle2@cref}{{[equation][2][3,3,13]3.3.13b}{44}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.2}Unicit\IeC {\'e} du point de selle}{44}{subsection.3.3.2}} +\newlabel{sec:unicite-du-point}{{3.3.2}{44}{Unicité du point de selle}{subsection.3.3.2}{}} +\newlabel{sec:unicite-du-point@cref}{{[subsection][2][3,3]3.3.2}{44}} +\newlabel{eq:limitesderiveeK}{{3.3.14}{44}{Unicité du point de selle}{equation.3.3.0.14}{}} +\newlabel{eq:limitesderiveeK@cref}{{[equation][14][3,3]3.3.14}{44}} +\newlabel{eq:6}{{3.3.16}{44}{Unicité du point de selle}{equation.3.3.0.16}{}} +\newlabel{eq:6@cref}{{[equation][16][3,3]3.3.16}{44}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.3}Approximation de la fonction de r\IeC {\'e}partition}{44}{subsection.3.3.3}} +\newlabel{sec:appr-de-prem-1}{{3.3.3}{44}{Approximation de la fonction de répartition}{subsection.3.3.3}{}} +\newlabel{sec:appr-de-prem-1@cref}{{[subsection][3][3,3]3.3.3}{44}} +\newlabel{eq:chengementytow}{{3.3.17}{45}{Approximation de la fonction de répartition}{equation.3.3.0.17}{}} +\newlabel{eq:chengementytow@cref}{{[equation][17][3,3]3.3.17}{45}} +\newlabel{eq:derivchengementytow}{{3.3.18}{45}{Approximation de la fonction de répartition}{equation.3.3.0.18}{}} +\newlabel{eq:derivchengementytow@cref}{{[equation][18][3,3]3.3.18}{45}} +\newlabel{eq:approxintegraleREPordre1}{{3.3.19}{45}{Approximation de la fonction de répartition}{equation.3.3.0.19}{}} +\newlabel{eq:approxintegraleREPordre1@cref}{{[equation][19][3,3]3.3.19}{45}} +\newlabel{eq:approximationsaddlepointREPordre1}{{3.3.20}{45}{Approximation de la fonction de répartition}{equation.3.3.0.20}{}} +\newlabel{eq:approximationsaddlepointREPordre1@cref}{{[equation][20][3,3]3.3.20}{45}} +\newlabel{eq:wuysaddlerepart2}{{3.3.21}{45}{Approximation de la fonction de répartition}{equation.3.3.0.21.21}{}} +\newlabel{eq:wuysaddlerepart2@cref}{{[equation][21][3,3]3.3.21}{45}} +\newlabel{eq:wysaddlerepart2}{{3.3.21a}{45}{Approximation de la fonction de répartition}{equation.3.3.0.21a}{}} +\newlabel{eq:wysaddlerepart2@cref}{{[equation][1][3,3,21]3.3.21a}{45}} +\newlabel{eq:uysaddlerepart2}{{3.3.21b}{45}{Approximation de la fonction de répartition}{equation.3.3.0.21b}{}} +\newlabel{eq:uysaddlerepart2@cref}{{[equation][2][3,3,21]3.3.21b}{45}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.4}Quelques propri\IeC {\'e}t\IeC {\'e}s des approximations}{45}{subsection.3.3.4}} +\newlabel{sec:proprietesaddle}{{3.3.4}{45}{Quelques propriétés des approximations}{subsection.3.3.4}{}} +\newlabel{sec:proprietesaddle@cref}{{[subsection][4][3,3]3.3.4}{45}} +\@writefile{toc}{\contentsline {section}{\numberline {3.4}Application de la m\IeC {\'e}thode du point de selle}{46}{section.3.4}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.1}Approximation de la densit\IeC {\'e}}{46}{subsection.3.4.1}} +\newlabel{sec:applicationsaddleGAL}{{3.4.1}{46}{Approximation de la densité}{subsection.3.4.1}{}} +\newlabel{sec:applicationsaddleGAL@cref}{{[subsection][1][3,4]3.4.1}{46}} +\newlabel{eq:derivcumulantGAL1}{{3.4.1}{46}{Approximation de la densité}{equation.3.4.0.1}{}} +\newlabel{eq:derivcumulantGAL1@cref}{{[equation][1][3,4]3.4.1}{46}} +\newlabel{eq:saddlepointGAL}{{3.4.3}{46}{Approximation de la densité}{equation.3.4.0.3}{}} +\newlabel{eq:saddlepointGAL@cref}{{[equation][3][3,4]3.4.3}{46}} +\newlabel{eq:derivcumulantGAL2}{{3.4.4}{47}{Approximation de la densité}{equation.3.4.0.4}{}} +\newlabel{eq:derivcumulantGAL2@cref}{{[equation][4][3,4]3.4.4}{47}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.2}Approximation de la fonction de r\IeC {\'e}partition}{47}{subsection.3.4.2}} +\@setckpt{chapitre3}{ +\setcounter{page}{48} +\setcounter{equation}{4} +\setcounter{enumi}{3} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{3} +\setcounter{section}{4} +\setcounter{subsection}{2} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{66} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{2} +\setcounter{lofdepth}{1} +\setcounter{table}{0} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{16} +\setcounter{Hfootnote}{1} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{48} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/chapitre3.tex b/memoire/chapitre3.tex new file mode 100644 index 0000000..4d79fbd --- /dev/null +++ b/memoire/chapitre3.tex @@ -0,0 +1,631 @@ +\chapter{Approximation de la densité et de la fonction de + répartition} % numéroté + +Il existe plusieurs situations où il n'est pas possible d'obtenir une +forme analytique des fonctions de densité et de répartition de la +distribution d'une variable aléatoire. C'est particulièrement le cas +pour la majorité des distributions issues d'un processus de Lévy, des +sommes de longueur finie ou aléatoire de variables aléatoires et des +mélanges de distributions. Cependant, dans la majorité de ces cas, il +est facile d'obtenir la fonction caractéristique ou encore la fonction +génératrice des moments. D'ailleurs, il pourrait être intéressant +d'utiliser l'intégration numérique afin d'obtenir les transformées de +Fourier ou de Laplace de ces fonctions. Par contre, la convergence de +ces méthodes d'intégration est souvent lente, c'est-à-dire qu'il faut +un pas de discrétisation très fin afin d'obtenir une approximation +satisfaisante \citep{lugannani1980saddle}. Il est aussi possible +d'utiliser l'algorithme de la transformée de Fourier rapide, mais +celui-ci a comme désavantage d'être très peu précis dans l'évaluation +des probabilités aux extrémités du support de la densité (figure +\ref{fig:probdroite}). +\begin{figure}[!ht] + \centering \input{../graphiques/probdroite.tex} + \caption{Probabilité à l'extrémité du support} + \label{fig:probdroite} +\end{figure} + +C'est dans cette optique qu'ont été développées les approximations par +la méthode du point de selle. Puisque celle-ci requiert l'utilisation +de l'approximation de Laplace pour la fonction de densité et de +l'approximation de Temme pour la fonction de répartition, on les +introduit avant de développer la méthode à proprement parler. Il est à +noter que la majorité de ces démonstrations sont tirées de +\cite{butler2007saddlepoint} qui présente une monographie assez +complète sur la méthode du point de selle. + +\section{L'approximation de Laplace} +\label{sec:appr-de-lapl} + +L'approximation de Laplace sera utilisée afin de développer la méthode +du point de selle pour la fonction de densité. Soit $g(x)$, une +fonction régulière sur l'intervalle $\left[c,d\right]$ ayant un +minimum global au point $\hat{x} \in +\left]c,d\right]$. L'approximation de Laplace de premier ordre prend +la forme suivante: +\begin{align} + \label{eq:approxlaplace1} + \int_{c}^{d} e^{-g(x)}dx &\approx + \frac{\sqrt{2\pi}e^{-g(\hat{x})}}{\sqrt{g^{\prime\prime}(\hat{x})}}. +\end{align} + +La démonstration se fait facilement en utilisant le développement de +Taylor de $g(x)$ autour du point $\hat{x}$: +\begin{align} + \label{eq:taylorgx} + g(x) &= g(\hat{x}) + g^{\prime}(\hat{x})(x-\hat{x}) + \frac{1}{2} + g^{\prime\prime}(\hat{x})(x-\hat{x})^2 + R. +\end{align} + +La valeur $R$ représente le reste du développement et est une quantité +relativement petite. Comme $\hat{x}$ est un minimum local, alors la +dérivée première $g^{\prime}(\hat{x})$ vaut $0$ et la dérivée seconde +$g^{\prime\prime}(\hat{x})$ est supérieure à 0. On obtient donc +l'intégrale suivante, qui ressemble à celle de la fonction de +répartition de la distribution normale: +\begin{align} + \label{eq:integraleapproxlaplace1} + e^{-g(\hat{x})}\int_{c}^{d} + \exp{\left\{-\frac{1}{2}g^{\prime\prime}(\hat{x})(x-\hat{x})^2 + \right\}}dx &\approx e^{-g(\hat{x})} + \sqrt{\frac{2\pi}{g^{\prime\prime}(\hat{x})}}. +\end{align} + +En utilisant un terme de plus du développement de Taylor +\eqref{eq:taylorgx}, on obtient l'approximation de Laplace de second +ordre, qui a une plus grande précision: +\begin{align} + \label{eq:approxlaplace2} + \int_{c}^{d} e^{-g(x)}dx &\approx + \frac{\sqrt{2\pi}e^{-g(\hat{x})}}{\sqrt{g^{\prime\prime}(\hat{x})}}\left\{\frac{5}{24}\hat{\kappa}_3^2 + - \frac{1}{8} \hat{\kappa}_4 \right\} +\end{align} + +où +\begin{align} + \hat{\kappa}_i &= + \frac{g^{(i)}(\hat{x})}{\left\{g^{\prime\prime}(\hat{x})\right\}^{i/2}} + ,\quad i\geq 3 \label{eq:kappailaplace}. +\end{align} + +\section{L'approximation de Temme} +\label{sec:lappr-de-temme} + +L'approximation de Temme sera utilisée afin de dériver la méthode du +point de selle pour la fonction de répartition. On considère une +variable aléatoire $Z_{w_0}$ de distribution normale standard tronquée +de telle sorte qu'elle ne puisse prendre des valeurs supérieures au +point $w_0$. +\begin{figure}[!ht] + \centering \input{../graphiques/normaletronque.tex} + \caption{Distribution normale tronquée à $w_0$} + \label{fig:normaletronque} +\end{figure} + +On définit l'espérance d'une fonction $h$ de cette variable aléatoire, +à partir des fonctions de densité $\phi(w)$ et de répartition +$\Phi(w)$ de la loi normale, comme suit: +\begin{align} + E[h(Z_{w_0})] &= \frac{\int_{-\infty}^{w_0} h(w)\phi(w) + dw}{\Phi(w_0)}. +\end{align} + +Une approximation du numérateur est donnée par le développement +suivant: +\begin{align*} + \int_{-\infty}^{w_0} h(w)\phi(w) dw &= h(0)\Phi(w_0)+\int_{-\infty}^{w_0} \frac{h(w)-h(0)}{w} w\phi(w)dw \nonumber\\ + &= h(0)\Phi(w_0)-\int_{-\infty}^{w_0} \frac{h(w)-h(0)}{w} d\phi(w)\quad \mbox{(car }w\phi(w)dw = -d\phi(w) \mbox{)} \nonumber\\ + &\approx h(0)\Phi(w_0) - \left[ \frac{h(w)-h(0)}{w} \phi(w) \right]_{w=-\infty}^{w=w_0}. \nonumber\\ +\end{align*} + +On ignore l'intégrale de l'intégration par parties, puisqu'elle prend +une petite valeur, pour obtenir l'approximation de Temme : +\begin{align} + \label{eq:temmeintegrale} + \int_{-\infty}^{w_0} h(w)\phi(w) dw &\approx h(0)\Phi(w_0) + + \phi(w_0) \left[ \frac{h(w_0)-h(0)}{w_0} \right]. +\end{align} + +\section{La méthode du point de selle} +\label{sec:pointcolGAL} + +L'approximation de la distribution d'une variable aléatoire par la +méthode du point de selle a été introduite par +\cite{daniels1954saddlepoint}. Il cherchait une façon d'estimer la +distribution de la moyenne de variables aléatoires. + +\subsection{Approximation de la densité} +\label{sec:appr-de-prem} + +Soit une variable aléatoire $Y$ dont on cherche à évaluer la fonction +de densité $f_Y(y)$. Les fonctions génératrices des moments $M_Y( s)$ +et des cumulants $K_Y( s)$ sont définies de manière équivalente par +les intégrales suivantes, où l'on cherche une forme qui permettra +d'utiliser l'approximation de Laplace: +\begin{align} + M_Y( s) = e^{K_Y( s)} &= \int_{-\infty}^{\infty} e^{ s y} f_Y(y) dy \label{eq:FGMdefinitionsaddle}\\ + &= \int_{-\infty}^{\infty} e^{ s y + \ln{f_Y(y)}} dy \label{eq:FGMdaniels}\\ + &= \int_{-\infty}^{\infty} e^{-g( s,y)} dy. +\end{align} + +Cette intégrale doit converger pour toute valeur $s$ comprise dans un +intervalle $\left]-c_1,c_2\right[$ contenant l'origine et où la somme +des bornes est positive: $c_1+c_2 > 0$. Cet intervalle doit être +choisi de sorte qu'il soit le plus grand possible. On considère pour +l'instant que la fonction $g( s,y) = - s y - \ln{f_Y(y)}$ répond aux +conditions de l'approximation de Laplace énoncées au début de la +section \ref{sec:appr-de-lapl}. On obtient alors l'approximation +suivante pour une valeur de $s$ fixée: +\begin{align} + \label{eq:laplacesaddle1} + e^{K(s)} &\approx \sqrt{\frac{2\pi}{g^{\prime\prime}( s,\hat{y}_{ + s})}}e^{ s \hat{y}_{ s}} f_Y(\hat{y}_{ s}). +\end{align} + +La valeur $\hat{y}_{s}$ est celle qui minimise la fonction $g(s,y)$ +pour un point $s$ fixé. C'est donc la solution de la condition de +premier ordre: +\begin{align} + 0 &= -g^{\prime}( s,\hat{y}_{ s}) \nonumber\\ + &= s + \frac{\partial \ln{(f_Y(\hat{y}_{ s})}}{\partial \hat{y}_{ s}} \nonumber\\ + s &= - \frac{\partial \ln{(f_Y(\hat{y}_{ s})}}{\partial \hat{y}_{ + s}}. \label{eq:premierordresaddle} +\end{align} + +L'hypothèse posée afin de faire l'approximation est que la fonction +$g(s)$ est convexe, ce qui implique que la dérivée seconde de celle-ci +est positive $(\frac{\partial^2g( s,y)}{\partial y^2} > 0)$ par +définition. On examine la relation entre le point de selle $s$ et le +point $\hat{y}_{ s}$ déterminé par l'équation +\eqref{eq:premierordresaddle}. En dérivant celle-ci par rapport au +point $\hat{y}_{ s}$, on constate qu'il existe une relation croissante +et monotone entre les deux: +\begin{align} + \label{eq:secondordresaddle} + \frac{d s}{d\hat{y}_{ s}} &= - \frac{\partial^2 \ln{(f_Y(\hat{y}_{ + s})}}{\partial \hat{y}_{ s}^2} > 0. +\end{align} + +On doit maintenant trouver une solution de l'équation de premier ordre +\eqref{eq:premierordresaddle}. Pour ce faire, on isole le terme +$\ln{f_Y(y)}$ dans l'équation \eqref{eq:laplacesaddle1}: +\begin{align} + \label{eq:lnfysaddlelaplace} + \ln{f_Y(y)} & \approx K( s) - s \hat{y}_{ s} - \frac{1}{2} + \ln{\left(\frac{2\pi}{-\left(\frac{\partial^2 \ln{(f_Y(\hat{y}_{ + s}))}}{\partial \hat{y}_{ s}^2}\right)}\right)}. +\end{align} + +Si l'on considère que le dernier terme de la soustraction est +relativement constant par rapport au point $\hat{y}_{ s}$, alors sa +dérivée sera pratiquement nulle. Ce terme peut donc être négligé dans +la prochaine étape, où on dérive \eqref{eq:lnfysaddlelaplace} par +rapport à $y_s$: +\begin{align} + \frac{\partial \ln{f_Y(\hat{y}_{s})}}{\partial \hat{y}_{s}} + &\approx \left(K^{\prime}(s) - \hat{y}_{s}\right)\frac{\partial + s}{\partial + \hat{y}_{s} } - s.\label{eq:dlnfysaddlelaplace} +\end{align} + +En combinant l'approximation \eqref{eq:dlnfysaddlelaplace} et la +condition de premier ordre \eqref{eq:premierordresaddle}, il en +résulte que le terme de gauche de la partie de droite est +approximativement égal à 0: +\begin{align} + \label{eq:premierodreapprox} + \left(K^{\prime}( s) - \hat{y}_{ s}\right)\frac{\partial s}{\partial + \hat{y}_{s} } &\approx 0. +\end{align} + +Cela conduit donc à l'équation qui met en relation le point de +selle $s$ et le point $\hat{y}_{ s}$: +\begin{align} + \label{eq:saddlepoint} + \hat{y}_{ s} &= K^{\prime}(s). +\end{align} + +Enfin, la dernière relation consiste à exprimer $g^{\prime\prime}( +s,\hat{y}_{ s})$ seulement en fonction du point de selle $s$ en +utilisant la condition de second ordre \eqref{eq:secondordresaddle} et +en dérivant l'équation précédente \eqref{eq:saddlepoint} en $s$: +\begin{align} + \label{eq:deriveesecondesaddlepoint} + g^{\prime\prime}( s,\hat{y}_{ s}) &= - \frac{\partial^2 + \ln{(f_Y(\hat{y_{s}}))}}{\partial \hat{y}_{ s}^2} \nonumber\\ + &= \frac{d s}{d\hat{y}_{ s}} \nonumber\\ + &= \left(\frac{d\hat{y}_{ s}}{d s}\right)^{-1} \nonumber\\ + &= \left(K^{\prime\prime}( s)\right)^{-1}. +\end{align} + +On peut donc réécrire l'approximation de Laplace +\eqref{eq:laplacesaddle1} en utilisant les relations +\eqref{eq:saddlepoint} et \eqref{eq:deriveesecondesaddlepoint}: +\begin{align} + e^{K( s)} &\approx \sqrt{2\pi K^{\prime\prime}( s)}\exp{\left\{ s + \hat{y}_{ s}\right\}} + f_Y(\hat{y_{s}}) \nonumber\\ + \hat{f_Y}(y_{s}) &\approx \frac{1}{\sqrt{2\pi K^{\prime\prime}( + \hat{s})}} \exp{\left\{K(\hat{s}) - \hat{s} y\right\}} = + \hat{f}_Y(y).\label{eq:approximationsaddlepointordre1} +\end{align} + +L'équation \eqref{eq:approximationsaddlepointordre1} est une +approximation par la méthode du point de selle de la densité de la +variable aléatoire $Y$ au point $y_{s}$. + +Par convention, on considère que l'approximation $\hat{f}_Y(y)$ est +une fonction de densité, mais, elle n'en est pas exactement une +puisqu'elle n'intègre pas à 1. Cependant, il est possible de la +normaliser en calculant la valeur de l'intégrale de celle-ci sur le +domaine $\chi$ de la distribution, puis en divisant l'approximation +$\hat{f}_Y(y)$ par celle-ci: +\begin{subequations}\label{eq:normalisationsaddle} + \begin{align} + c &= \int_{\chi} \hat{f}_Y(y) dy \neq 1 \label{eq:normalisationsaddle1}\\ + \overline{f}_Y(y) &= c^{-1} \hat{f}_Y(y),\quad \forall y \in + \chi. \label{eq:normalisationsaddle2} + \end{align} +\end{subequations} + +\subsection{Unicité du point de selle} +\label{sec:unicite-du-point} + +\cite{daniels1954saddlepoint} démontre que l'équation du point de +selle \eqref{eq:saddlepoint} a une solution unique dans l'intervalle +$\left[-c_1,c_2 \right]$ sélectionné précédemment lorsque $y_0$ est +situé dans un intervalle $\left[a,b\right]$. + +Pour ce faire, il présente une condition essentielle. Les limites +inférieure et supérieure de la dérivée première de la fonction +génératrice des cumulants doivent correspondre aux points $a$ et $b$ +respectivement \eqref{eq:limitesderiveeK}: +\begin{align} + \label{eq:limitesderiveeK} + \lim_{s\to c_2} K'(s) = b \\ + \lim_{s\to -c_1} K'(s) = a. +\end{align} + +On définit la fonction génératrice des moments de la différence entre +la variable aléatoire $Y$ et le point $y_0$ par $M(s,y)$. + +Quand $a < y_0 < b$, la dérivée première prend la forme suivante: +\begin{align} + \label{eq:6} + M'(s,y_0) = \int_{a}^{b} (y-y_0) e^{s(y-y_0)} dF(y). +\end{align} + +Les limites inférieure et supérieure de la fonction $M'(s,y_0)$ sont +respectivement $M'(-\infty,y_0) = -\infty$ et $M'(\infty,y_0) = +\infty$. De plus, la dérivée de celle-ci est toujours positive +$M''(s,y_0)>0$. Donc, pour chaque valeur $a < y_0 < b$, une racine +unique $\hat{s}$ de l'équation $M'(s,y_0)=0$ existe +nécessairement. Par conséquent, l'équation du point de selle +\eqref{eq:saddlepoint}, qui lui est équivalente, a aussi une solution +réelle unique. + +% \subsection{Approximation de second ordre de la densité} +% \label{sec:appr-de-second} + +% On reprend la définition de la fonction génératrice des moments +% \eqref{eq:FGMdefinitionsaddle}. En réorganisant les termes, on définit +% une fonction de densité décalée pour chaque valeur $ s \in +% \left]a,b\right[$: +% \begin{align} +% \label{eq:nouvelledensiteordre2} +% f_Y(y; s) = \exp{\left\{ s y - K( s)\right\}} f_Y(y) +% \end{align} + +% Cet ensemble est défini comme étant la famille de densités décalées +% par rapport au point de selle $s$, et a été développé par +% Esscher. Soit $Y_{ s}$ la variable aléatoire ayant cette densité, dont +% la moyenne et la variance sont définies comme suit: +% \begin{align} +% E[Y_{ s}] &= K^{\prime}( s) \label{eq:moyennetilted}\\ +% Var[Y_{ s}] &= K^{\prime\prime}( s) \label{eq:vartilted} +% \end{align} + +% On remarque que lorsque l'on situe le point de selle à l'origine +% $(s=0)$, on retrouve la définition usuelle des deux premiers +% cumulants. On définit les cumulants standardisés de la variable +% aléatoire $Y_{ s}$, qui sont basés sur les coefficients de +% l'approximation de Laplace du second ordre \eqref{eq:kappailaplace}. +% \begin{align} +% \hat{\kappa}_i &= +% \frac{K^{(i)}(\hat{x})}{\left\{K^{\prime\prime}(\hat{x})\right\}^{i/2}} +% ,\quad i\geq 3 \label{eq:kappaitilting} +% \end{align} + +% On définit l'expansion d'Edgeworth d'ordre 2 +% \nocite{abramowitz1965handbook} autour de la moyenne comme étant +% l'approximation suivante, basée sur la distribution normale +% $\mathcal{N}(\mu,\sigma^2)$: +% \begin{align} +% \label{eq:edgeworthmoyenne} +% f_Y(\mu) &\approx +% \frac{1}{\sigma\sqrt{2\pi}}\left(1+\frac{1}{8}\kappa_4 - +% \frac{5}{24} \kappa_3^2 \right) +% \end{align} + +% La technique développée par Esscher utilise indirectement cette +% expansion. Elle consiste en deux étapes: +% \begin{enumerate} +% \item Représenter la fonction de densité de la variable aléatoire $Y$ +% en l'isolant dans l'expression \eqref{eq:nouvelledensiteordre2}: +% \begin{align} +% \label{eq:etape1esscher} +% f_Y(y) = \exp{\left\{K( s)- s y \right\}} f_Y(y; s) +% \end{align} +% \item Utiliser l'expansion \eqref{eq:edgeworthmoyenne} de la densité +% $f_Y(y; s)$ autour d'un point $ s $ optimal, qui se trouve à être la +% moyenne de la variable $Y_{ s}$ définie par +% \eqref{eq:moyennetilted}, ce qui revient à résoudre l'équation du +% point de selle \eqref{eq:saddlepoint} en $s$: +% \begin{align*} +% E[Y_{\hat{s}}] &= K^{\prime}(\hat{s}) = y +% \end{align*} +% \end{enumerate} + +% L'expansion prend donc la forme suivante en remplaçant l'écart-type +% $\sigma$ par la variance de la variable aléatoire $Y_{ s}$, exprimée à +% l'aide de la fonction génératrice des cumulants \eqref{eq:vartilted}. +% \begin{align} +% \label{eq:edgeworthordre2} +% f_Y(y;\hat{s}) &\approx \frac{1}{\sqrt{2\pi +% K^{\prime\prime}(\hat{s})}} \left(1+\frac{1}{8}\kappa_4(\hat{s}) +% - \frac{5}{24} \kappa_3^2(\hat{s}) \right) +% \end{align} +% En substituant l'expansion \eqref{eq:edgeworthordre2} dans l'équation +% \eqref{eq:etape1esscher}, on obtient l'approximation par la méthode du +% point de selle d'ordre 2 de la densité de la variable aléatoire $Y$. +% \begin{align} +% \label{eq:approximationsaddlepointordre2} +% f_Y(y) &\approx \frac{1}{\sqrt{2\pi K^{\prime\prime}(\hat{s})}} +% \exp{\left\{K(\hat{s})-\hat{s} y \right\}} +% \left(1+\frac{1}{8}\kappa_4(\hat{s}) - \frac{5}{24} +% \kappa_3^2(\hat{s}) \right) +% \end{align} + +\subsection{Approximation de la fonction de + répartition} +\label{sec:appr-de-prem-1} + +La fonction de répartition de la variable aléatoire $Y$ au point $y$ +est définie comme étant la probabilité $P(y \leq Y)$ et s'obtient en +intégrant la densité sur l'intervalle $\left[ -\infty,y \right]$. On +considère le changement de variable $y \mapsto \hat{w}$ défini comme +suit: +\begin{align} + \label{eq:chengementytow} + \frac{\hat{w}^2}{2} = \hat{s} y - K(\hat{s}). +\end{align} + +La valeur $\hat{w}_y$ est obtenue en résolvant l'équation du point de +selle pour la racine $\hat{s}$ et en remplaçant celle-ci dans +l'équation \eqref{eq:chengementytow}. Ce changement de variable est +une transformation monotone, continue et croissante dont la dérivée +est donnée par l'équation suivante: +\begin{align} + \label{eq:derivchengementytow} + \frac{dy}{d\hat{w}} = \begin{cases} + \hat{w} / \hat{s}, & \hat{s} \neq 0 \\ + \sqrt{K^{\prime\prime}(0)}, & \hat{s} = 0. + \end{cases} +\end{align} + +L'approximation de premier ordre de la fonction de répartition +s'obtient en intégrant celle de premier ordre de la densité +\eqref{eq:approximationsaddlepointordre1} sur ce même intervalle à +l'aide du changement de variable \eqref{eq:chengementytow}: +\begin{align} + \label{eq:approxintegraleREPordre1} + F_Y(y) &\approx \int_{-\infty}^{y} \frac{1}{\sqrt{2\pi + K^{\prime\prime}( s)}} + \exp{\left\{K( s) - s x\right\}} dx \nonumber\\ + &= \int_{-\infty}^{y} \frac{1}{\sqrt{K^{\prime\prime}( s)}} + \phi(\hat{w}) dx \nonumber\\ + &= \int_{-\infty}^{\hat{w}_y} + \frac{\hat{w}}{\hat{s}\sqrt{K^{\prime\prime}( s)}}\phi(\hat{w}) + d\hat{w}. +\end{align} + +On applique l'approximation de Temme à l'intégrale précédente avec +l'équation: +\begin{align*} + h(\hat{w})=\frac{\hat{w}}{\hat{s}\sqrt{K^{\prime\prime}( s)}}. +\end{align*} + +On obtient ainsi l'approximation de premier ordre de la fonction de +répartition: +\begin{align} + \label{eq:approximationsaddlepointREPordre1} + F_Y(y) &\approx \Phi(\hat{w}_y) + + \phi(\hat{w}_y)\frac{1-h(\hat{w}_y)}{\hat{w}_y} \nonumber\\ + & \approx \Phi(\hat{w}_y) + \phi(\hat{w}_y) + \left(\frac{1}{\hat{w}_y}-\frac{1}{\hat{u}_y} \right). +\end{align} + +Les deux constantes $\hat{w}_y$ et $\hat{u}_y$ sont respectivement +définies comme suit: +\begin{subequations}\label{eq:wuysaddlerepart2} + \begin{align} + \hat{w}_y &= \sgn(\hat{s})\sqrt{2\left\{\hat{s} y - K(\hat{s})\right\}} \label{eq:wysaddlerepart2}\\ + \hat{u}_y &= + \hat{s}\sqrt{K^{\prime\prime}(\hat{s})} \label{eq:uysaddlerepart2}. + \end{align} +\end{subequations} + +% \subsection{Approximation de second ordre de la fonction de +% répartition} +% \label{sec:appr-de-second-1} + +% On reprend le développement précédent +% \eqref{eq:approxintegraleREPordre1}, mais avec l'approximation de +% second ordre de la densité \eqref{eq:approximationsaddlepointordre2} +% \citep{daniels1987tail}. +% \begin{align} +% \label{eq:approxintegraleREPordre2} +% F_Y(y) &\approx \int_{-\infty}^{y} \frac{\exp{\left\{K( s) - s +% x\right\}}}{\sqrt{2\pi K^{\prime\prime}( s)}} +% \left(1+\frac{1}{8}\kappa_4(\hat{s}) - \frac{5}{24} +% \kappa_3^2(\hat{s}) \right) dx \nonumber\\ +% &= \int_{-\infty}^{\hat{w}_y} +% \frac{\hat{w}}{\hat{s}\sqrt{K^{\prime\prime}( +% s)}}\left(1+\frac{1}{8}\kappa_4(\hat{s}) - \frac{5}{24} +% \kappa_3^2(\hat{s}) \right)\phi(\hat{w}) d\hat{w} +% \end{align} + +% La fonction à utiliser pour l'approximation de Temme est alors +% \begin{align*} +% h(\hat{w})=\frac{\hat{w}}{\hat{s}\sqrt{K^{\prime\prime}( +% s)}}\left(1+\frac{1}{8}\kappa_4(\hat{s}) - \frac{5}{24} +% \kappa_3^2(\hat{s}) \right)\phi(\hat{w}) +% \end{align*} + +% Ainsi, on obtient l'approximation de second ordre de la fonction de +% répartition. +% \begin{align} +% \label{eq:approximationsaddlepointREPordre2} +% F_Y(y) & \approx \Phi(\hat{w}_y) + \phi(\hat{w}_y) +% \left(\frac{1}{\hat{w}_y}-\frac{1}{\hat{u}_y}(1-\frac{1}{8} +% \hat{\kappa}_4+\frac{5}{24}\hat{\kappa}_3^2) \right) +% \end{align} + +\subsection{Quelques propriétés des approximations} +\label{sec:proprietesaddle} + +Les approximations par la méthode du point de selle ont pour +principale propriété de respecter le principe d'invariance. Cette +propriété permet notamment d'obtenir une approximation de la densité +ainsi que la fonction de répartition en utilisant des paramètres +estimés à partir d'un échantillon de données centrées et +réduites. Puis, elle permet d'appliquer la transformation +linéaire $Y = \sigma{X}+\mu$, telle qu'utilisée à la section +\ref{sec:transGAL}, tout en conservant la même approximation. + +Une autre propriété de ces approximations est qu'elle respecte la +symétrie de la distribution. Ainsi, lorsque la distribution est +asymétrique, on doit s'attendre à ce que l'approximation le soit +aussi. + +\section{Application de la méthode du point de selle} + +\subsection{Approximation de la densité} +\label{sec:applicationsaddleGAL} + +On applique la méthode du point de selle à la distribution de Laplace +asymétrique généralisée, afin d'obtenir une approximation de la +densité \eqref{eq:densitekotz} et aussi pouvoir représenter la +fonction de répartition, car la première n'est pas intégrable +analytiquement. Cependant, pour des fins de simplification, la +paramétrisation $\mu$ sera utilisée. On rappelle qu'on peut passer +d'une forme à l'autre à l'aide des équations \eqref{eq:mukappa} et +\eqref{eq:kappamu}. + +On évalue d'abord la dérivée première de la fonction génératrice des +cumulants \eqref{eq:fgcGAL} par rapport à la variable de +transformation $s$: +\begin{align} + \label{eq:derivcumulantGAL1} + K^{\prime}_Y({s}) &= + \frac{\partial}{\partial{s}}\ln\left(\frac{e^{\theta + {s}}}{\left(1-\frac{1}{2} \sigma^2 {s}^2 - \mu {s} + \right)^{\tau}}\right) \nonumber\\ + &= \frac{{\sigma}^{2}\theta{{s}}^{2}+\left( + 2\mu\theta-2{\sigma}^{2}\tau\right) + {s}-2\theta-2\mu\tau}{{\sigma}^{2}{{s}}^{2}+2\mu{s}-2}, \\ & \quad + \mbox{où } 1-\frac{1}{2} \sigma^2 {s}^2 - \mu {s} > 0. +\end{align} + +On détermine ensuite le point de selle $\hat{s}_y$ correspondant à la +valeur $y$ pour laquelle on veut estimer la fonction de densité ou de +répartition, à l'aide de l'équation \eqref{eq:saddlepoint} et de la +première dérivée de la fonction génératrice des cumulants +\eqref{eq:derivcumulantGAL1}: +\begin{align*} + y &= K^{\prime}_Y({\hat{s}_y}) \\ + &= \frac{{\sigma}^{2}\theta{\hat{s}_y}^{2}+\left( + 2\mu\theta-2{\sigma}^{2}\tau\right) + {\hat{s}_y}-2\theta-2\mu\tau}{{\sigma}^{2}{{\hat{s}_y}}^{2}+2\mu{\hat{s}_y}-2}. +\end{align*} + +L'équation précédente a deux solutions dont une seule correspond à un +minimum. Pour déterminer quelle solution est appropriée, on doit +évaluer la dérivée seconde en ces deux points: +\begin{align} + \label{eq:saddlepointGAL} + \hat{s}_y &= + \frac{\pm\sqrt{2{\sigma}^{2}{{y}}^{2}+{\mu}^{2}{{y}}^{2}-4{\sigma}^{2}\theta{y}-2{\mu}^{2}\theta{y}+2{\sigma}^{2}{\theta}^{2}+{\mu}^{2}{\theta}^{2}+{\sigma}^{4}{\tau}^{2}}}{{\sigma}^{2}(y-\theta)} + -\frac{\mu}{\sigma^{2}} -\frac{\tau}{y-\theta}. +\end{align} + +On évalue la dérivée seconde de la fonction génératrice des cumulants: +\begin{align} + \label{eq:derivcumulantGAL2} + K^{\prime\prime}_Y({s}) &= \frac{2\left( + {s}^{2}{\sigma}^{4}+2\mu{s}{\sigma}^{2}+2{\sigma}^{2}+2{\mu}^{2}\right) + \tau}{{\left( {s}^{2}{\sigma}^{2}+2\mu{s}-2\right) }^{2}} > 0. +\end{align} + +On peut maintenant déterminer quelle valeur $\hat{s}_y$ utiliser, car +la fonction $K^{\prime\prime}_Y({s})$ sera positive pour celle-ci. De manière équivalente, on utilise la condition +\ref{eq:fgmGALcond}, qui permet l'existence de la fonction génératrice +des moments, afin d'identifier plus aisément la solution unique. + +On peut maintenant évaluer \eqref{eq:approximationsaddlepointordre1}, +pour obtenir l'approximation de premier ordre de la fonction de +densité de la distribution de Laplace asymétrique généralisée. + +% \subsection{Approximation de second ordre de la densité} + +% L'évaluation de l'approximation de second ordre nécessite les +% dérivées troisième et quatrième de la fonction génératrice des +% cumulants: +% \begin{align} +% \label{eq:K3K4GAL} +% K^{\prime\prime\prime}({s}) &= -\frac{4\tau\left( {\sigma}^{2}{s}+\mu\right) \left( {\sigma}^{4}{{s}}^{2}+2\mu{\sigma}^{2}{s}+6{\sigma}^{2}+4{\mu}^{2}\right) }{{\left( {\sigma}^{2}{{s}}^{2}+2\mu{s}-2\right) }^{3}} \\ +% K^{iv}({s}) &= \frac{12\tau\left( +% {\sigma}^{8}{s}^{4}+4\mu{\sigma}^{6}{s}^{3}+12{\sigma}^{6}{s}^{2}+12{\mu}^{2}{\sigma}^{4}{s}^{2}+24\mu{\sigma}^{4}{s}+16{\mu}^{3}{\sigma}^{2}{s}+4{\sigma}^{4}+16{\mu}^{2}{\sigma}^{2}+8{\mu}^{4}\right) +% }{{\left( {\sigma}^{2}{s}^{2}+2\mu{s}-2\right) }^{4}} +% \end{align} + +% On peut ainsi évaluer les cumulants normalisés $\kappa_3(s)$ et +% $\kappa_4(s)$ à l'aide de la définition \eqref{eq:kappaitilting}. +% \begin{align} +% \kappa_3(s) &= -\frac{\sqrt{2}\left( s{\sigma}^{2}+\mu\right) \left( {s}^{2}{\sigma}^{4}+2\mu{s}{\sigma}^{2}+6{\sigma}^{2}+4{\mu}^{2}\right) \left| {s}^{2}{\sigma}^{2}+2\mu{s}-2\right| \tau}{\left( {s}^{2}{\sigma}^{2}+2\mu{s}-2\right) {\left( \left( {s}^{2}{\sigma}^{4}+2\mu{s}{\sigma}^{2}+2{\sigma}^{2}+2{\mu}^{2}\right) \tau\right) }^{\frac{3}{2}}} \label{eq:kappa3gal}\\ +% \kappa_4(s) &= \frac{3\left( +% {s}^{4}{\sigma}^{8}+4\mu{s}^{3}{\sigma}^{6}+12{s}^{2}{\sigma}^{6}+12{\mu}^{2}{s}^{2}{\sigma}^{4}+24\mu{s}{\sigma}^{4}+4{\sigma}^{4}+16{\mu}^{3}{s}{\sigma}^{2}+16{\mu}^{2}{\sigma}^{2}+8{\mu}^{4}\right) +% }{{\left( +% {s}^{2}{\sigma}^{4}+2\mu{s}{\sigma}^{2}+2{\sigma}^{2}+2{\mu}^{2}\right) +% }^{2}\tau} \label{eq:kappa4gal} +% \end{align} + +% En remplaçant ces valeurs dans l'équation +% \eqref{eq:approximationsaddlepointordre2}, on obtient +% l'approximation de second ordre de la densité de la distribution de +% Laplace asymétrique généralisée. + +\subsection{Approximation de la fonction de répartition} + +En évaluant les valeurs $\hat{w}_y \mbox{ et } \hat{u}_y$ +\eqref{eq:wuysaddlerepart2} à l'aide de celles du point de selle +\eqref{eq:saddlepointGAL}, de la fonction génératrice des cumulants +\eqref{eq:fgcGAL} et de la dérivée seconde de cette dernière +\eqref{eq:derivcumulantGAL2}, on obtient l'approximation de premier +ordre de la fonction de répartition. + +% Pour évaluer l'approximation de second ordre de la fonction de +% répartition \eqref{eq:approximationsaddlepointREPordre2}, on reprend +% les cumulants normalisés $\kappa_3(s) \text{ et } \kappa_4(s)$ +% \eqref{eq:kappa3gal} \eqref{eq:kappa4gal} et on les remplace dans +% l'expression \eqref{eq:approximationsaddlepointREPordre2}. + +Tout au long de ce chapitre, on a supposé que les paramètres de la +distribution étaient connus. Cependant, ceux-ci doivent être estimés à +partir des données d'un échantillon représentatif de la population +étudiée. C'est d'ailleurs ce qui sera fait dans les chapitres +suivants. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/chapitre4.aux b/memoire/chapitre4.aux new file mode 100644 index 0000000..41c218a --- /dev/null +++ b/memoire/chapitre4.aux @@ -0,0 +1,264 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{hansen1982large} +\citation{hamilton1994time} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {chapter}{\chapternumberline {4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{49}{chapter.4}} +\@writefile{toc}{\contentsline {section}{\numberline {4.1}Introduction}{49}{section.4.1}} +\newlabel{sec:intromethodeGMM}{{4.1}{49}{Introduction}{section.4.1}{}} +\newlabel{sec:intromethodeGMM@cref}{{[section][1][4]4.1}{49}} +\newlabel{eq:vraisemblance}{{4.1.1}{49}{Introduction}{equation.4.1.0.1}{}} +\newlabel{eq:vraisemblance@cref}{{[equation][1][4,1]4.1.1}{49}} +\newlabel{eq:thetavraisemblance}{{4.1.2}{49}{Introduction}{equation.4.1.0.2}{}} +\newlabel{eq:thetavraisemblance@cref}{{[equation][2][4,1]4.1.2}{49}} +\newlabel{eq:EEvraisemblance}{{4.1.3}{49}{Introduction}{equation.4.1.0.3}{}} +\newlabel{eq:EEvraisemblance@cref}{{[equation][3][4,1]4.1.3}{49}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.1.1}M\IeC {\'e}thode classique des moments}{50}{subsection.4.1.1}} +\newlabel{sec:methodemoments}{{4.1.1}{50}{Méthode classique des moments}{subsection.4.1.1}{}} +\newlabel{sec:methodemoments@cref}{{[subsection][1][4,1]4.1.1}{50}} +\newlabel{eq:momentspopulation}{{4.1.4}{50}{Méthode classique des moments}{equation.4.1.0.4}{}} +\newlabel{eq:momentspopulation@cref}{{[equation][4][4,1]4.1.4}{50}} +\newlabel{eq:momentsechantillon}{{4.1.5}{50}{Méthode classique des moments}{equation.4.1.0.5}{}} +\newlabel{eq:momentsechantillon@cref}{{[equation][5][4,1]4.1.5}{50}} +\@writefile{toc}{\contentsline {section}{\numberline {4.2}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{50}{section.4.2}} +\newlabel{sec:methodeGMM}{{4.2}{50}{Méthode des moments généralisée}{section.4.2}{}} +\newlabel{sec:methodeGMM@cref}{{[section][2][4]4.2}{50}} +\citation{hansen1982large} +\citation{berkson1980minimum} +\citation{wolfowitz1957minimum} +\newlabel{eq:1}{{4.2.1}{51}{Méthode des moments généralisée}{equation.4.2.0.1}{}} +\newlabel{eq:1@cref}{{[equation][1][4,2]4.2.1}{51}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.2.1}D\IeC {\'e}finition}{51}{subsection.4.2.1}} +\newlabel{sec:definitionGMM}{{4.2.1}{51}{Définition}{subsection.4.2.1}{}} +\newlabel{sec:definitionGMM@cref}{{[subsection][1][4,2]4.2.1}{51}} +\newlabel{eq:def1condmoment}{{4.2.3}{51}{Définition}{equation.4.2.0.3}{}} +\newlabel{eq:def1condmoment@cref}{{[equation][3][4,2]4.2.3}{51}} +\newlabel{eq:estimateurfonctiongh}{{4.2.5}{51}{Définition}{equation.4.2.0.5}{}} +\newlabel{eq:estimateurfonctiongh@cref}{{[equation][5][4,2]4.2.5}{51}} +\citation{hansen1982large} +\newlabel{eq:objectifGMM1}{{4.2.6}{52}{Définition}{equation.4.2.0.6}{}} +\newlabel{eq:objectifGMM1@cref}{{[equation][6][4,2]4.2.6}{52}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.2.2}Convergence}{52}{subsection.4.2.2}} +\newlabel{sec:convergenceGMM}{{4.2.2}{52}{Convergence}{subsection.4.2.2}{}} +\newlabel{sec:convergenceGMM@cref}{{[subsection][2][4,2]4.2.2}{52}} +\newlabel{eq:paramegalecondmomentsGMM}{{4.2.7}{52}{Convergence}{equation.4.2.0.7}{}} +\newlabel{eq:paramegalecondmomentsGMM@cref}{{[equation][7][4,2]4.2.7}{52}} +\newlabel{eq:normequadratique}{{4.2.8}{52}{Convergence}{equation.4.2.0.8}{}} +\newlabel{eq:normequadratique@cref}{{[equation][8][4,2]4.2.8}{52}} +\newlabel{eq:conditionGMM1.1}{{4.2.9}{52}{Convergence}{equation.4.2.0.9}{}} +\newlabel{eq:conditionGMM1.1@cref}{{[equation][9][4,2]4.2.9}{52}} +\newlabel{eq:conditionGMM1.2}{{4.2.10}{53}{Convergence}{equation.4.2.0.10}{}} +\newlabel{eq:conditionGMM1.2@cref}{{[equation][10][4,2]4.2.10}{53}} +\newlabel{eq:conditionGMM2.1}{{4.2.11}{53}{Convergence}{equation.4.2.0.11}{}} +\newlabel{eq:conditionGMM2.1@cref}{{[equation][11][4,2]4.2.11}{53}} +\newlabel{eq:conditionGMM2.2}{{4.2.12}{53}{Convergence}{equation.4.2.0.12}{}} +\newlabel{eq:conditionGMM2.2@cref}{{[equation][12][4,2]4.2.12}{53}} +\newlabel{eq:estimateurGMM}{{4.2.13}{53}{Convergence}{equation.4.2.0.13}{}} +\newlabel{eq:estimateurGMM@cref}{{[equation][13][4,2]4.2.13}{53}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.2.3}Matrice de pond\IeC {\'e}ration optimale}{53}{subsection.4.2.3}} +\newlabel{sec:matriceWoptimaleGMM}{{4.2.3}{53}{Matrice de pondération optimale}{subsection.4.2.3}{}} +\newlabel{sec:matriceWoptimaleGMM@cref}{{[subsection][3][4,2]4.2.3}{53}} +\newlabel{eq:matricevcov1}{{4.2.14}{53}{Matrice de pondération optimale}{equation.4.2.0.14}{}} +\newlabel{eq:matricevcov1@cref}{{[equation][14][4,2]4.2.14}{53}} +\citation{hamilton1994time} +\citation{wooldridge2001econometric} +\citation{hall2005generalized} +\newlabel{eq:matriceWinvercevcov}{{4.2.16}{54}{Matrice de pondération optimale}{equation.4.2.0.16}{}} +\newlabel{eq:matriceWinvercevcov@cref}{{[equation][16][4,2]4.2.16}{54}} +\newlabel{eq:objectifGMM2}{{4.2.17}{54}{Matrice de pondération optimale}{equation.4.2.0.17}{}} +\newlabel{eq:objectifGMM2@cref}{{[equation][17][4,2]4.2.17}{54}} +\newlabel{eq:estimateurGMM2}{{4.2.18}{54}{Matrice de pondération optimale}{equation.4.2.0.18}{}} +\newlabel{eq:estimateurGMM2@cref}{{[equation][18][4,2]4.2.18}{54}} +\newlabel{eq:matponderationproduith}{{4.2.19}{54}{Matrice de pondération optimale}{equation.4.2.0.19}{}} +\newlabel{eq:matponderationproduith@cref}{{[equation][19][4,2]4.2.19}{54}} +\newlabel{eq:matvcovGMM}{{4.2.21}{54}{Matrice de pondération optimale}{equation.4.2.0.21}{}} +\newlabel{eq:matvcovGMM@cref}{{[equation][21][4,2]4.2.21}{54}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.2.4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e it\IeC {\'e}rative}{55}{subsection.4.2.4}} +\newlabel{sec:GMMtwostep}{{4.2.4}{55}{Méthode des moments généralisée itérative}{subsection.4.2.4}{}} +\newlabel{sec:GMMtwostep@cref}{{[subsection][4][4,2]4.2.4}{55}} +\newlabel{eq:criterearret}{{4.2.23}{55}{Méthode des moments généralisée itérative}{equation.4.2.0.23}{}} +\newlabel{eq:criterearret@cref}{{[equation][23][4,2]4.2.23}{55}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.2.5}Distribution asymptotique des estimateurs}{55}{subsection.4.2.5}} +\newlabel{sec:matvcovGMM}{{4.2.5}{55}{Distribution asymptotique des estimateurs}{subsection.4.2.5}{}} +\newlabel{sec:matvcovGMM@cref}{{[subsection][5][4,2]4.2.5}{55}} +\newlabel{eq:premierordreGMM}{{4.2.24}{55}{Distribution asymptotique des estimateurs}{equation.4.2.0.24}{}} +\newlabel{eq:premierordreGMM@cref}{{[equation][24][4,2]4.2.24}{55}} +\citation{gourieroux1989statistique} +\newlabel{eq:gradientGMM}{{4.2.25}{56}{Distribution asymptotique des estimateurs}{equation.4.2.0.25}{}} +\newlabel{eq:gradientGMM@cref}{{[equation][25][4,2]4.2.25}{56}} +\newlabel{eq:TCL-GMM}{{4.2.26}{56}{Distribution asymptotique des estimateurs}{equation.4.2.0.26}{}} +\newlabel{eq:TCL-GMM@cref}{{[equation][26][4,2]4.2.26}{56}} +\newlabel{eq:varasympt.1}{{4.2.27}{56}{Distribution asymptotique des estimateurs}{equation.4.2.0.27}{}} +\newlabel{eq:varasympt.1@cref}{{[equation][27][4,2]4.2.27}{56}} +\newlabel{eq:varasympt.3}{{4.2.28}{56}{Distribution asymptotique des estimateurs}{equation.4.2.0.28}{}} +\newlabel{eq:varasympt.3@cref}{{[equation][28][4,2]4.2.28}{56}} +\newlabel{eq:taylorfonction.g}{{4.2.29}{57}{Distribution asymptotique des estimateurs}{equation.4.2.0.29}{}} +\newlabel{eq:taylorfonction.g@cref}{{[equation][29][4,2]4.2.29}{57}} +\newlabel{eq:taylorfonction.gprod}{{4.2.30}{57}{Distribution asymptotique des estimateurs}{equation.4.2.0.30}{}} +\newlabel{eq:taylorfonction.gprod@cref}{{[equation][30][4,2]4.2.30}{57}} +\newlabel{eq:taylorfonction.gprod2}{{4.2.31}{57}{Distribution asymptotique des estimateurs}{equation.4.2.0.31}{}} +\newlabel{eq:taylorfonction.gprod2@cref}{{[equation][31][4,2]4.2.31}{57}} +\newlabel{eq:taylorfonction.gprod3}{{4.2.32}{57}{Distribution asymptotique des estimateurs}{equation.4.2.0.32}{}} +\newlabel{eq:taylorfonction.gprod3@cref}{{[equation][32][4,2]4.2.32}{57}} +\newlabel{eq:taylorfonction.gprod4}{{4.2.33}{57}{Distribution asymptotique des estimateurs}{equation.4.2.0.33}{}} +\newlabel{eq:taylorfonction.gprod4@cref}{{[equation][33][4,2]4.2.33}{57}} +\newlabel{matricevcovparamGMMnc}{{4.2.35}{58}{Distribution asymptotique des estimateurs}{equation.4.2.0.35}{}} +\newlabel{matricevcovparamGMMnc@cref}{{[equation][35][4,2]4.2.35}{58}} +\@writefile{toc}{\contentsline {section}{\numberline {4.3}Estimation sous contraintes}{58}{section.4.3}} +\newlabel{sec:estimGMMcontraint}{{4.3}{58}{Estimation sous contraintes}{section.4.3}{}} +\newlabel{sec:estimGMMcontraint@cref}{{[section][3][4]4.3}{58}} +\newlabel{eq:contraintelin0}{{4.3.1}{58}{Estimation sous contraintes}{equation.4.3.0.1}{}} +\newlabel{eq:contraintelin0@cref}{{[equation][1][4,3]4.3.1}{58}} +\newlabel{eq:contraintelin}{{4.3.2}{58}{Estimation sous contraintes}{equation.4.3.0.2}{}} +\newlabel{eq:contraintelin@cref}{{[equation][2][4,3]4.3.2}{58}} +\newlabel{eq:gradientcontrainte}{{4.3.3}{58}{Estimation sous contraintes}{equation.4.3.0.3}{}} +\newlabel{eq:gradientcontrainte@cref}{{[equation][3][4,3]4.3.3}{58}} +\newlabel{eq:estimateurGMMlagrange}{{4.3.4}{59}{Estimation sous contraintes}{equation.4.3.0.4}{}} +\newlabel{eq:estimateurGMMlagrange@cref}{{[equation][4][4,3]4.3.4}{59}} +\newlabel{eq:lagrangienGMMcontraint}{{4.3.5}{59}{Estimation sous contraintes}{equation.4.3.0.5}{}} +\newlabel{eq:lagrangienGMMcontraint@cref}{{[equation][5][4,3]4.3.5}{59}} +\newlabel{eq:premierordreGMMlagrange1}{{4.3.6}{59}{Estimation sous contraintes}{equation.4.3.0.6}{}} +\newlabel{eq:premierordreGMMlagrange1@cref}{{[equation][6][4,3]4.3.6}{59}} +\newlabel{eq:premierordreGMMlagrange2}{{4.3.7}{59}{Estimation sous contraintes}{equation.4.3.0.7}{}} +\newlabel{eq:premierordreGMMlagrange2@cref}{{[equation][7][4,3]4.3.7}{59}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.3.1}Distribution asymptotique des estimateurs contraints}{59}{subsection.4.3.1}} +\newlabel{sec:matvcovGMMconst}{{4.3.1}{59}{Distribution asymptotique des estimateurs contraints}{subsection.4.3.1}{}} +\newlabel{sec:matvcovGMMconst@cref}{{[subsection][1][4,3]4.3.1}{59}} +\newlabel{eq:premierordreGMMlagrange1.1-2}{{4.3.8}{59}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.8.8}{}} +\newlabel{eq:premierordreGMMlagrange1.1-2@cref}{{[equation][8][4,3]4.3.8}{59}} +\newlabel{eq:premierordreGMMlagrange1.1}{{4.3.8a}{59}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.8a}{}} +\newlabel{eq:premierordreGMMlagrange1.1@cref}{{[equation][1][4,3,8]4.3.8a}{59}} +\newlabel{eq:premierordreGMMlagrange1.2}{{4.3.8b}{59}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.8b}{}} +\newlabel{eq:premierordreGMMlagrange1.2@cref}{{[equation][2][4,3,8]4.3.8b}{59}} +\newlabel{eq:fisherJGMMlagrange}{{4.3.9}{59}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.9}{}} +\newlabel{eq:fisherJGMMlagrange@cref}{{[equation][9][4,3]4.3.9}{59}} +\newlabel{eq:premierordreGMMnc1.1}{{4.3.10}{60}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.10}{}} +\newlabel{eq:premierordreGMMnc1.1@cref}{{[equation][10][4,3]4.3.10}{60}} +\newlabel{eq:premierordreGMMlagrange2.1}{{4.3.11}{60}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.11}{}} +\newlabel{eq:premierordreGMMlagrange2.1@cref}{{[equation][11][4,3]4.3.11}{60}} +\newlabel{eq:premierordreGMMlagrange2.2}{{4.3.12}{60}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.12}{}} +\newlabel{eq:premierordreGMMlagrange2.2@cref}{{[equation][12][4,3]4.3.12}{60}} +\newlabel{eq:premierordreGMMlagrange3.1}{{4.3.13}{60}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.13}{}} +\newlabel{eq:premierordreGMMlagrange3.1@cref}{{[equation][13][4,3]4.3.13}{60}} +\newlabel{eq:premierordreGMMlagrange3.2}{{4.3.14}{60}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.14}{}} +\newlabel{eq:premierordreGMMlagrange3.2@cref}{{[equation][14][4,3]4.3.14}{60}} +\newlabel{eq:LagrangienJ.GMM}{{4.3.15}{60}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.15}{}} +\newlabel{eq:LagrangienJ.GMM@cref}{{[equation][15][4,3]4.3.15}{60}} +\newlabel{eq:contraintvsncGMM}{{4.3.16}{60}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.16}{}} +\newlabel{eq:contraintvsncGMM@cref}{{[equation][16][4,3]4.3.16}{60}} +\newlabel{eq:VcontraintGMM}{{4.3.17}{60}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.17}{}} +\newlabel{eq:VcontraintGMM@cref}{{[equation][17][4,3]4.3.17}{60}} +\citation{newey1994large} +\newlabel{eq:distcontraintGMM}{{4.3.18}{61}{Distribution asymptotique des estimateurs contraints}{equation.4.3.0.18}{}} +\newlabel{eq:distcontraintGMM@cref}{{[equation][18][4,3]4.3.18}{61}} +\@writefile{toc}{\contentsline {section}{\numberline {4.4}Tests d'hypoth\IeC {\`e}ses param\IeC {\'e}triques}{61}{section.4.4}} +\newlabel{sec:testparam}{{4.4}{61}{Tests d'hypothèses paramétriques}{section.4.4}{}} +\newlabel{sec:testparam@cref}{{[section][4][4]4.4}{61}} +\newlabel{eq:hypcontraintelin}{{4.4.1}{61}{Tests d'hypothèses paramétriques}{equation.4.4.0.1}{}} +\newlabel{eq:hypcontraintelin@cref}{{[equation][1][4,4]4.4.1}{61}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.1}Test de Wald}{61}{subsection.4.4.1}} +\newlabel{sec:testwald}{{4.4.1}{61}{Test de Wald}{subsection.4.4.1}{}} +\newlabel{sec:testwald@cref}{{[subsection][1][4,4]4.4.1}{61}} +\newlabel{eq:7}{{4.4.2}{61}{Test de Wald}{equation.4.4.0.2}{}} +\newlabel{eq:7@cref}{{[equation][2][4,4]4.4.2}{61}} +\newlabel{eq:moyennevariancesomme}{{4.4.3}{61}{Test de Wald}{equation.4.4.0.3}{}} +\newlabel{eq:moyennevariancesomme@cref}{{[equation][3][4,4]4.4.3}{61}} +\citation{newey1987hypothesis} +\newlabel{eq:statistiqueWald}{{4.4.5}{62}{Test de Wald}{equation.4.4.0.5}{}} +\newlabel{eq:statistiqueWald@cref}{{[equation][5][4,4]4.4.5}{62}} +\newlabel{eq:statistiqueWald2}{{4.4.6}{62}{Test de Wald}{equation.4.4.0.6}{}} +\newlabel{eq:statistiqueWald2@cref}{{[equation][6][4,4]4.4.6}{62}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.2}Test du multiplicateur de Lagrange}{62}{subsection.4.4.2}} +\newlabel{sec:testscore}{{4.4.2}{62}{Test du multiplicateur de Lagrange}{subsection.4.4.2}{}} +\newlabel{sec:testscore@cref}{{[subsection][2][4,4]4.4.2}{62}} +\newlabel{statistiqueLM}{{4.4.7}{62}{Test du multiplicateur de Lagrange}{equation.4.4.0.7}{}} +\newlabel{statistiqueLM@cref}{{[equation][7][4,4]4.4.7}{62}} +\newlabel{eq:statistiqueLM2}{{4.4.8}{63}{Test du multiplicateur de Lagrange}{equation.4.4.0.8}{}} +\newlabel{eq:statistiqueLM2@cref}{{[equation][8][4,4]4.4.8}{63}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.3}Test bas\IeC {\'e} sur la statistique de m\IeC {\'e}trique de distance}{63}{subsection.4.4.3}} +\newlabel{statistiqueD}{{4.4.9}{63}{Test basé sur la statistique de métrique de distance}{equation.4.4.0.9}{}} +\newlabel{statistiqueD@cref}{{[equation][9][4,4]4.4.9}{63}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.4}En r\IeC {\'e}sum\IeC {\'e}}{63}{subsection.4.4.4}} +\newlabel{sec:resumetests}{{4.4.4}{63}{En résumé}{subsection.4.4.4}{}} +\newlabel{sec:resumetests@cref}{{[subsection][4][4,4]4.4.4}{63}} +\@writefile{lot}{\contentsline {table}{\numberline {4.1}{\ignorespaces Tests d'hypoth\IeC {\`e}se param\IeC {\'e}triques pour la m\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{63}{table.caption.25}} +\newlabel{tab:testsparamGMM}{{4.1}{63}{Tests d'hypothèse paramétriques pour la méthode des moments généralisée\relax }{table.caption.25}{}} +\newlabel{tab:testsparamGMM@cref}{{[table][1][4]4.1}{63}} +\@setckpt{chapitre4}{ +\setcounter{page}{64} +\setcounter{equation}{9} +\setcounter{enumi}{3} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{4} +\setcounter{section}{4} +\setcounter{subsection}{4} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{82} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{1} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{24} +\setcounter{Hfootnote}{1} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{64} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/chapitre4.tex b/memoire/chapitre4.tex new file mode 100644 index 0000000..7debc62 --- /dev/null +++ b/memoire/chapitre4.tex @@ -0,0 +1,1033 @@ +\chapter{Méthode des moments généralisée} % numéroté + +Si l'on pose l'hypothèse selon laquelle un échantillon issu d'une +population a une certaine distribution de probabilité, on doit +ensuite déterminer quels sont les paramètres de celle-ci. Ces +paramètres peuvent être estimés à l'aide de différentes méthodes +statistiques. La méthode des moments généralisée est une de celles-ci. + +\section{Introduction} +\label{sec:intromethodeGMM} + +On considère un échantillon de taille $T$ formé de plusieurs +réalisations $\mathbf{y} = y(1),\ldots,y(T)$ indépendantes entre elles +et identiquement distribuées. Selon le modèle paramétrique, cet +échantillon est formé de réalisations d'une variable aléatoire $Y$ +suivant une distribution particulière. Le vecteur de paramètres de +celle-ci, $\theta$, de longueur $a$, appartenant à l'espace $\Omega +\subset \mathbb{R}^a$, doit être estimé à partir de l'échantillon. Une +première approche consiste à maximiser la fonction de vraisemblance +$LL(\theta;\mathbf{y})$, qui équivaut au produit de la densité +$f(y;\theta)$ évaluée à chacune des réalisations $y(t)$: +\begin{align} + \label{eq:vraisemblance} + LL(\theta;\mathbf{y}) = \prod_{t=1}^T + f(y(t);\theta),\quad\theta\in\Omega. +\end{align} + +L'estimateur $\hat\theta$ est, dans ce cas, la valeur pour +laquelle l'échantillon a la plus grande probabilité d'être +observé. Cependant, puisque cela apporte plusieurs simplifications +intéressantes, on cherchera à maximiser la fonction de +log-vraisemblance $l(\theta;\mathbf{y}) = +\ln{(L(\theta;\mathbf{y}))}$: +\begin{align} + \label{eq:thetavraisemblance} + \hat\theta = \underset{\theta}{\operatorname{arg\,max}} \, + l(\theta;\mathbf{y}). +\end{align} + +Le maximum est obtenu en résolvant la condition de premier ordre, +c'est-à-dire en égalant le gradient de la fonction de +log-vraisemblance à 0: +\begin{align} + \label{eq:EEvraisemblance} + \frac{\partial{l(\theta;\mathbf{y})}}{\partial{\theta}} &= 0. +\end{align} + +Cette méthode permet d'obtenir un estimateur convergent de variance +minimale, car elle utilise l'ensemble de l'information contenue dans +l'échantillon. Cependant, la fonction de vraisemblance doit pouvoir +être spécifiée sous une forme analytique, et de plus, celle-ci doit +être différentiable afin de pouvoir utiliser cette méthode, ce qui +n'est pas le cas avec la distribution de Laplace asymétrique +généralisée. Lorsque cette situation se présente, une méthode +alternative est préférée, celle des moments généralisée, comme +proposée par \cite{hansen1982large}. Celle-ci est décrite par +\cite{hamilton1994time} dans le contexte de l'étude des séries +chronologiques. Elle a pour avantage de nécessiter seulement la +spécification de certaines conditions de moment. Par contre, elle +n'utilise pas toute l'information fournie par l'échantillon, ce qui ne +permettra pas d'obtenir un estimateur de variance minimale. + +\subsection{Méthode classique des moments} +\label{sec:methodemoments} + +Pour certaines distributions, on ne peut directement estimer les +paramètres. On cherchera alors des fonctions des paramètres qui sont +facilement estimables de manière convergente. L'ensemble de fonctions +le plus commun qui répond à cette condition est celui des moments, +d'où le nom de la méthode. + +La méthode classique des moments a été introduite par Pearson en +1894. On considère un échantillon de taille $T$ dont les observations +seront notées $y(t)$. On veut estimer le vecteur de paramètres +$\theta$, de longueur $a$ de la distribution. On définit les moments +$\mathbf{m} = m_1, \ldots, m_a$ de la population totale, +représentée par la variable aléatoire $Y$, comme étant l'espérance des +puissances de celle-ci, et donc une fonction des paramètres de la +distribution. On considèrera le même nombre de moments que de +paramètres à estimer. +\begin{align} + \label{eq:momentspopulation} + m_i\left( \theta \right) = E \left[ Y^i \right] ,\qquad i=1, + \ldots, a +\end{align} + +Cette méthode consiste à résoudre un système d'équations +\eqref{eq:momentsechantillon} où l'on égale les moments empiriques +$m_i(\theta)$ à ceux de la distribution $\hat{m}(\mathbf{y})$. +\begin{align} + \label{eq:momentsechantillon} + \left\{\begin{array}{rcl} + m_1(\theta) &=& \frac{1}{T}\sum_{t=1}^T y_t\\ + m_2(\theta) &=& \frac{1}{T}\sum_{t=1}^T y_t^2\\ + &\vdots& \\ + m_a(\theta) &=& \frac{1}{T}\sum_{t=1}^T y_t^a + \end{array}\right\} +\end{align} + +L'estimateur des moments $\hat\theta_T$ est celui qui résout ce +système. + +\section{Méthode des moments généralisée} +\label{sec:methodeGMM} + +La méthode classique des moments utilise le même nombre d'équations +d'estimation que de paramètres $(r=a)$. De plus, le système +d'équations formé par celles-ci doit admettre une solution réelle +appartenant à l'espace des paramètres $\Omega$, ce qui n'est pas +toujours le cas. Lorsque ces deux conditions ne sont pas réunies, on +doit choisir un vecteur de paramètres $\theta$ pour lequel les moments +de la population $m_i$ ont une valeur la plus près possible de ceux +de l'échantillon $\hat{m}_i$ correspondants. Cette distance est notée +par le vecteur $g(\theta;\mathbf{y})$ et correspond au cas le plus +simple de la méthode des moments généralisée. +\begin{align} + \label{eq:1} + g(\theta;\mathbf{y}) &= \begin{bmatrix} + m_1 - \hat{m}_1\\ + \vdots\\ + m_r - \hat{m}_r\\ + \end{bmatrix}. +\end{align} + +Pour obtenir ces estimateurs, on cherchera plutôt à minimiser une +fonction objectif notée $Q\left(\theta;\mathbf{y} \right)$, qui +correspond à une norme quadratique pondérée par une matrice définie +positive $W$: +\begin{align} + Q\left(\theta;\mathbf{y} \right) \equiv g(\theta;\mathbf{y})' W + g(\theta;\mathbf{y}). +\end{align} + +\cite{hansen1982large} nomme cette procédure «méthode des moments +généralisée». Elle est aussi nommée «méthode du $\chi^2$ minimum» par +\cite{berkson1980minimum}, bien qu'elle n'en soit qu'un cas +particulier. On retrouve aussi le nom d'estimateur de distance +minimale, par \cite{wolfowitz1957minimum}. + +\subsection{Définition} +\label{sec:definitionGMM} + +On considère un vecteur $\mathbf{y}_{T}$ de longueur $T$ contenant les +données $y(t)$ de l'échantillon tirées d'une population représentée +par la variable aléatoire $Y$. On considère de plus un vecteur de +paramètres $\theta \in \Omega$ de longueur $a$ dont la vraie valeur, +qui est inconnue, est représentée par la constante $\theta_0$. Soit +une fonction vectorielle de longueur $r$ de la variable aléatoire $Y$ +appelée condition de moment ou d'orthogonalité: +\begin{align} + \label{eq:def1condmoment} + h\left(\theta,Y\right):\left(\mathbb{R}^a \times \mathbb{R}\right) + \longrightarrow \mathbb{R}^r. +\end{align} + +L'espérance de cette fonction, sous l'hypothèse $\theta = +\mathbf{\theta_0}$, est un vecteur nul noté $\mathbf{0}_r$. +\begin{align} + E \left[h\left(\theta_0,Y \right) \right] = \mathbf{0}_r. +\end{align} + +On définit aussi la fonction $g(\theta,\mathbf{y}_{T}):\mathbb{R}^a +\longrightarrow \mathbb{R}^r$ comme étant la moyenne empirique des +conditions de moment $h\left(\theta,y \right)$: +\begin{align} + \label{eq:estimateurfonctiongh} + g(\theta,\mathbf{y}_{T}) \equiv \frac{1}{T} \sum_{t=1}^T + h\left(\theta,y(t) \right). +\end{align} + +L’idée derrière la méthode des moments généralisée est de choisir un +ensemble de paramètres $\theta$ de sorte que la valeur de la fonction +$g(\theta,\mathbf{y}_{T})$ soit aussi près que possible du vecteur nul +$\mathbf{0}_r$. Selon la norme utilisée pour mesurer cette distance, +les propriétés de l'estimateur $\hat{\theta}_T$ vont varier. Étant +donné les nombreuses propriétés bien établies dans le domaine des +statistiques, on utilise la norme quadratique avec pondération, +appelée aussi moindres carrés généralisés, qui prend la forme +suivante: +\begin{align} + \label{eq:objectifGMM1} + Q(\theta,\mathbf{y}_{T}) = \left[g(\theta,\mathbf{y}_{T}) \right]' + W_T \left[g(\theta,\mathbf{y}_{T}) \right]. +\end{align} + +Cette fonction de minimisation permettra d'utiliser des tests +statistiques basés sur la distribution $\chi^2$ de Pearson. La matrice +carrée $W_T$ de dimension $r\times r$ est définie positive et est +habituellement une fonction des données de l'échantillon +$\mathbf{y}_{T}$ et des paramètres $\theta$. Une matrice de +pondération optimale sera déterminée à la section +\ref{sec:matriceWoptimaleGMM}. + +\subsection{Convergence} +\label{sec:convergenceGMM} + +Si le nombre de paramètres $a$ est égal à celui de conditions de +moments $r$, alors la fonction objectif atteindra un minimum de 0 au +point $\mathbf{\hat\theta}$. On obtiendra ce dernier en résolvant +l'équation suivante pour le paramètre $\theta$: +\begin{align} + g(\hat\theta_T,\mathbf{y}_{T})=0. \label{eq:paramegalecondmomentsGMM} +\end{align} + +Lorsque le nombre de conditions de moments est plus grand que celui +des paramètres, on ne pourra pas obtenir une solution pour l'équation +précédente \eqref{eq:paramegalecondmomentsGMM}. La proximité entre la +valeur de chaque condition de moment et $0$ sera déterminée par la +matrice de pondération $W_T$. Étant donné que la fonction +$g(\hat\theta_T,\mathbf{y}_{T})$ est la moyenne échantillonnale de la +fonction aléatoire $h\left(\theta_0,Y \right)$, on a, par la loi des +grands nombres, la relation suivante entre ces deux quantités: +\begin{align} + g(\theta,\mathbf{y}_{T}) \stackrel{P}{\longrightarrow} E + \left[h\left(\theta,Y \right) \right]. \label{eq:normequadratique} +\end{align} + +On considère la suite d'observations $\left\{y(t) \right\}_{t=1}^T$ +comme un processus stochastique. L'ensemble des conditions de +régularité suivantes permet d'obtenir un estimateur convergent +\citep{hansen1982large}. +\begin{enumerate} +\item Le processus stochastique $\left\{y(t) \right\}_{t=1}^T$ est + \textbf{stationnaire}, dont la distribution conjointe de plusieurs + observations ne change pas dans le temps: + \begin{align} + \label{eq:conditionGMM1.1} + F_{Y}(y({t(1)+\tau}) ,\ldots, y({t(k)+\tau})) = + F_{Y}(y({t(1)}),\ldots, y({t(k)})), \quad \forall \tau \in + \mathbb{R}. + \end{align} + + Il est aussi \textbf{ergodique}, c'est-à-dire que l'on peut déduire + les propriétés du processus à partir d'un échantillon (ou + réalisation) suffisamment long de celui-ci. Entre autres, la moyenne + et la variance d'un échantillon recueilli sur une période en + particulier sont représentatives de celles de n'importe quel autre + intervalle de temps de ce processus. Dans ces deux dernières + situations, on parle aussi de convergence en moyenne + ($\mathbb{L}^1$) et en norme quadratique ($\mathbb{L}^2$): + \begin{align} + \label{eq:conditionGMM1.2} + \lim_{T\longrightarrow\infty}\mathrm{E}\left(\left|Y_T-Y\right|^r\right)=0, + \quad r\in\left\{1,2\right\}. + \end{align} + +\item L'\textbf{espace métrique} $(\Omega,\sigma)$, défini par + l'espace des paramètres $\Omega$ et la norme valeur absolue est + \textbf{séparable}: + \begin{align} + \label{eq:conditionGMM2.1} + \sigma=\left|Y_T-Y\right|^r. + \end{align} + + Cette condition définit l'unicité du vecteur de paramètres, car si + la distance est nulle, les estimateurs correspondent aux vrais paramètres: + \begin{align} + \label{eq:conditionGMM2.2} + \sigma = 0 &\Longleftrightarrow \mathbf{\hat\theta} = \theta_0, + \quad \theta_0 \in \Omega. + \end{align} + +\item La fonction $h\left(\theta,\mathbf{y}\right)$ est + \textbf{mesurable au sens de Borel} pour chaque vecteur $\theta$, + c'est-à-dire qu'un sous-ensemble de l'espace des paramètres $\Omega$ + existe pour chaque valeur qu'elle peut prendre. De plus, la fonction + est continue sur l'ensemble $\Omega$ pour chaque échantillon + $\mathbf{y}$. + +\item L'espérance de la fonction $h\left(\theta, Y \right)$ existe et + est définie pour toute valeur $\theta \in \Omega$. De plus, par + définition, l'espérance de la fonction pour les vrais paramètres est + de 0: + \begin{align*} + E\left[h\left(\theta_0, Y \right)\right] &= 0. + \end{align*} + +\item La séquence de matrices de pondération $\left\{ W_T + \right\}_{T=1}^{\infty}$ converge presque sûrement, élément par + élément, vers une constante $W_0$, en utilisant la norme valeur + absolue définie précédemment \eqref{eq:conditionGMM2.1}. +\end{enumerate} + +$\hat\theta_T \in \Omega$ est un estimateur convergent de $\theta$ +lorsque les conditions précédentes sont respectées. +\begin{align} + \hat{\theta}_T &= \operatorname{arg}\min_{\theta\in\Omega} Q(\theta,\mathbf{y}_{T}) \nonumber\\ + &= \operatorname{arg}\min_{\theta\in\Omega} + \left[g(\theta,\mathbf{y}_{T}) \right]' W_T + \left[g(\theta,\mathbf{y}_{T}) \right]. \label{eq:estimateurGMM} +\end{align} + +\subsection{Matrice de pondération optimale} +\label{sec:matriceWoptimaleGMM} + +On définit la variance-covariance $\mathbf{S}(\theta;\mathbf{y})$ de +la moyenne échantillonnale de la fonction +$h(\theta,\mathbf{y})$. Cette matrice est formée par l'espérance, +élément par élément, du produit extérieur de l'estimateur par sa +transposée, multiplié par la taille de l'échantillon $T$ +\begin{align} + \label{eq:matricevcov1} + \mathbf{S}(\theta;\mathbf{y}) = T \cdot E\left\{ \left[ + h(\theta,\mathbf{y})\right] \left[ h(\theta,\mathbf{y}) \right]' + \right\} . +\end{align} + +La variance-covariance asymptotique de la moyenne échantillonnale de +la fonction $h(\theta_0,y(t))$ est obtenue en évaluant la matrice +\eqref{eq:matricevcov1} au point $\mathbf{\theta_0}$. +\begin{align} + \mathbf{S}(\theta_0;\mathbf{y}) = T \left[\cdot \lim_{T + \longrightarrow \infty} E \left\{ \left[ + h(\theta,\mathbf{y})\right] \left[ h(\theta,\mathbf{y}) + \right]' \right\} \right]_{\theta=\theta_0} +\end{align} + +La valeur optimale de la matrice de pondération $W_T$ de l'équation +\eqref{eq:estimateurGMM} est obtenue en inversant la +variance-covariance asymptotique: +\begin{align} + \label{eq:matriceWinvercevcov} + W_T = \mathbf{S}^{-1}(\theta_0;\mathbf{y}_T). +\end{align} + +Cependant, comme on ne connaît pas la valeur de $\theta_0$, on +utilisera l'estimateur convergent ${\mathbf{\hat\theta}}$, qui +minimise la condition suivante: +\begin{align} + \label{eq:objectifGMM2} + Q_T(\theta,\mathbf{y}_{T}) = \left[g(\theta,\mathbf{y}_{T}) \right]' + \mathbf{S}_T^{-1}(\theta;\mathbf{y}_T) + \left[g(\theta,\mathbf{y}_{T}) \right]. +\end{align} + +Le problème d'optimisation se note alors comme suit: +\begin{align} + \label{eq:estimateurGMM2} + \hat{\theta}_T &= \operatorname{arg}\min_{\theta\in\Omega} + Q_T(\theta,\mathbf{y}_{T}). +\end{align} + +Comme la séquence $\left\{ h(\theta_0,y(t)) +\right\}_{t=-\infty}^{\infty}$ ne présente pas de corrélation +sérielle, on pourrait estimer la variance-covariance $S_T$ de manière +convergente en évaluant la moyenne empirique du produit extérieur de +la condition de moment \citep[p.413]{hamilton1994time}: +\begin{align} + \label{eq:matponderationproduith} + \mathbf{S}_T^{*}(\theta;\mathbf{y}_T) = \frac{1}{T} \sum_{t=1}^T + \left[g\left(\theta_0,y(t) \right) \right] + \left[g\left(\theta_0,y(t) \right) \right]^{\prime}. +\end{align} + +L'estimateur $\mathbf{\hat{S}}_T({\mathbf{\hat\theta}};\mathbf{y}_T)$ +converge en probabilité vers la vraie valeur de la matrice +$\mathbf{S}(\theta_0;\mathbf{y}_T)$. Étant donné que l'on estime la +fonction $h(\theta,\mathbf{y}_{T})$ à l'aide de la fonction +$g(\mathbf{\hat\theta};\mathbf{y}_T)$, on a aussi la convergence en +probabilité: +\begin{align} + \mathbf{\hat{S}}_T({\mathbf{\hat\theta}};\mathbf{y}_T) = \frac{1}{T} + \sum_{t=1}^T \left[g\left(\hat{\theta}_T,w(t) \right) \right] + \left[g\left(\hat{\theta}_T,w(t) \right) \right]^{\prime} + \stackrel{P}{\longrightarrow} \mathbf{S}(\theta_0;\mathbf{y}_T). +\end{align} + +L'estimateur de la matrice de pondération optimale est alors défini +comme étant l'inverse de la variance-covariance estimée: +\begin{align} + \hat{W}_T &= + \mathbf{\hat{S}}_T^{-1}(\mathbf{\hat\theta};\mathbf{y}_T). \label{eq:matvcovGMM} +\end{align} + +Comme cette matrice dépend de l'estimateur $\hat\theta$, qui est pour +l'instant inconnu, elle ne pourra pas être utilisée pour une première +optimisation de l'équation de minimisation +\eqref{eq:estimateurGMM}. Cependant, elle pourra être utilisée dans +une procédure itérative à la section \ref{sec:GMMtwostep}. On devra +considérer l'utilisation d'un point de départ alternatif pour le +vecteur de paramètres dans l'algorithme de minimisation. + +\subsection{Méthode des moments généralisée itérative} +\label{sec:GMMtwostep} +\nocite{wooldridge2001econometric} + +La méthode des moments généralisée itérative de +\cite{hall2005generalized} permet de contourner le problème de +l'estimation de la matrice +$\mathbf{\hat{S}}_T(\mathbf{\hat\theta};\mathbf{y}_T)$. Elle consiste, +en premier lieu, à calculer un estimateur préliminaire +$\mathbf{\hat{\theta}}^{(0)}$, en utilisant la matrice identité $W_T = +I_r$ dans l'équation \eqref{eq:estimateurGMM}. On suggère d'utiliser +un vecteur de valeurs initiales $\hat{\theta}^{I}$ obtenues par une +autre méthode d'estimation, lorsque possible, comme point de départ de +l'optimisation numérique. + +À l'aide de l'estimateur initial $\mathbf{\hat{\theta}}^{(0)}$, on +obtient une première évaluation de la matrice de pondération: +\begin{align} + W_T = \left[\hat{S}_T(\mathbf{\hat{\theta}}^{(0)};\mathbf{y}_T) + \right]^{-1}. +\end{align} + +En utilisant la matrice $W_T$ comme pour pondérer la fonction objectif +\eqref{eq:estimateurGMM}, on obtiendra un nouvel estimateur +$\hat{\theta}_T^{(1)}$. + +Par la suite, on répète cette procédure jusqu'à ce qu'on obtienne deux +estimateurs consécutifs ($\hat{\theta}_T^{(j)}$ et +$\hat{\theta}_T^{(j+1)}$) qui ne sont pas significativement +différents, selon un critère d'arrêt $\epsilon_i < \epsilon$. Ce +critère d'arrêt prendra ici la forme suivante, qui correspond à la +norme euclidienne de la différence entre les deux derniers estimateurs +obtenus: +\begin{align} + \label{eq:criterearret} + \epsilon_i = + \sqrt{\left[\hat\theta^{(i)}_T-\hat\theta^{(i-1)}_T\right]^{\prime}\left[\hat\theta^{(i)}_T-\hat\theta^{(i-1)}_T\right]}. +\end{align} + +On peut aussi fixer un nombre maximal d'itérations $i_{max}$ si l'on +ne parvient pas à obtenir le niveau de précision voulu. Par contre, +dans cette situation, on préfère utiliser un autre point de départ. + +Chaque estimateur $\hat{\theta}_T^{(i)}$ a la même distribution +asymptotique. La méthode itérative a pour avantage, en pratique, que +les estimateurs produits sont invariants d'échelle et ne dépendent pas +de la matrice $W_T$ initiale. + +\subsection{Distribution asymptotique des estimateurs} +\label{sec:matvcovGMM} + +On considère $\hat{\theta}_T$, la valeur qui minimise la fonction +objectif $Q(\theta,\mathbf{y}_{T})$ \eqref{eq:objectifGMM2}. Cette +minimisation équivaut à résoudre le système non linéaire où l'on égale +la dérivée de l'équation d'optimisation à 0: +\begin{align} \label{eq:premierordreGMM} + \frac{d}{d\theta^{\prime}}Q(\theta,\mathbf{y}_{T}) &= + \left[\left.\frac{d}{d\theta^{\prime}}g(\theta,\mathbf{y}_{T})\right|_{\theta=\hat{\theta}} + \right]^{\prime} \cdot \hat{W}_T \cdot g(\theta,\mathbf{y}_{T}) \\ + &= 0. \nonumber +\end{align} + +où le gradient $D'(\theta,\mathbf{y}_{T})$ est +\begin{align} \label{eq:gradientGMM} D'(\theta,\mathbf{y}_{T}) &= + \left[\frac{d}{d\theta^{\prime}}g(\theta,\mathbf{y}_{T}) + \right]^{\prime} \nonumber\\ + &= \begin{bmatrix} + \frac{d}{d\theta_1}g_1(\theta,\mathbf{y}_{T})& \cdots & \frac{d}{d\theta_1}g_a(\theta,\mathbf{y}_{T}) \\ + \vdots & \ddots & \vdots \\ + \frac{d}{d\theta_k}g_1(\theta,\mathbf{y}_{T})& \cdots & + \frac{d}{d\theta_k}g_a(\theta,\mathbf{y}_{T}) + \end{bmatrix}. +\end{align} + +En appliquant le théorème central limite multivarié +\eqref{eq:TCLmulti2} à l'estimateur des conditions de moments +$g(\theta,\mathbf{y}_{T})$, on obtient que, lorsque la taille $T$ est +suffisamment grande, celui-ci converge en loi vers un vecteur +aléatoire de distribution normale multivariée de moyenne +$\mathbf{0}_a$ et de variance-covariance +$T^{-1}\mathbf{S}(\mathbf{\theta};\mathbf{y}_T)$: +\begin{align} + \label{eq:TCL-GMM} + \sqrt{T} g(\theta,\mathbf{y}_{T}) \stackrel{L}{\longrightarrow} + N\left(\mathbf{0}_a,\mathbf{S}(\mathbf{\theta};\mathbf{y}_T)\right). +\end{align} + +Cette relation permet de conclure que l'estimateur $\hat\theta$ est +gaussien. On n'a donc qu'à calculer sa variance-covariance +asymptotique. + +Soit $\left\{\mathbf{\hat{S}}_T \right\}_{T=1}^{\infty}$, une séquence +de matrices $(r \times r)$ définies positives qui convergent en +probabilité vers la variance-covariance asymptotique: +\begin{align*} + \mathbf{\hat{S}}_T \stackrel{P}{\longrightarrow} \mathbf{S}. +\end{align*} + +On ajoute que la fonction $g(\theta,\mathbf{y}_{T})$ doit être +différentiable par rapport au vecteur $\theta$ pour tout échantillon +$\mathbf{y}_{T}$. + +On doit préalablement poser un ensemble de conditions supplémentaires +de régularité: +\begin{enumerate} +\item L'estimateur $\hat\theta_T$ converge en probabilité vers la + vraie valeur des paramètres $\theta_0$: + \begin{align} + \label{eq:varasympt.1} + \hat\theta_T \stackrel{P}{\longrightarrow} \theta_0. + \end{align} +\item Le théorème central limite s'applique pour la fonction + $g(\theta,\mathbf{y}_{T})$ \eqref{eq:TCL-GMM}. +\item Pour toute séquence d'estimateurs $\left\{ \theta_T^{*} + \right\}_{T=1}^{\infty}$ convergents en probabilité $\theta_T^{*} + \stackrel{P}{\longrightarrow} \theta_0$, on peut évaluer le gradient + $D^{\prime}(\theta,\mathbf{y}_{T})$ de l'équation + $g(\theta,\mathbf{y}_{T})$ \eqref{eq:gradientGMM} à l'aide de + limites: + \begin{align} + \label{eq:varasympt.3} + D^{\prime}_T(\theta,\mathbf{y}_{T}) &\equiv plim \left\{\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}|_{\theta=\theta_T^{*}} \right\} \nonumber\\ + &= plim + \left\{\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}|_{\theta=\theta_0} + \right\}. + \end{align} + On note que les colonnes de la matrice $D$ sont linéairement + indépendantes. +\end{enumerate} + +Pour obtenir la distribution asymptotique de l'estimateur +$\hat\theta$, on utilise le premier ordre du développement de Taylor +de la fonction $g(\hat\theta,\mathbf{y}_{T})$ autour de la valeur du +vrai paramètre $\theta_0$, tel qu'avancé par +\cite{gourieroux1989statistique}: +\begin{align} + \label{eq:taylorfonction.g} + g(\hat\theta,\mathbf{y}_{T}) &= g(\theta_0,\mathbf{y}_{T}) + + D^{\prime}_T(\theta,\mathbf{y}_{T}) \left(\hat\theta-\theta_0 + \right). +\end{align} + +On multiplie de part et d'autre par la matrice $ +\left(\left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T \right)$ de dimension $(a \times r)$: +\begin{align} + \label{eq:taylorfonction.gprod} + \left(\left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T\right) \times g(\hat\theta,\mathbf{y}_{T}) &= + \left(\left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T\right) \times g(\theta_0,\mathbf{y}_{T})\nonumber\\ + &+ + \left(\left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T\right) \times D^{\prime}_T(\theta,\mathbf{y}_{T}) + \left(\hat\theta-\theta_0 \right). +\end{align} + +L'équation de premier ordre \eqref{eq:premierordreGMM} nous indique +que le côté gauche de l'égalité précédente +\eqref{eq:taylorfonction.gprod} vaut 0. On retrouve alors une +expression de la distance entre l'estimateur et la vraie valeur des +paramètres, qui dépend de la matrice de pondération, de la fonction +$g(\theta_0,\mathbf{y}_{T})$ et du gradient +$D^{\prime}_T(\theta,\mathbf{y}_{T})$: +\begin{align} + \label{eq:taylorfonction.gprod2} + \left(\hat\theta-\theta_0 \right) &= - + \left[\left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T \times D^{\prime}_T(\theta,\mathbf{y}_{T}) \right]^{-1} \nonumber\\ + &\quad\times + \left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T \times g(\theta_0,\mathbf{y}_{T}). +\end{align} + +La condition de régularité \eqref{eq:varasympt.3} permet la +convergence de chaque rangée de l'estimateur +$D^{\prime}_T(\theta,\mathbf{y}_{T})$ vers celles du gradient +$D^{\prime}(\theta_0,\mathbf{y}_{T})$. De plus, l'équation +\eqref{eq:taylorfonction.gprod2} implique la relation de convergence +suivante: +\begin{align} + \label{eq:taylorfonction.gprod3} + \sqrt{T} \left(\hat\theta-\theta_0 \right) + &\stackrel{P}{\longrightarrow} + -\left\{D(\theta,\mathbf{y}_{T})W_TD^{\prime}(\theta,\mathbf{y}_{T}) + \right\}^{-1} \nonumber\\ + &\quad\times \left\{D(\theta,\mathbf{y}_{T})W_T\sqrt{T} \cdot + g(\theta_0,\mathbf{y}_{T}) \right\}. +\end{align} + +Afin de simplifier la notation, on définit la constante +$C(\theta,\mathbf{y}_{T})$: +\begin{align*} + C(\theta,\mathbf{y}_{T})=-\left\{D(\theta,\mathbf{y}_{T})W_TD^{\prime} + (\theta,\mathbf{y}_{T}) \right\}^{-1} \times + D(\theta,\mathbf{y}_{T})W_T. +\end{align*} + +L'équation \eqref{eq:taylorfonction.gprod3} devient +\begin{align} + \label{eq:taylorfonction.gprod4} + \sqrt{T} \left(\hat\theta-\theta_0 \right) + &\stackrel{P}{\longrightarrow} C(\theta,\mathbf{y}_{T})\sqrt{T} + \cdot g(\theta_0,\mathbf{y}_{T}). +\end{align} + +En combinant la relation \eqref{eq:TCL-GMM}, où l'on applique le +théorème central limite à la fonction $g(\theta_0,\mathbf{y}_{T})$, +avec la méthode delta multivariée \eqref{eq:deltamethodmult}, on +retrouve la forme suivante, avec une convergence en loi cependant, +puisque celle-ci est moins forte que celle en probabilité: +\begin{align} + \sqrt{T} (\hat{\theta}-\theta_0) \stackrel{L}{\longrightarrow} + N(0,\mathcal{J}_0^{-1}) +\end{align} +où +\begin{align} + \mathcal{J}_0^{-1} &= + C(\theta,\mathbf{y}_{T})\left\{W_T\right\}^{-1}C(\theta,\mathbf{y}_{T}) + \nonumber\\ + &= \left\{D(\theta,\mathbf{y}_{T})W_TD^{\prime} + (\theta,\mathbf{y}_{T}) \right\}^{-1} D(\theta,\mathbf{y}_{T})W_T + \left\{W_T\right\}^{-1}\nonumber\\ + &\quad\times W_T D^{\prime}(\theta,\mathbf{y}_{T}) + \left\{D(\theta,\mathbf{y}_{T})W_TD^{\prime} (\theta,\mathbf{y}_{T}) + \right\}^{-1}\nonumber\\ + &= + \left\{D(\theta,\mathbf{y}_{T})W_TD^{\prime}(\theta,\mathbf{y}_{T})\right\}^{-1}. \label{matricevcovparamGMMnc} +\end{align} + +\section{Estimation sous contraintes} +\label{sec:estimGMMcontraint} + +La méthode des moments généralisée suppose que le vrai vecteur de +paramètres $\theta_0$ appartient à l'ensemble $\Omega$. En pratique, +les paramètres sont souvent soumis à certaines contraintes à +l'égalité. + +On définit un ensemble de $q$ contraintes linéaires implicites +appliquées au vecteur de paramètres $\theta$ de longueur $a$: +\begin{align} + \label{eq:contraintelin0} + \left\{ + \begin{array}{rcl} + r_{(1,1)}\theta_1 + \ldots + r_{(1,a)}\theta_a &=& r_{(1,0)}\\ + \ldots \\ + r_{(q,1)}\theta_1 + \ldots + r_{(q,a)}\theta_a &=& r_{(q,0)} + \end{array} + \right\}. +\end{align} + +On peut les présenter sous la forme d'un système matriciel: +\begin{align} + \label{eq:contraintelin} + \underbrace{ + \begin{bmatrix} + r_{(1,1)}&\ldots&r_{(1,a)}\\ + \vdots&\ddots&\vdots\\ + r_{(q,1)}&\ldots&r_{(q,a)}\\ + \end{bmatrix}}_{\mathbf{R}} \times \underbrace{\begin{bmatrix} + \theta_1\\ + \vdots\\ + \theta_a + \end{bmatrix}}_{\mathbf{\theta}} &= \underbrace{\begin{bmatrix} + r_{(1,0)}\\ + \vdots\\ + r_{(q,0)} + \end{bmatrix}}_{\mathbf{r}}. +\end{align} + +Afin de les inclure dans un problème de minimisation, on préfèrera +utiliser la notation $a(\theta) = R\theta-r$. On notera au passage +que le gradient du vecteur de contraintes équivaut à la matrice de +coefficients: +\begin{align} + \label{eq:gradientcontrainte} + \frac{\partial}{\partial\theta^{\prime}} a(\theta) = R. +\end{align} + +Ainsi, on peut estimer les paramètres de la distribution contrainte à +l'aide de la méthode des moments généralisée, de manière analogue à la +distribution non contrainte, comme il a été présenté à la section +précédente. On utilisera la technique du multiplicateur de Lagrange +afin d'inclure la contrainte $a(\theta)$ dans l'équation de +minimisation \eqref{eq:estimateurGMM2}. Le vecteur $\gamma$ associe un +multiplicateur à chaque contrainte linéaire. On définit le lagrangien +$\mathcal{L}(\tilde{\theta})$ mettant en relation la fonction objectif +$Q_T(\theta)$ \eqref{eq:objectifGMM2} et les contraintes $a(\theta)$: +\begin{equation} + \label{eq:estimateurGMMlagrange} + \mathcal{L}(\theta) = - Q_T(\theta) - a(\theta)^{\prime} \gamma. +\end{equation} + +L'estimateur contraint $\tilde{\theta}$ est obtenu en maximisant ce +lagrangien: +\begin{align} + \label{eq:lagrangienGMMcontraint} + \tilde{\theta} = \operatorname{arg}\max_{\theta\in\Omega} + \mathcal{L}(\theta). +\end{align} + +La solution optimale s'obtient en résolvant les conditions de premier +ordre par rapport au vecteur de paramètres $\theta$ et celui des +multiplicateurs de Lagrange $\gamma$: + +\begin{align} + \frac{\partial}{\partial\theta}Q_T(\tilde\theta) - + \frac{\partial}{\partial\theta^{\prime}}a(\tilde\theta)^{\prime}\gamma_{\scriptscriptstyle + T} &= 0\label{eq:premierordreGMMlagrange1} \\ a(\tilde\theta) &= + 0. \label{eq:premierordreGMMlagrange2} +\end{align} + +On s'intéresse aussi à la distribution asymptotique de cet estimateur +contraint. Pour ce faire, on doit développer les conditions de premier +ordre comme il a été fait à la section \ref{sec:matvcovGMM} pour +l'estimateur non contraint. + +\subsection{Distribution asymptotique des estimateurs contraints} +\label{sec:matvcovGMMconst} + +Supposons que les conditions de premier ordre sont deux fois +continûment dérivables par rapport au vecteur $\theta$. On développe +les équations \eqref{eq:premierordreGMMlagrange1} et +\eqref{eq:premierordreGMMlagrange2} autour de la vraie valeur du +paramètre contraint $\theta_0$. Puis, on les multiplie par les facteurs +$\frac{1}{\sqrt{T}}$ et $\sqrt{T}$ respectivement. Notons que la +fonction $a(\theta)$ vaut 0 au point $\theta_0$, ce qui permettra de +simplifier la seconde équation: +\begin{subequations} \label{eq:premierordreGMMlagrange1.1-2} + \begin{align} + \frac{1}{\sqrt{T}} \frac{\partial}{\partial\theta}Q_T(\theta_0) + + \frac{1}{T} + \frac{\partial^2}{\partial\theta\partial\theta^{\prime}}Q_T(\theta_0) + \sqrt{T} (\tilde\theta - \theta_0) - + \frac{\partial}{\partial\theta^{\prime}}a(\tilde\theta)^{\prime}\frac{\gamma_{\scriptscriptstyle + T}}{\sqrt{T}} &\approx 0 \label{eq:premierordreGMMlagrange1.1}\\ + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) \sqrt{T} + (\tilde\theta - \theta_0) &\approx + 0. \label{eq:premierordreGMMlagrange1.2} + \end{align} +\end{subequations} + +On définit la matrice d'information de Fisher comme étant la limite de +l'espérance de la valeur de la dérivée seconde de la fonction +objectif. Au point $\theta_0$, on identifie l'estimateur de cette +matrice par $\mathcal{J}_0$, la variance-covariance de l'estimateur +non contraint \eqref{matricevcovparamGMMnc}: +\begin{align} + \label{eq:fisherJGMMlagrange} + \mathcal{J}_0 &= \lim_{T\to\infty} -\frac{1}{T} + \frac{\partial^2}{\partial\theta\partial\theta^{\prime}}Q_T(\theta_0). +\end{align} + +On reprend l'équivalent asymptotique de l'équation +\eqref{eq:premierordreGMMlagrange1.1} pour l'estimateur non contraint: +\begin{align} + \frac{1}{\sqrt{T}} \frac{\partial}{\partial\theta}Q_T(\theta_0) - + \mathcal{J}_0 \sqrt{T} (\hat\theta - \theta_0) &\approx + 0\label{eq:premierordreGMMnc1.1}. +\end{align} + +En combinant ces deux dernières expressions, on peut formuler les +conditions de premier ordre \eqref{eq:premierordreGMMlagrange1.1-2} +comme étant asymptotiquement des fonctions linéaires de l'estimateur +non contraint $\sqrt{T} (\hat\theta - \theta_0)$: +\begin{align} + \mathcal{J}_0 \sqrt{T} (\hat\theta - \theta_0) - \mathcal{J}_0 + \sqrt{T} (\tilde\theta - \theta_0) - + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\frac{\gamma_{\scriptscriptstyle + T}}{\sqrt{T}} &\approx 0 \label{eq:premierordreGMMlagrange2.1}\\ + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) \sqrt{T} + (\tilde\theta - \theta_0) &\approx + 0. \label{eq:premierordreGMMlagrange2.2} +\end{align} + +En réorganisant la première équation +\eqref{eq:premierordreGMMlagrange2.1}, on obtient: +\begin{align} + \label{eq:premierordreGMMlagrange3.1} + \sqrt{T} (\tilde\theta - \theta_0) \approx \sqrt{T} (\hat\theta - + \theta_0) - \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\frac{\gamma_{\scriptscriptstyle + T}}{\sqrt{T}}. +\end{align} + +En la reportant dans la seconde équation +\eqref{eq:premierordreGMMlagrange2.2}, on obtient: +\begin{align} + \label{eq:premierordreGMMlagrange3.2} + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) \sqrt{T} + (\hat\theta - \theta_0) - \frac{\partial}{\partial\theta^{\prime}} + a(\theta_0) \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\frac{\gamma_{\scriptscriptstyle + T}}{\sqrt{T}} + \frac{\partial}{\partial\theta^{\prime}} + a(\theta_0) \sqrt{T} (\tilde\theta - \theta_0) &\approx 0. +\end{align} + +Comme le rang de la matrice $\frac{\partial}{\partial\theta^{\prime}} +a(\theta_0)$ est égal au nombre de contraintes $r$, alors +$\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) +\mathcal{J}_0^{-1} +\frac{\partial}{\partial\theta}a(\theta_0)^{\prime}$ est inversible, +et l'on peut donc isoler le multiplicateur de Lagrange +$\frac{\gamma_{\scriptscriptstyle T}}{\sqrt{T}}$ en fonction des +estimateurs contraints $\sqrt{T} (\tilde\theta - \theta_0)$ et non +contraints $\sqrt{T} (\hat\theta - \theta_0)$: +\begin{align} + \label{eq:LagrangienJ.GMM} + \frac{\gamma_{\scriptscriptstyle T}}{\sqrt{T}} & \approx + \left(\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime} \right)^{-1} + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) \sqrt{T} + (\hat\theta - \theta_0). +\end{align} + +On définit l'estimateur contraint en fonction de l'estimateur non +contraint en utilisant le lagrangien \eqref{eq:LagrangienJ.GMM} dans +la condition \eqref{eq:premierordreGMMlagrange3.1}: +\begin{align} + \label{eq:contraintvsncGMM} + \sqrt{T} (\tilde\theta - \theta_0) &\approx + \left(I-\underbrace{\mathcal{J}_0^{-1}\frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\left(\frac{\partial}{\partial\theta^{\prime}} + a(\theta_0) \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime} + \right)^{-1}\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + }_{P}\right)\mathcal{J}_0^{-1}\sqrt{T} (\hat\theta - \theta_0) \\ + & \approx \left(I-P\right)\mathcal{J}_0^{-1}\sqrt{T} (\hat\theta - + \theta_0). \nonumber +\end{align} + +La variance asymptotique de l'estimateur contraint est donc, à partir +du résultat précédent \eqref{eq:contraintvsncGMM} et de la définition +\eqref{eq:premierordreGMMnc1.1}: +\begin{align} + \label{eq:VcontraintGMM} + V\left[\sqrt{T} (\tilde\theta - \theta_0) \right] = + \left(I-P\right)\mathcal{J}_0^{-1}. +\end{align} + +L'estimateur $\tilde\theta$ suit donc asymptotiquement une +distribution normale multivariée de moyenne $\theta_0$ et de variance +$T\left(I-P\right)\mathcal{J}_0^{-1}$: +\begin{align} + \label{eq:distcontraintGMM} + \tilde\theta &\sim + \mathcal{N}\left(\theta_0,T\left(I-P\right)\mathcal{J}_0^{-1}\right). +\end{align} + +\section{Tests d'hypothèses paramétriques} +\label{sec:testparam} + +Les tests d'hypothèses paramétriques sont utilisés afin d'évaluer une +hypothèse concernant les paramètres d'une distribution, en fonction +d'un échantillon de données. Afin d'effectuer ces tests, on présume +que la différence entre l'estimateur $\hat\theta$ et la vraie valeur +des paramètres $\theta_0$ suit une distribution normale +multivariée. Les hypothèses sont habituellement formulées sous la +forme de contraintes linéaires, ainsi, les statistiques de test sont +obtenues à partir du calcul matriciel. Les trois tests les plus +couramment utilisés dans le cadre de l'estimation par maximum de +vraisemblance peuvent être adaptés à la méthode des moments +généralisée \citep{newey1994large}. Pour l'ensemble de ces tests, +l'hypothèse nulle correspond à la contrainte linéaire suivante: +\begin{equation} + \label{eq:hypcontraintelin} + H_0: a(\theta) = R\theta - r = 0. +\end{equation} + +\subsection{Test de Wald} +\label{sec:testwald} + +Le test de Wald permet de vérifier si la différence entre l'estimateur +non contraint $\hat\theta$ et l'estimateur contraint $\tilde\theta$ +est significative. Les contraintes linéaires posées ne seront pas +applicables lorsque le résultat est positif. Pour ce faire, on doit +connaître la distribution asymptotique de celle-ci. Comme la +distribution asymptotique des deux estimateurs est normale, alors +celle de cette différence l'est aussi: +\begin{align} + \label{eq:7} + (\hat\theta - \tilde\theta) \sim \mathcal{N}(0,TP\mathcal{J}_0^{-1}). +\end{align} + +On obtient l'espérance et la variance de la statistique +$\sqrt{T}\left(\hat\theta - \tilde\theta\right)$ en utilisant le fait +que la somme de deux variables aléatoires normales l'est aussi: +\begin{align} + \label{eq:moyennevariancesomme} + E\left[\sqrt{T}\left(\hat\theta - \tilde\theta\right) \right] &= + E\left[\sqrt{T}\left(\hat\theta - \theta_0\right) \right] - + E\left[\sqrt{T}\left(\tilde\theta - \theta_0\right) \right]\nonumber\\ + &= \theta_0 - \theta_0 \nonumber\\ + &= 0 \\ + V\left[\sqrt{T}\left(\hat\theta - \tilde\theta\right) \right] &= + V\left[\sqrt{T}\left(\hat\theta - \theta_0\right) \right] - + V\left[\sqrt{T}\left(\tilde\theta - \theta_0\right) \right]\nonumber\\ + &= \left(I-(I-P)\right)\mathcal{J}_0^{-1}\nonumber\\ + &= P\mathcal{J}_0^{-1}. +\end{align} + +On définit la statistique $\chi^{WALD,1}$, qui a asymptotiquement une +distribution $\chi^2$ avec $q$ degrés de liberté: +\begin{align} + \label{eq:statistiqueWald} + \chi^{WALD,1} &= T \left(\hat\theta - \tilde\theta\right)^{\prime} P + \left(\hat\theta - \tilde\theta\right) \\ + &=T \left(\hat\theta - \tilde\theta\right)^{\prime} + \mathcal{J}_0^{-1}\frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\left(\frac{\partial}{\partial\theta^{\prime}} + a(\theta_0) \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime} + \right)^{-1}\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \left(\hat\theta - \tilde\theta\right). \nonumber +\end{align} + + +On rejettera l'hypothèse nulle \eqref{eq:hypcontraintelin} lorsque la +valeur de la statistique $\chi^{WALD,1}$ sera supérieure à un seuil +critique $\chi_{q,1-\alpha}^2$ au niveau de confiance $1-\alpha$. + +Une version asymptotiquement équivalente de ce test qui ne requiert +pas de calculer la valeur de l'estimateur contraint existe. Ce test +est équivalent lorsque les contraintes définissent certains paramètres +comme des constantes. On vérifie si un cas particulier d'une +distribution s'applique, par exemple avec celle de Laplace asymétrique +généralisée. + +On définit alors la statistique $\chi^{WALD,2}$, qui a aussi +asymptotiquement une distribution $\chi^2$ avec $q$ degrés de +liberté. Par contre, ici, on teste si la valeur de la contrainte +linéaire est significativement différente de 0: +\begin{align} + \label{eq:statistiqueWald2} + \chi^{WALD,2} &= T a^{\prime}(\hat\theta) + \left(\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a^{\prime}(\theta_0) \right)^{-1} + a(\hat\theta). +\end{align} + + +\subsection{Test du multiplicateur de Lagrange} +\label{sec:testscore} + +Le test du multiplicateur de Lagrange, ou du score, introduit par +\cite{newey1987hypothesis}, est basé uniquement sur l'estimateur +contraint et est équivalent asymptotiquement au test de Wald présenté +à la section précédente. Il vérifie l'application de la contrainte +\eqref{eq:contraintelin} à l'estimateur $\tilde\theta$. Selon la +définition du lagrangien \eqref{eq:estimateurGMMlagrange}, si la +contrainte est vérifiée, alors la restriction $a(\tilde\theta)$ vaudra +0. Selon la condition de premier ordre +\eqref{eq:premierordreGMMlagrange2}, la dérivée de la fonction +objectif de l'estimateur non contraint +$\frac{\partial}{\partial\theta}Q_T(\theta)$, le score, $\tilde\theta$ +devrait aussi être égale à 0. On cherchera donc à tester si cette +valeur est significativement différente de 0. + +On définit la statistique du multiplicateur de Lagrange $\chi^{LM,1}$, +qui a asymptotiquement une distribution $\chi^2_q$ avec $q$ degrés de +liberté: +\begin{align} + \label{statistiqueLM} + \chi^{LM,1} &= T \frac{\partial}{\partial\theta}Q_T(\tilde\theta) P + \frac{\partial}{\partial\theta}Q_T(\tilde\theta). +\end{align} + +On rejettera l'hypothèse nulle \eqref{eq:hypcontraintelin} lorsque la +valeur de la statistique $\chi^{LM,1}$ sera supérieure à un seuil +critique $\chi_{q,1-\alpha}^2$ au niveau de confiance $1-\alpha$. + +On peut construire un test équivalent, basé sur la valeur du +multiplicateur, dont la matrice de variance-covariance est l'inverse +de celle de la contrainte. On définit alors la statistique +$\chi^{LM,2}$ suivante: +\begin{align} + \label{eq:statistiqueLM2} + \chi^{LM,2} &= T \gamma^{\prime} + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a^{\prime}(\theta_0) \gamma. +\end{align} + +\subsection{Test basé sur la statistique de métrique de distance} + +La statistique de métrique de distance est basée sur la différence +entre les valeurs minimales de la fonction objectif $Q_T(\theta)$ +obtenues lors de l'optimisation avec contraintes +\eqref{eq:estimateurGMMlagrange} et sans contraintes +\eqref{eq:estimateurGMM}. + +On définit la statistique $\chi^{DM}$: +\begin{align} + \label{statistiqueD} + \chi^{DM} &= -T \left[Q_T(\tilde\theta) - Q_T(\hat\theta)\right]. +\end{align} + +Cette statistique a asymptotiquement une distribution $\chi^2_q$ avec +$q$ degrés de liberté. Elle est l'analogue de la statistique du ratio +de vraisemblance dans le cadre de l'estimation par la méthode du +maximum de vraisemblance. + +Le test basé sur la métrique de distance vérifie que la contrainte +\eqref{eq:contraintelin} posée lors de l'estimation du vecteur +$\tilde\theta$ est valide. Un des désavantages de ce test est qu'il +requiert deux optimisations. Par contre, on peut facilement récupérer +les valeurs de $Q_T(\theta)$ lors de l'estimation. + +On rejettera l'hypothèse nulle \eqref{eq:hypcontraintelin} lorsque la +valeur de la statistique $\chi^{DM}$ sera supérieure à un seuil +critique $\chi_{q,1-\alpha}^2$ au niveau de confiance $1-\alpha$. + +\subsection{En résumé} +\label{sec:resumetests} + +On rassemble les différentes statistiques permettant d'effectuer un +test d'hypothèse paramétrique à la table \ref{tab:testsparamGMM}. +\begin{table}[!ht] + \centering + \begin{tabular}{cc} + \hline + \textbf{Statistique} & \textbf{Valeur} \\ + \hline + $\chi^{WALD,1}$ & $T \left(\hat\theta - \tilde\theta\right)^{\prime} + \mathcal{J}_0^{-1}\frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\left(\frac{\partial}{\partial\theta^{\prime}} + a(\theta_0) \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime} + \right)^{-1}\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \left(\hat\theta - \tilde\theta\right)$ \\ + $\chi^{WALD,2}$ & $T a^{\prime}(\hat\theta) + \left(\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a^{\prime}(\theta_0) \right)^{-1} + a(\hat\theta)$ \\ + $\chi^{LM,1}$ & $T \frac{\partial}{\partial\theta}Q_T(\tilde\theta) P + \frac{\partial}{\partial\theta}Q_T(\tilde\theta)$ \\ + $\chi^{LM,2}$ & $T \gamma^{\prime} + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a^{\prime}(\theta_0) \gamma$ \\ + $\chi^{DM}$ & $-T \left[Q_T(\tilde\theta) - Q_T(\hat\theta)\right]$ \\ + \hline + \end{tabular} + \caption{Tests d'hypothèse paramétriques pour la méthode des moments généralisée} + \label{tab:testsparamGMM} +\end{table} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/chapitre5.aux b/memoire/chapitre5.aux new file mode 100644 index 0000000..690aa36 --- /dev/null +++ b/memoire/chapitre5.aux @@ -0,0 +1,167 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{everitt2006cambridge} +\citation{crowder1986consistency} +\citation{crowder1987linear} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {chapter}{\chapternumberline {5}M\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale}{65}{chapter.5}} +\newlabel{eq:generalEE86}{{5.0.1}{65}{Méthode de l'équation d'estimation optimale}{equation.5.0.0.1}{}} +\newlabel{eq:generalEE86@cref}{{[equation][1][5,0]5.0.1}{65}} +\newlabel{eq:generalEE87}{{5.0.2}{65}{Méthode de l'équation d'estimation optimale}{equation.5.0.0.2}{}} +\newlabel{eq:generalEE87@cref}{{[equation][2][5,0]5.0.2}{65}} +\newlabel{eq:EEsansbiais}{{5.0.3}{65}{Méthode de l'équation d'estimation optimale}{equation.5.0.0.3}{}} +\newlabel{eq:EEsansbiais@cref}{{[equation][3][5,0]5.0.3}{65}} +\citation{fox1986large} +\citation{crowder1987linear} +\newlabel{eq:scoreEMV}{{5.0.4}{66}{Méthode de l'équation d'estimation optimale}{equation.5.0.0.4}{}} +\newlabel{eq:scoreEMV@cref}{{[equation][4][5,0]5.0.4}{66}} +\newlabel{eq:derivmom}{{5.0.5}{66}{Méthode de l'équation d'estimation optimale}{equation.5.0.0.5.5}{}} +\newlabel{eq:derivmom@cref}{{[equation][5][5,0]5.0.5}{66}} +\newlabel{eq:derivmom1}{{5.0.5a}{66}{Méthode de l'équation d'estimation optimale}{equation.5.0.0.5a}{}} +\newlabel{eq:derivmom1@cref}{{[equation][1][5,0,5]5.0.5a}{66}} +\newlabel{eq:derivmom2}{{5.0.5b}{66}{Méthode de l'équation d'estimation optimale}{equation.5.0.0.5b}{}} +\newlabel{eq:derivmom2@cref}{{[equation][2][5,0,5]5.0.5b}{66}} +\newlabel{eq:generalquad}{{5.0.6}{66}{Méthode de l'équation d'estimation optimale}{equation.5.0.0.6}{}} +\newlabel{eq:generalquad@cref}{{[equation][6][5,0]5.0.6}{66}} +\newlabel{eq:vecteuroptimal.ab}{{5.0.7}{66}{Méthode de l'équation d'estimation optimale}{equation.5.0.0.7}{}} +\newlabel{eq:vecteuroptimal.ab@cref}{{[equation][7][5,0]5.0.7}{66}} +\newlabel{eq:demogeneralquad-convergent}{{5.0.8}{66}{Méthode de l'équation d'estimation optimale}{equation.5.0.0.8}{}} +\newlabel{eq:demogeneralquad-convergent@cref}{{[equation][8][5,0]5.0.8}{66}} +\citation{crowder1986consistency} +\@writefile{lot}{\contentsline {table}{\numberline {5.1}{\ignorespaces M\IeC {\'e}thodes d'estimation repr\IeC {\'e}sentables par la forme quadratique \textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {eq:generalquad}\unskip \@@italiccorr )}} \relax }}{67}{table.caption.26}} +\newlabel{tab:methodesquad}{{5.1}{67}{Méthodes d'estimation représentables par la forme quadratique \eqref {eq:generalquad} \relax }{table.caption.26}{}} +\newlabel{tab:methodesquad@cref}{{[table][1][5]5.1}{67}} +\@writefile{toc}{\contentsline {section}{\numberline {5.1}\IeC {\'E}quation d'estimation optimale}{67}{section.5.1}} +\newlabel{sec:equationsquadopt}{{5.1}{67}{Équation d'estimation optimale}{section.5.1}{}} +\newlabel{sec:equationsquadopt@cref}{{[section][1][5]5.1}{67}} +\newlabel{eq:Crowder86-th4.1-def}{{5.1.1}{67}{Équation d'estimation optimale}{equation.5.1.0.1}{}} +\newlabel{eq:Crowder86-th4.1-def@cref}{{[equation][1][5,1]5.1.1}{67}} +\newlabel{eq:Crowder86-th3.3-def}{{5.1.3}{67}{Équation d'estimation optimale}{equation.5.1.0.3.3}{}} +\newlabel{eq:Crowder86-th3.3-def@cref}{{[equation][3][5,1]5.1.3}{67}} +\newlabel{eq:Crowder86-th3.3-def-1}{{5.1.3a}{67}{Équation d'estimation optimale}{equation.5.1.0.3a}{}} +\newlabel{eq:Crowder86-th3.3-def-1@cref}{{[equation][1][5,1,3]5.1.3a}{67}} +\newlabel{eq:Crowder86-th3.3-def-2}{{5.1.3b}{67}{Équation d'estimation optimale}{equation.5.1.0.3b}{}} +\newlabel{eq:Crowder86-th3.3-def-2@cref}{{[equation][2][5,1,3]5.1.3b}{67}} +\newlabel{eq:VarAsymptEstEE}{{5.1.4}{67}{Équation d'estimation optimale}{equation.5.1.0.4}{}} +\newlabel{eq:VarAsymptEstEE@cref}{{[equation][4][5,1]5.1.4}{67}} +\newlabel{eq:Crowdervecteuroptimal}{{5.1.5}{68}{Équation d'estimation optimale}{equation.5.1.0.5}{}} +\newlabel{eq:Crowdervecteuroptimal@cref}{{[equation][5][5,1]5.1.5}{68}} +\newlabel{eq:13}{{5.1.6}{68}{Équation d'estimation optimale}{equation.5.1.0.6}{}} +\newlabel{eq:13@cref}{{[equation][6][5,1]5.1.6}{68}} +\newlabel{eq:15}{{5.1.8}{68}{Équation d'estimation optimale}{equation.5.1.0.8}{}} +\newlabel{eq:15@cref}{{[equation][8][5,1]5.1.8}{68}} +\newlabel{eq:16}{{5.1.9}{68}{Équation d'estimation optimale}{equation.5.1.0.9}{}} +\newlabel{eq:16@cref}{{[equation][9][5,1]5.1.9}{68}} +\newlabel{eq:matvcov-u-EE}{{5.1.10}{68}{Équation d'estimation optimale}{equation.5.1.0.10}{}} +\newlabel{eq:matvcov-u-EE@cref}{{[equation][10][5,1]5.1.10}{68}} +\newlabel{eq:momentsupgamma}{{5.1.11}{69}{Équation d'estimation optimale}{equation.5.1.0.11.11}{}} +\newlabel{eq:momentsupgamma@cref}{{[equation][11][5,1]5.1.11}{69}} +\newlabel{eq:momentsupgamma1}{{5.1.11a}{69}{Équation d'estimation optimale}{equation.5.1.0.11a}{}} +\newlabel{eq:momentsupgamma1@cref}{{[equation][1][5,1,11]5.1.11a}{69}} +\newlabel{eq:momentsupgamma2}{{5.1.11b}{69}{Équation d'estimation optimale}{equation.5.1.0.11b}{}} +\newlabel{eq:momentsupgamma2@cref}{{[equation][2][5,1,11]5.1.11b}{69}} +\newlabel{eq:momentsupgamma3}{{5.1.11c}{69}{Équation d'estimation optimale}{equation.5.1.0.11c}{}} +\newlabel{eq:momentsupgamma3@cref}{{[equation][3][5,1,11]5.1.11c}{69}} +\newlabel{eq:18}{{5.1.12}{69}{Équation d'estimation optimale}{equation.5.1.0.12}{}} +\newlabel{eq:18@cref}{{[equation][12][5,1]5.1.12}{69}} +\newlabel{eq:19}{{5.1.15}{69}{Équation d'estimation optimale}{equation.5.1.0.15}{}} +\newlabel{eq:19@cref}{{[equation][15][5,1]5.1.15}{69}} +\newlabel{eq:coefficientscrowder}{{5.1.17}{70}{Équation d'estimation optimale}{equation.5.1.0.17.17}{}} +\newlabel{eq:coefficientscrowder@cref}{{[equation][17][5,1]5.1.17}{70}} +\newlabel{eq:acrowder}{{5.1.17a}{70}{Équation d'estimation optimale}{equation.5.1.0.17a}{}} +\newlabel{eq:acrowder@cref}{{[equation][1][5,1,17]5.1.17a}{70}} +\newlabel{eq:bcrowder}{{5.1.17b}{70}{Équation d'estimation optimale}{equation.5.1.0.17b}{}} +\newlabel{eq:bcrowder@cref}{{[equation][2][5,1,17]5.1.17b}{70}} +\newlabel{eq:Moptimalestimetheta}{{5.1.18}{70}{Équation d'estimation optimale}{equation.5.1.0.18}{}} +\newlabel{eq:Moptimalestimetheta@cref}{{[equation][18][5,1]5.1.18}{70}} +\newlabel{eq:eqnobjectifEE}{{5.1.19}{70}{Équation d'estimation optimale}{equation.5.1.0.19}{}} +\newlabel{eq:eqnobjectifEE@cref}{{[equation][19][5,1]5.1.19}{70}} +\@writefile{toc}{\contentsline {section}{\numberline {5.2}\IeC {\'E}quation d'estimation optimale modifi\IeC {\'e}e}{70}{section.5.2}} +\newlabel{sec:eqoptmodif}{{5.2}{70}{Équation d'estimation optimale modifiée}{section.5.2}{}} +\newlabel{sec:eqoptmodif@cref}{{[section][2][5]5.2}{70}} +\newlabel{eq:vecteurcrowdermod}{{5.2.1}{70}{Équation d'estimation optimale modifiée}{equation.5.2.0.1.1}{}} +\newlabel{eq:vecteurcrowdermod@cref}{{[equation][1][5,2]5.2.1}{70}} +\newlabel{eq:acrowdermod}{{5.2.1a}{70}{Équation d'estimation optimale modifiée}{equation.5.2.0.1a}{}} +\newlabel{eq:acrowdermod@cref}{{[equation][1][5,2,1]5.2.1a}{70}} +\newlabel{eq:bcrowdermod}{{5.2.1b}{70}{Équation d'estimation optimale modifiée}{equation.5.2.0.1b}{}} +\newlabel{eq:bcrowdermod@cref}{{[equation][2][5,2,1]5.2.1b}{70}} +\newlabel{eq:coeffemp}{{5.2.1c}{70}{Équation d'estimation optimale modifiée}{equation.5.2.0.1c}{}} +\newlabel{eq:coeffemp@cref}{{[equation][3][5,2,1]5.2.1c}{70}} +\@setckpt{chapitre5}{ +\setcounter{page}{72} +\setcounter{equation}{1} +\setcounter{enumi}{3} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{5} +\setcounter{section}{2} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{90} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{1} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{24} +\setcounter{Hfootnote}{1} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{67} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/chapitre5.log b/memoire/chapitre5.log new file mode 100644 index 0000000..a47b2cd --- /dev/null +++ b/memoire/chapitre5.log @@ -0,0 +1,25 @@ +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.3.10) 5 MAY 2013 21:14 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**\input chapitre5.tex +(./chapitre5.tex +! Undefined control sequence. +l.1 \chapter + {Estimation du modèle Laplace asymétrique généralisé} +? X + +Here is how much of TeX's memory you used: + 5 strings out of 493485 + 141 string characters out of 3143526 + 47818 words of memory out of 3000000 + 3412 multiletter control sequences out of 15000+200000 + 3640 words of font info for 14 fonts, out of 3000000 for 9000 + 957 hyphenation exceptions out of 8191 + 4i,0n,3p,91b,8s stack positions out of 5000i,500n,10000p,200000b,50000s +No pages of output. +PDF statistics: + 0 PDF objects out of 1000 (max. 8388607) + 0 named destinations out of 1000 (max. 500000) + 1 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/memoire/chapitre5.tex b/memoire/chapitre5.tex new file mode 100644 index 0000000..6833381 --- /dev/null +++ b/memoire/chapitre5.tex @@ -0,0 +1,391 @@ +\chapter{Méthode de l'équation d'estimation + optimale} % numéroté + +Une équation d'estimation est une fonction des données de +l'échantillon et des paramètres d'un modèle qui spécifie de quelle +manière on doit procéder pour estimer ces derniers, lorsque la +distribution de la population est inconnue. Cette approche a un +avantage sur les méthodes de vraisemblance, car elle ne requiert pas +l'utilisation de la fonction de densité ou de répartition, mais +seulement les moments de la distribution. \citep{everitt2006cambridge} + +\cite{crowder1986consistency} définit l'équation d'estimation sous une +forme générale, composée d'une matrice d'échelle $S(\theta;Y)$, un +vecteur de fonctions aléatoires $u(\theta;Y)$ et un vecteur de +pondération $\mathbf{w}(\theta;Y)$: +\begin{align} + \label{eq:generalEE86} + g(\theta;Y) = S^{-1}(\theta;Y) \sum_{t=1}^T u(\theta;Y) + \mathbf{w}(\theta;Y). +\end{align} + +En faisant abstraction du vecteur de pondération +$\mathbf{w}(\theta;Y)$, on retrouve une forme qui rappelle la méthode +des moments généralisée. La fonction d'estimation $u(\theta;Y)$ +définit, dans ce cas, des conditions de moment. Toutefois, en faisant +abstraction de la matrice $S(\theta;Y)$, on obtient une classe +d'équations d'estimation $g(\theta;Y)$ qui généralise plusieurs +méthodes connues, comme développé par \cite{crowder1987linear}: +\begin{align} + \label{eq:generalEE87} + g(\theta;Y) = \sum_{t=1}^T u(\theta;Y) \mathbf{w}(\theta;Y). +\end{align} + +La propriété fondamentale de toute équation d'estimation est que son +espérance est nulle. L'équation d'estimation \eqref{eq:generalEE86}, +et donc \eqref{eq:generalEE87}, produit des estimateurs sans biais: +\begin{align} + \label{eq:EEsansbiais} + E\left[ g(\theta;Y) \right] = 0. +\end{align} + +Par analogie avec la méthode du maximum de vraisemblance, on pourra +alors la considérer comme une équation de quasi-score. On peut en +effet représenter l'équation précédente par celle du score +\eqref{eq:scoreEMV} donnée par l'espérance de la dérivée de la +fonction de log-vraisemblance par rapport au vecteur de paramètres +$\theta$: +\begin{align} + \label{eq:scoreEMV} + E\left[g\left(\theta;Y \right)^{EMV}\right] &= E\left[\frac{d \ln + L\left(\theta;Y\right)}{d\theta}\right] = 0. +\end{align} + +On désigne la moyenne et la variance de la distribution par les +notations $\mu\left(\theta\right)$ et $\sigma^2(\theta)$ +respectivement. On définit aussi les dérivées premières de la moyenne +et de l'écart-type par rapport au vecteur de paramètres: +\begin{subequations}\label{eq:derivmom} + \begin{align} + \mu^{\prime}\left( \theta \right) &= \frac{d \mu\left(\theta\right)}{d\theta} \label{eq:derivmom1}\\ + \sigma^{\prime}\left( \theta \right) &= \frac{d + \sqrt{\sigma^2(\theta)}}{d\theta}.\label{eq:derivmom2} + \end{align} +\end{subequations} + +Dans ce chapitre, on considère une équation d'estimation quadratique +de la forme suivante: +\begin{equation} + \label{eq:generalquad} + g\left(\theta;Y \right) = \sum_{t=1}^n \left[ \mathbf{a}(\theta;y_t)(y_t-\mu\left(\theta\right)) + \mathbf{b}(\theta;y_t)\left((y_t-\mu\left(\theta\right))^2-\sigma^2\left(\theta\right) \right)\right]. +\end{equation} + +Le vecteur de pondération $\mathbf{w}(\theta;Y)$ de l'équation +\eqref{eq:generalEE87} est composé de deux fonctions déterministes: +$\mathbf{a}(\theta;Y)$ et $\mathbf{b}(\theta;Y)$: +\begin{align} + \label{eq:vecteuroptimal.ab} + \mathbf{w}(\theta;Y) &= \begin{bmatrix} \mathbf{a}(\theta;Y) \\ + \mathbf{b}(\theta;Y) \end{bmatrix}. +\end{align} + +On démontre facilement que c'est une équation d'estimation en +calculant son espérance: +\begin{align} + \label{eq:demogeneralquad-convergent} + E\left[g\left(\theta;Y \right)\right] &= E\left[ \sum_{t=1}^n \left[ \mathbf{a}(\theta;y_t)(y_t-\mu\left(\theta\right)) + \mathbf{b}(\theta;y_t)\left((y_t-\mu\left(\theta\right))^2-\sigma^2\left(\theta\right) \right)\right]\right] \nonumber\\ + &= \sum_{t=1}^n \left( \mathbf{a}(\theta;y_t)\cdot E\left[y_t-\mu\left(\theta\right)\right] + \mathbf{b}(\theta;y_t)\cdot E\left[(y_t-\mu\left(\theta\right))^2-\sigma^2\left(\theta\right) \right]\right) \nonumber\\ + &= 0. +\end{align} + +La forme quadratique regroupe certaines méthodes d'estimation bien +établies, recensées à la table \ref{tab:methodesquad}. +\begin{table}[!ht] + + \centering + \begin{tabular}{ccc} + \hline + \textbf{Méthode} & $\mathbf{a}(\theta;y_t)$ & $\mathbf{b}(\theta;y_t)$ \\ + \hline + Moindres carrés non pondérés & $\mu^{\prime}\left( \theta \right)$ & $0$ \\ + Maximum de quasi-vraisemblance & $\frac{\mu^{\prime}\left( \theta \right)}{ \sigma^2\left( \theta \right)}$ & $0$ \\ + Estimation gaussienne de Whittle + \citep{fox1986large} & $\frac{\mu^{\prime}\left( \theta \right)}{ \sigma^2\left( \theta \right)}$ & $\frac{\sigma^{\prime}\left( \theta \right)}{ \sigma^2\left( \theta \right)^{3/2}}$ \\ + \hline + \end{tabular} + \caption{Méthodes d'estimation représentables par la forme quadratique \eqref{eq:generalquad} + }\label{tab:methodesquad} +\end{table} + +La méthode d'estimation gaussienne de Whittle présente un avantage par +rapport aux modèles linéaires conventionnels (moindres carrés pondérés +et maximum de quasi-vraisemblance) du fait qu'elle prend en +considération la variance comme critère d'optimisation et non +seulement comme pondération. + +\cite{crowder1987linear} développe, à partir de cette dernière, +une équation d'estimation optimale qui vise à +remplacer la méthode de quasi-vraisemblance basée sur la moyenne et la +variance jusqu'alors utilisée. + +\section{Équation d'estimation optimale} +\label{sec:equationsquadopt} + +Soit une équation d'estimation de la forme générale +\eqref{eq:generalEE87}. On a, par définition, que l'espérance de la +composante stochastique $u(\theta;Y_t)$ de celles-ci sous les vrais +paramètres $\theta_0$ est nulle. De plus, si l'on considère que les +données ne sont pas corrélées, l'espérance du produit de deux de ces +fonctions est nulle, sauf lorsqu'elle est évaluée au même point $Y_t$: +\begin{align} + \label{eq:Crowder86-th4.1-def} + E \left[u(\theta_0;Y_t) \right] &= 0 \\ + E \left[u(\theta_0;Y_s)u(\theta_0;Y_t) \right] &= + \delta_{s,t}Var\left[u(\theta_0;Y_t)\right]. +\end{align} + +Les matrices non singulières $M(\theta_0) \mbox{ et } V(\theta_0)$ +représentent respectivement, pour l'équation d'estimation, l'espérance +de sa dérivée par rapport à $\theta$ et l'inverse de sa +variance-covariance, évaluée à la vraie valeur des paramètres +$\theta_0$: +\begin{subequations}\label{eq:Crowder86-th3.3-def} + \begin{align} + M(\theta_0;Y) &= \sum_{t=1}^{T} \mathbf{w}(\theta_0) E\left[u^{\prime}(\theta_0;Y_t) \right] \label{eq:Crowder86-th3.3-def-1}\\ + V(\theta_0;Y) &= \sum_{t=1}^{T} \mathbf{w}(\theta_0) + Var\left[u(\theta_0;Y_t)\right] + \mathbf{w}^{T}(\theta_0)\label{eq:Crowder86-th3.3-def-2}. + \end{align} +\end{subequations} + +La variance asymptotique de l'estimateur $\hat\theta$ prend alors la +forme suivante, en utilisant la méthode delta multivariée: +\begin{align} + \label{eq:VarAsymptEstEE} + Var\left[\hat\theta\right] &= + M^{-1}(\theta_0)V(\theta_0)\left[M^{-1}(\theta_0)\right]^{T}. +\end{align} + +On désire obtenir des équations d'estimation optimales de telle sorte +que la variance soit minimale. Le \emph{théorème 4.1} de +\cite{crowder1986consistency} permet d'établir quel vecteur de +pondération $\mathbf{w}^{*}(\theta)$ utiliser. + +Selon ce théorème, lorsqu'on utilise le vecteur de pondération optimal +\eqref{eq:Crowdervecteuroptimal}, on obtient que la dérivée de la +fonction d'estimation $M^{*}(\theta)$ soit égale à l'inverse de la +matrice de variance-covariance $V^{*}(\theta)$: +\begin{align} + \label{eq:Crowdervecteuroptimal} + \mathbf{w}^{*}(\theta) &= \left\{E \left[u^{\prime}(\theta_0;Y_t) + \right]\right\}^{T}\left\{Var\left[u(\theta;Y_t)\right] + \right\}^{-1}. +\end{align} + +La variance asymptotique \eqref{eq:VarAsymptEstEE} du vecteur de +paramètres optimal $\hat\theta^{*}$ qui résout l'équation d'estimation +$g^{*}(\theta)$ se simplifie alors sous la forme: +$\left[V^{*}(\theta_0;y_t)\right]^{-1}$ +\begin{align} + \label{eq:13} + M^{*}(\theta_0) &= \sum_{t=1}^{T} \left\{E + \left[u^{\prime}(\theta_0;Y_t) + \right]\right\}^{T}\left\{Var\left[u(\theta_0;Y_t)\right] + \right\}^{-1} E\left[u^{\prime}(\theta_0;Y_t) \right] \\ + V^{*}(\theta_0) &= \sum_{t=1}^{T} \left\{E + \left[u^{\prime}(\theta_0;Y_t) + \right]\right\}^{T}\left\{Var\left[u(\theta_0;Y_t)\right] + \right\}^{-1} Var\left[u(\theta_0;Y_t)\right] \nonumber\\ + &\quad \times \left[\left\{E \left[u^{\prime}(\theta_0;Y_t) + \right]\right\}^{T}\left\{Var\left[u(\theta_0;Y_t)\right] + \right\}^{-1}\right]^{T} \nonumber\\ + &= \sum_{t=1}^{T} \left\{E \left[u^{\prime}(\theta_0;Y_t) + \right]\right\}^{T}\left\{Var\left[u(\theta_0;Y_t)\right] + \right\}^{-1} E\left[u^{\prime}(\theta_0;Y_t) \right] \nonumber\\ + \Leftrightarrow M^{*}(\theta_0) &= V^{*}(\theta_0) \nonumber\\ + \Leftrightarrow Var(\hat\theta^{*}) &= \left[M^{*}(\theta_0)\right]^{-1} V^{*}(\theta_0) \left[\left[M^{*}(\theta_0)\right]^{-1}\right]^{T} \nonumber\\ + &= \left[V^{*}(\theta_0)\right]^{-1}. +\end{align} + +On considère la fonction $u(\theta;Y_t)$ de l'équation quadratique +\eqref{eq:generalquad} , dont on cherche le vecteur de pondération +optimal correspondant selon la proposition +\eqref{eq:Crowdervecteuroptimal}: +\begin{align} + \label{eq:15} + u(\theta;Y_t) &= \begin{bmatrix} + Y_t-\mu(\theta)\\ + (Y_t-\mu(\theta))^2-\sigma^2(\theta) + \end{bmatrix}. +\end{align} + +On évalue l'espérance, sous le vecteur de vrais paramètres $\theta_0$, +de la dérivée par rapport à $\theta$ de la fonction $u(\theta;Y)$: +\begin{align} + \label{eq:16} + E\left[u^{\prime}(\theta;Y_t)\right] &= \begin{bmatrix} + E \left[-\mu^{\prime}(\theta)\right] \\ + E + \left[-2(Y-\mu(\theta))\mu^{\prime}(\theta)-2\sqrt{\sigma^2(\theta)}\sigma^{\prime}(\theta)\right] + \end{bmatrix} \nonumber\\ + &= \begin{bmatrix} + -\mu^{\prime}(\theta) \\ + -2 \sqrt{\sigma^2(\theta)}\sigma^{\prime}(\theta). + \end{bmatrix}. +\end{align} + +Puis, on évalue la variance-covariance du vecteur $u(\theta)$. Pour ce +faire, on devra évaluer séparément chaque élément composant cette +matrice: +\begin{align} + \label{eq:matvcov-u-EE} + Var\left[u(\theta;Y_t) \right] &= + \begin{bmatrix} + Var\left[Y-\mu(\theta)\right] & Cov\left[Y-\mu(\theta),(Y-\mu(\theta))^2-\sigma^2(\theta) \right] \\ + Cov\left[Y-\mu(\theta),(Y-\mu(\theta))^2-\sigma^2(\theta) \right] + & Var\left[(Y-\mu(\theta))^2-\sigma^2(\theta)\right] + \end{bmatrix}. +\end{align} + +On définit les coefficients d'asymétrie et d'aplatissement normalisés, +ainsi qu'une constante $\gamma_3(\theta)$ qui en découle, afin de +simplifier les expressions qui seront obtenues conséquemment: +\begin{subequations}\label{eq:momentsupgamma} + \begin{align} + \gamma_1\left(\theta\right) &= \frac{m_3\left(\theta\right)}{m_2\left(\theta\right)^{3/2}} \nonumber\\ + &= \frac{E\left[\left(Y-\mu\left(\theta\right)\right)^3\right]}{\sigma^2\left(\theta\right)^{3/2}} \label{eq:momentsupgamma1}\\ + \gamma_2\left(\theta\right) &= \frac{m_4\left(\theta\right)}{m_2\left(\theta\right)^{2}} - 3 \nonumber\\ + &=\frac{E\left[\left(Y-\mu\left(\theta\right)\right)^4\right]}{\sigma^2\left(\theta\right)^2}-3 \label{eq:momentsupgamma2}\\ + \gamma_3(\theta) &= \gamma_2\left(\theta\right) + 2 - + \gamma_1\left(\theta\right)^2. \label{eq:momentsupgamma3} + \end{align} +\end{subequations} + +On développe les différentes composantes de la matrice: +\eqref{eq:matvcov-u-EE} +\begin{align} + \label{eq:18} + Var\left[Y-\mu(\theta)\right] &= E\left[(Y-\mu(\theta))^2\right] - + E\left[Y-\mu(\theta)\right]^2 \nonumber\\ + &=\sigma^2(\theta)\\ + Cov\left[Y-\mu(\theta),(Y-\mu(\theta))^2-\sigma^2(\theta) \right] &= + E\left[\left(Y-\mu(\theta)\right)\left((Y-\mu(\theta))^2-\sigma^2(\theta)\right) + \right] \nonumber\\ &\quad - E\left[Y-\mu(\theta) \right] + E\left[(Y-\mu(\theta))^2-\sigma^2(\theta) \right] \nonumber\\ + &= E\left[(Y-\mu(\theta))^3 - \sigma^2(\theta)(Y-\mu(\theta))\right] \nonumber\\ + &= \sigma^2(\theta)^{3/2}\gamma_1(\theta)\\ + Var\left[(Y-\mu(\theta))^2-\sigma^2(\theta)\right] &= + E\left[\left((Y-\mu(\theta))^2-\sigma^2(\theta) + \right)\left((Y-\mu(\theta))^2-\sigma^2(\theta) \right) \right]\nonumber\\ + &= E\left[(Y-\mu(\theta))^4 - 2\sigma^2(\theta)(Y-\mu(\theta))^2 + + \sigma^2(\theta)^2 \right] \nonumber\\ + &= \sigma^2(\theta)^2\left(\gamma_2(\theta)+3\right) - 2\sigma^2(\theta)^2 + \sigma^2(\theta)^2 \nonumber\\ + &=\sigma^2(\theta)^2\left(\gamma_2(\theta)+2 \right). +\end{align} + +On obtient alors la matrice de variance-covariance de la fonction +$u(\theta)$, dont on évalue par la suite l'inverse: +\begin{align} + \label{eq:19} + Var\left[u(\theta;Y_t) \right] &= \begin{bmatrix} + \sigma^2(\theta) & \sigma^2(\theta)^{3/2}\gamma_1(\theta)\\ + \sigma^2(\theta)^{3/2}\gamma_1(\theta) & + \sigma^2(\theta)^2\left(\gamma_2(\theta)+2 \right) + \end{bmatrix} \\ + Var^{-1}\left[u(\theta;Y_t) \right] &= + \frac{1}{\sigma^2(\theta)^3\gamma_3(\theta)}\begin{bmatrix} + \sigma^2(\theta)^2\left(\gamma_2(\theta)+2 \right) & -\sigma^2(\theta)^{3/2}\gamma_1(\theta)\\ + -\sigma^2(\theta)^{3/2}\gamma_1(\theta) & \sigma^2(\theta) + \end{bmatrix}. +\end{align} + +On peut enfin évaluer l'expression \eqref{eq:Crowdervecteuroptimal} à +l'aide des résultats précédents: +\begin{align*} + \mathbf{w}^{*}(\theta) &= \begin{bmatrix} -\mu^{\prime}(\theta) & -2 + \sqrt{\sigma^2(\theta)}\sigma^{\prime}(\theta). + \end{bmatrix} \times + \frac{1}{\sigma^2(\theta)^3\gamma_3(\theta)} \begin{bmatrix} + \sigma^2(\theta)^2\left(\gamma_2(\theta)+2 \right) & -\sigma^2(\theta)^{3/2}\gamma_1(\theta)\\ + -\sigma^2(\theta)^{3/2}\gamma_1(\theta) & \sigma^2(\theta) + \end{bmatrix}. +\end{align*} + +On obtient alors le vecteur de pondération optimal +\eqref{eq:vecteuroptimal.ab} formé des fonctions +$\mathbf{a}^{*}(\theta)$ et $\mathbf{b}^{*}(\theta)$: +\begin{subequations}\label{eq:coefficientscrowder} + \begin{align} + \mathbf{a}^{*}(\theta) &= \frac{\left\{ -\left( \gamma_2\left(\theta\right)+2 \right) \mu^{\prime}\left( \theta \right) + 2\gamma_1\left(\theta\right) \sigma^{\prime}\left( \theta \right) \right\}}{\sigma^2\left(\theta\right)\gamma_3(\theta)} \label{eq:acrowder}\\ + \mathbf{b}^{*}(\theta) &= + \frac{\gamma_1\left(\theta\right)\mu^{\prime}\left( \theta + \right)-2\sigma^{\prime}\left( \theta \right)}{\sigma^2\left( + \theta \right)^{3/2}\gamma_3(\theta)}. \label{eq:bcrowder} + \end{align} +\end{subequations} + +On obtient aisément les matrices $M^{\star}\left(\theta\right) \mbox{ + et } V^{\star}\left(\theta\right)$ à partir de leur définition +\eqref{eq:Crowder86-th3.3-def}: +\begin{align} + M^{\star}\left(\theta\right) &= V^{\star}\left(\theta\right) \nonumber\\ + &= T \cdot \sigma(\theta)^{-2} \left\{ \left(\mu^{\prime}\left( + \theta \right)\right)\left(\mu^{\prime}\left( \theta + \right)\right)^T+\gamma_3(\theta)^{-1}\left(\gamma_1\left(\theta\right)\mu^{\prime}\left( + \theta \right)-2\sigma^{\prime}\left( \theta + \right)\right)\left(\gamma_1\left(\theta\right)\mu^{\prime}\left( + \theta \right)-2\sigma^{\prime}\left( \theta \right)\right)^T + \right\}. \label{eq:Moptimalestimetheta} +\end{align} + +Afin d'obtenir la solution optimale $\theta^{\star}$, on définit la +fonction objectif $\Lambda\left(\theta;\mathbf{y}_T\right)$: +\begin{align} + \label{eq:eqnobjectifEE} + \Lambda\left(\theta;\mathbf{y}_T\right) &= + g\left(\theta;\mathbf{y}_T \right) W\left( \theta;\mathbf{y}_T + \right) g\left( \theta;\mathbf{y}_T \right)^T. +\end{align} + +$W\left( \theta ;\mathbf{y}_T \right)$ est une matrice définie +positive. En premier lieu, on suggère d'utiliser la matrice +identité. Puis, on peut raffiner l'estimation en lui substituant la +matrice $V^{\star}(\hat\theta)$ de manière itérative, de la même +manière qu'avec la méthode des moments généralisée itérative, à la +section \ref{sec:GMMtwostep}. + +\section{Équation d'estimation optimale modifiée} +\label{sec:eqoptmodif} + +Il est possible que, pour certaines distributions, les expressions +formant le vecteur de pondération optimal $\mathbf{w}^{*}(\theta)$ +soient particulièrement complexes. On pourra toujours les utiliser, +cependant, il peut être intéressant de développer une approximation de +celles-ci. Pour ce faire, on substitue les valeurs théoriques des +coefficients d'asymétrie et d'aplatissement par une valeur estimée à +partir de l'échantillon: +\begin{subequations}\label{eq:vecteurcrowdermod} + \begin{align} + \mathbf{a}_{mod}^{*}(\theta) &= \frac{\left\{ -\left( \hat\gamma_2\left(\mathbf{y}_T\right)+2 \right) \mu^{\prime}\left( \theta \right) + 2\hat\gamma_1\left(\mathbf{y}_T\right) \sigma^{\prime}\left( \theta \right) \right\}}{\sigma^2\left(\theta\right)\hat\gamma_3(\mathbf{y}_T)} \label{eq:acrowdermod}\\ + \mathbf{b}_{mod}^{*}(\theta) &= + \frac{\hat\gamma_1\left(\mathbf{y}_T\right)\mu^{\prime}\left( + \theta \right)-2\sigma^{\prime}\left( \theta + \right)}{\sigma^2\left( \theta + \right)^{3/2}\hat\gamma_3(\mathbf{y}_T)} \label{eq:bcrowdermod} + \end{align} + + où + \begin{align} + \label{eq:coeffemp} + \hat\gamma_1\left(\mathbf{y}_T\right) &= \frac{\sum_{t=1}^{T}(y_t-\overline{y})^3}{\left(\sum_{t=1}^{T}(y_t-\overline{y})^2\right)^{3/2}} \\ + \hat\gamma_2\left(\mathbf{y}_T\right) &= + \frac{\sum_{t=1}^{T}(y_t-\overline{y})^4}{\left(\sum_{t=1}^{T}(y_t-\overline{y})^2\right)^{2}} \\ + \hat\gamma_3\left(\mathbf{y}_T\right) &= + \hat\gamma_2\left(\mathbf{y}_T\right) + 2 - + \hat\gamma_1\left(\mathbf{y}_T\right)^2. + \end{align} +\end{subequations} + +On pourra ainsi réduire considérablement la taille des expressions à +évaluer tout en conservant les propriétés asymptotiques des +estimateurs, puisque l'espérance de chacune des statistiques utilisées +est égale à la valeur que l'on a remplacée. + +Au prochain chapitre, nous appliquerons la méthode des moments +généralisée et la méthode de l'équation d'estimation optimale à la +distribution de Laplace asymétrique généralisée. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/chapitre6.aux b/memoire/chapitre6.aux new file mode 100644 index 0000000..b7a2a1a --- /dev/null +++ b/memoire/chapitre6.aux @@ -0,0 +1,132 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{seneta2004fitting} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {chapter}{\chapternumberline {6}Estimation des param\IeC {\`e}tres de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{73}{chapter.6}} +\@writefile{toc}{\contentsline {section}{\numberline {6.1}Vecteur de param\IeC {\`e}tres initiaux}{73}{section.6.1}} +\newlabel{sec:condinitGAL}{{6.1}{73}{Vecteur de paramètres initiaux}{section.6.1}{}} +\newlabel{sec:condinitGAL@cref}{{[section][1][6]6.1}{73}} +\newlabel{eq:hypotheseMoM}{{6.1.1}{74}{Vecteur de paramètres initiaux}{equation.6.1.0.1}{}} +\newlabel{eq:hypotheseMoM@cref}{{[equation][1][6,1]6.1.1}{74}} +\newlabel{eq:ptdepartGAL}{{6.1.2}{74}{Vecteur de paramètres initiaux}{equation.6.1.0.2.2}{}} +\newlabel{eq:ptdepartGAL@cref}{{[equation][2][6,1]6.1.2}{74}} +\newlabel{eq:ptdepartGAL1}{{6.1.2a}{74}{Vecteur de paramètres initiaux}{equation.6.1.0.2a}{}} +\newlabel{eq:ptdepartGAL1@cref}{{[equation][1][6,1,2]6.1.2a}{74}} +\newlabel{eq:ptdepartGAL2}{{6.1.2b}{74}{Vecteur de paramètres initiaux}{equation.6.1.0.2b}{}} +\newlabel{eq:ptdepartGAL2@cref}{{[equation][2][6,1,2]6.1.2b}{74}} +\newlabel{eq:ptdepartGAL3}{{6.1.2c}{74}{Vecteur de paramètres initiaux}{equation.6.1.0.2c}{}} +\newlabel{eq:ptdepartGAL3@cref}{{[equation][3][6,1,2]6.1.2c}{74}} +\newlabel{eq:ptdepartGAL4}{{6.1.2d}{74}{Vecteur de paramètres initiaux}{equation.6.1.0.2d}{}} +\newlabel{eq:ptdepartGAL4@cref}{{[equation][4][6,1,2]6.1.2d}{74}} +\@writefile{toc}{\contentsline {section}{\numberline {6.2}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{74}{section.6.2}} +\newlabel{eq:condmom}{{6.2.1}{74}{Méthode des moments généralisée}{equation.6.2.0.1}{}} +\newlabel{eq:condmom@cref}{{[equation][1][6,2]6.2.1}{74}} +\newlabel{eq:condmomGAL}{{6.2.2}{74}{Méthode des moments généralisée}{equation.6.2.0.2}{}} +\newlabel{eq:condmomGAL@cref}{{[equation][2][6,2]6.2.2}{74}} +\newlabel{eq:condmomEMP}{{6.2.3}{75}{Méthode des moments généralisée}{equation.6.2.0.3}{}} +\newlabel{eq:condmomEMP@cref}{{[equation][3][6,2]6.2.3}{75}} +\newlabel{eq:condmomEMPGAL}{{6.2.4}{75}{Méthode des moments généralisée}{equation.6.2.0.4}{}} +\newlabel{eq:condmomEMPGAL@cref}{{[equation][4][6,2]6.2.4}{75}} +\@writefile{toc}{\contentsline {subsection}{\numberline {6.2.1}Matrice de pond\IeC {\'e}ration optimale}{75}{subsection.6.2.1}} +\newlabel{eq:matvcovGMMemp1}{{6.2.7}{76}{Matrice de pondération optimale}{equation.6.2.0.7}{}} +\newlabel{eq:matvcovGMMemp1@cref}{{[equation][7][6,2]6.2.7}{76}} +\@writefile{toc}{\contentsline {subsection}{\numberline {6.2.2}Variance-covariance des param\IeC {\`e}tres}{76}{subsection.6.2.2}} +\@writefile{toc}{\contentsline {subsection}{\numberline {6.2.3}Contraintes lin\IeC {\'e}aires}{77}{subsection.6.2.3}} +\@writefile{lot}{\contentsline {table}{\numberline {6.1}{\ignorespaces Contraintes lin\IeC {\'e}aires pour les cas particuliers de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e $GAL(\theta ,\sigma ,\mu ,\tau )$\relax }}{77}{table.caption.27}} +\newlabel{tab:contraintesGAL}{{6.1}{77}{Contraintes linéaires pour les cas particuliers de la distribution de Laplace asymétrique généralisée $GAL(\theta ,\sigma ,\mu ,\tau )$\relax }{table.caption.27}{}} +\newlabel{tab:contraintesGAL@cref}{{[table][1][6]6.1}{77}} +\@writefile{toc}{\contentsline {section}{\numberline {6.3}M\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale}{78}{section.6.3}} +\newlabel{sec:CrowderGAL}{{6.3}{78}{Méthode de l'équation d'estimation optimale}{section.6.3}{}} +\newlabel{sec:CrowderGAL@cref}{{[section][3][6]6.3}{78}} +\newlabel{eq:derivmomGAL}{{6.3.1}{78}{Méthode de l'équation d'estimation optimale}{equation.6.3.0.1.1}{}} +\newlabel{eq:derivmomGAL@cref}{{[equation][1][6,3]6.3.1}{78}} +\newlabel{eq:derivmom1GAL}{{6.3.1a}{78}{Méthode de l'équation d'estimation optimale}{equation.6.3.0.1a}{}} +\newlabel{eq:derivmom1GAL@cref}{{[equation][1][6,3,1]6.3.1a}{78}} +\newlabel{eq:derivmom2GAL}{{6.3.1b}{78}{Méthode de l'équation d'estimation optimale}{equation.6.3.0.1b}{}} +\newlabel{eq:derivmom2GAL@cref}{{[equation][2][6,3,1]6.3.1b}{78}} +\newlabel{eq:vecteurcrowdermodGAL}{{6.3.2}{79}{Méthode de l'équation d'estimation optimale}{equation.6.3.0.2.2}{}} +\newlabel{eq:vecteurcrowdermodGAL@cref}{{[equation][2][6,3]6.3.2}{79}} +\newlabel{eq:generalquadmod}{{6.3.3}{79}{Méthode de l'équation d'estimation optimale}{equation.6.3.0.3}{}} +\newlabel{eq:generalquadmod@cref}{{[equation][3][6,3]6.3.3}{79}} +\newlabel{eq:defcentrereduite}{{6.3.4}{79}{Méthode de l'équation d'estimation optimale}{equation.6.3.0.4}{}} +\newlabel{eq:defcentrereduite@cref}{{[equation][4][6,3]6.3.4}{79}} +\newlabel{eq:paramnonreduit}{{6.3.5}{79}{Méthode de l'équation d'estimation optimale}{equation.6.3.0.5}{}} +\newlabel{eq:paramnonreduit@cref}{{[equation][5][6,3]6.3.5}{79}} +\@setckpt{chapitre6}{ +\setcounter{page}{80} +\setcounter{equation}{5} +\setcounter{enumi}{3} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{6} +\setcounter{section}{3} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{98} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{1} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{24} +\setcounter{Hfootnote}{1} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{74} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/chapitre6.tex b/memoire/chapitre6.tex new file mode 100644 index 0000000..257592e --- /dev/null +++ b/memoire/chapitre6.tex @@ -0,0 +1,393 @@ +\chapter{Estimation des paramètres de la distribution de Laplace + asymétrique généralisée} + +Ce chapitre présente les différents éléments qui permettent d'obtenir +des estimateurs convergents à l'aide de la méthode des moments +généralisée et de l'équation d'estimation optimale. Premièrement, une +méthode pour obtenir un point de départ efficace pour l'algorithme +d'optimisation sera détaillée. Enfin, les résultats découlant des deux +chapitres précédents seront présentés. + +\section{Vecteur de paramètres initiaux} +\label{sec:condinitGAL} + +Comme chacune des deux méthodes d'estimation étudiées requiert +l'utilisation d'un algorithme d'optimisation numérique, on doit être +en mesure de fournir un vecteur de paramètres initiaux qui favorisera +la convergence de celui-ci. Ce vecteur doit faire partie de l'espace +des paramètres $\Omega$, tel que défini à la table +\ref{tab:roleparamGAL}. Ce vecteur est plus facile à obtenir de +manière empirique, par exemple, avec la méthode des moments, lorsque +c'est possible. Cependant, pour la distribution de Laplace asymétrique +généralisée, le système d'équations formé à partir des quatre premiers +moments centraux empiriques et théoriques \eqref{eq:momentsGAL} n'a +pas de solution analytique. + +\cite{seneta2004fitting} propose une technique pour obtenir un vecteur +de paramètres initiaux basé sur la méthode des moments. En utilisant +la même démarche que ce dernier, on peut obtenir un vecteur pour la +forme en $\mu$ de la fonction caractéristique +\eqref{eq:fncaractGALmu}. On pose comme hypothèse dans les équations +des quatre premiers moments \eqref{eq:moments1GAL}, +\eqref{eq:moments2GAL}, \eqref{eq:moments5GAL} et +\eqref{eq:moments6GAL}, que le paramètre $\mu$ est significativement +nul, donc que la distribution est presque symétrique. On impose donc +que les puissances de ce paramètre plus grandes ou égales à deux +soient nulles: +\begin{align} + \label{eq:hypotheseMoM} + \mu^k=0, \quad k\in\{2,3,4\}. +\end{align} + +En posant l'égalité entre les moments théoriques et leur estimateur +correspondant, on obtient le système d'équations suivant: +\begin{align*} + \hat{m_1} &= \theta+\tau\,\mu \\ + \hat{m_2} &= {\sigma}^{2}\,\tau \\ + \hat{\gamma_1} &= \frac{3\,\mu}{\left| \sigma\right| \,\sqrt{\tau}} \\ + \hat{\gamma_2} &= \frac{3}{\tau}. +\end{align*} + +Ce système possède deux solutions analytiques, dont une respecte la +condition selon laquelle le paramètre $\sigma$ doit être positif. On +note que ce résultat ne constitue qu'un point de départ pour +l'algorithme d'optimisation numérique et qu'on ne peut, en aucun cas, +utiliser celui-ci à des fins statistiques. C'est un estimateur biaisé +et non convergent étant donné la condition \eqref{eq:hypotheseMoM}: +\begin{subequations}\label{eq:ptdepartGAL} + \begin{align} + \hat\theta &=-\frac{\hat{\gamma_1}\,\sqrt{\hat{m_2}}-\hat{\gamma_2}\,\hat{m_1}}{\hat{\gamma_2}} \label{eq:ptdepartGAL1}\\ + \hat\sigma &= \frac{\sqrt{\hat{\gamma_2}\,\hat{m_2}}}{\sqrt{3}} > 0 \label{eq:ptdepartGAL2}\\ + \hat\mu &= \frac{\hat{\gamma_1}\,\sqrt{\hat{m_2}}}{3} \label{eq:ptdepartGAL3}\\ + \hat\tau &= \frac{3}{\hat{\gamma_2}}. \label{eq:ptdepartGAL4} + \end{align} +\end{subequations} + +\section{Méthode des moments généralisée} + +En premier lieu, on utilise la méthode des moments généralisée afin +d'estimer les paramètres de la distribution de Laplace asymétrique +généralisée. Aux fins d'illustration, seulement deux conditions de +moments seront utilisées. Cependant, pour obtenir des résultats +optimaux en termes de convergence asymptotique, on doit utiliser au +moins autant de conditions de moment que de paramètres. Ainsi, en +considérant la différence entre les moyennes et variances empirique et +théorique, on obtient les conditions de moment suivantes, qui ont une +espérance nulle sous les vrais paramètres $\theta_0$: +\begin{align} + h(\theta;Y) &= \left[\begin{array}[]{c} + Y - m_1 \\ + \left(Y - m_1 \right)^2 - m_2 + \end{array}\right] \label{eq:condmom}\\ + &= \left[\begin{array}[]{c} + Y - \left(\theta+\mu\tau\right) \\ + \left(Y - \left(\theta+\mu\tau\right) \right)^2 - + \tau\left(\sigma^2+\mu^2\right) + \end{array}\right]. \label{eq:condmomGAL} +\end{align} + +On définit la contrepartie empirique de ces conditions de moment +comme suit: +\begin{align} + g(\theta;\mathbf{y}_T) &= \frac{1}{T} \left[\begin{array}[]{c} + g_1(\theta;\mathbf{y}_T)\\ + g_2(\theta;\mathbf{y}_T) + \end{array}\right] \nonumber\\ + &= \frac{1}{T} \left[\begin{array}[]{c} + \sum_{t=1}^T y_t - m_1 \\ + \sum_{t=1}^T \left(y_t - m_1 \right)^2 - m_2 + \end{array}\right] \label{eq:condmomEMP}\\ + &= \frac{1}{T} \left[\begin{array}[]{c} + \sum_{t=1}^Ty_t - \left(\theta+\mu\tau\right) \\ + \sum_{t=1}^T\left(y_t - \left(\theta+\mu\tau\right) \right)^2 - + \tau\left(\sigma^2+\mu^2\right) + \end{array}\right]. \label{eq:condmomEMPGAL} +\end{align} + +On doit définir la matrice de pondération optimale. + +\subsection{Matrice de pondération optimale} + +On s'intéresse à la forme analytique de la matrice de pondération +optimale \eqref{eq:matricevcov1}. Celle-ci est obtenue en prenant +l'espérance du produit extérieur du vecteur des conditions de moment +théoriques sur elles-mêmes. Pour des fins de simplification, on +définit le produit des conditions $i$ et $j$ par la notation +$H_{i,j}(\theta;Y) = h_i(\theta;Y)h_j(\theta;Y)$. +\begin{align} + S(\theta;Y) &= E \left[\begin{array}{cc} + H_{(1,1)}(\theta;Y) & H_{(1,2)}(\theta;Y) \\ + H_{(2,1)}(\theta;Y) & H_{(2,2)}(\theta;Y) + \end{array} \right] \nonumber\\ + &= \left[\begin{array}{cc} + E \left[H_{(1,1)}(\theta;Y)\right] & E \left[H_{(1,2)}(\theta;Y)\right] \\ + E \left[H_{(2,1)}(\theta;Y)\right] & E \left[H_{(2,2)}(\theta;Y)\right] + \end{array} \right] +\end{align} +où +\begin{align*} + H_{(1,1)}(\theta;Y) &=\left( Y-\theta-\mu\,\tau\right)^{2} \\ + H_{(1,2)}(\theta;Y) &=\left( {Y}^{2}-2\,\theta\,Y-2\,\mu\,\tau\,Y+{\theta}^{2}+2\,\mu\,\tau\,\theta+{\mu}^{2}\,{\tau}^{2}-{\sigma}^{2}\,\tau-{\mu}^{2}\,\tau\right)^{2} \\ + H_{(2,2)}(\theta;Y) &=\left( Y-\theta-\mu\,\tau\right) \,\left( + {Y}^{2}-2\,\theta\,Y-2\,\mu\,\tau\,Y+{\theta}^{2}+2\,\mu\,\tau\,\theta+{\mu}^{2}\,{\tau}^{2}-{\sigma}^{2}\,\tau-{\mu}^{2}\,\tau\right). +\end{align*} + +On évalue ensuite l'espérance de chacun des éléments de la matrice, où +l'on définit de manière analogue celle du produit des conditions de +moments, $W_{(i,j)}(\theta;Y) = E[H_{(i,j)}(\theta;Y)]$. +\begin{align} + W(\theta;Y) &= \left[\begin{array}{cc} + W_{(1,1)}(\theta;Y) & W_{(1,2)}(\theta;Y) \\ + W_{(2,1)}(\theta;Y) & W_{(2,2)}(\theta;Y) + \end{array} \right] +\end{align} +où +\begin{align*} + W_{(1,1)} &= {\mu}^{2}\,{\tau}^{2}-2\,{\mu}^{2}\,\nu\,\tau+\nu\,{\sigma}^{2}+{\mu}^{2}\,{\nu}^{2}+{\mu}^{2}\,\nu \\ + W_{(2,2)} &= {\mu}^{4}\,{\tau}^{4}+\left( -2\,{\mu}^{2}\,{\sigma}^{2}-4\,{\mu}^{4}\,\nu-2\,{\mu}^{4}\right) \,{\tau}^{3}+\left( {\sigma}^{4}+\left( 10\,{\mu}^{2}\,\nu+2\,{\mu}^{2}\right) \,{\sigma}^{2}+6\,{\mu}^{4}\,{\nu}^{2}+10\,{\mu}^{4}\,\nu+{\mu}^{4}\right) \,{\tau}^{2}\\ + &+\left( -2\,\nu\,{\sigma}^{4}+ \left( -14\,{\mu}^{2}\,{\nu}^{2}-16\,{\mu}^{2}\,\nu\right) \,{\sigma}^{2}-4\,{\mu}^{4}\,{\nu}^{3}-14\,{\mu}^{4}\,{\nu}^{2}-10\,{\mu}^{4}\,\nu\right) \,\tau+\left( 3\,{\nu}^{2}+3\,\nu\right) \,{\sigma}^{4}\\ + &+\left( 6\,{\mu}^{2}\,{\nu}^{3}+18\,{\mu}^{2}\,{\nu}^{2}+12\,{\mu}^{2}\,\nu\right) \,{\sigma}^{2}+{\mu}^{4}\,{\nu}^{4}+6\,{\mu}^{4}\,{\nu}^{3}+11\,{\mu}^{4}\,{\nu}^{2}+6\,{\mu}^{4}\,\nu \\ + W_{(1,2)} &= W_{(2,1)} = 3\,{\theta}^{4}+\left( 6\,\mu\,\nu-3\right) \,{\theta}^{3}+\left( 3\,\nu\,{\sigma}^{2}+3\,{\mu}^{2}\,{\nu}^{2}+\left( 3\,{\mu}^{2}-3\,\mu\right) \,\nu\right) \,{\theta}^{2}-{\mu}^{3}\,{\tau}^{3} \\ + &+\left( \mu\,{\sigma}^{2}+3\,{\mu}^{3}\,\nu+{\mu}^{3}\right) \,{\tau}^{2}+\left( -4\,\mu\,\nu\,{\sigma}^{2}-3\,{\mu}^{3}\,{\nu}^{2}-4\,{\mu}^{3}\,\nu\right) \,\tau+\left( 3\,\mu\,{\nu}^{2}+3\,\mu\,\nu\right) \,{\sigma}^{2}\\ + &+{\mu}^{3}\,{\nu}^{3}+3\,{\mu}^{3}\,{\nu}^{2}+2\,{\mu}^{3}\,\nu. +\end{align*} + +On évalue quelle serait la valeur de la variance-covariance sous des +paramètres optimaux. Ce résultat permet par la suite d'évaluer la +distribution asymptotique des estimateurs. + +On obtient l'estimateur de la matrice optimale \eqref{eq:matvcovGMM} +en effectuant le produit extérieur du vecteur de conditions de moment +empirique \eqref{eq:condmomGAL}, puis en évaluant la moyenne des +matrices résultantes de l'équation \eqref{eq:matponderationproduith}: +\begin{align} + \label{eq:matvcovGMMemp1} + \hat{S}(\theta;\mathbf{y}_T) &= \frac{1}{T} \sum_{t=1}^T + \left[\begin{array}{cc} + G_{(1,1)}(\theta;y_t) & G_{(1,2)}(\theta;y_t) \\ + G_{(2,1)}(\theta;y_t) & G_{(2,2)}(\theta;y_t) + \end{array} \right] +\end{align} +où +\begin{align*} + G_{(1,1)}(\theta;y_t) &=\left( y_t-\theta-\mu\,\tau\right)^{2} \\ + G_{(1,2)}(\theta;y_t) &=\left( {y_t}^{2}-2\,\theta\,y_t-2\,\mu\,\tau\,y_t+{\theta}^{2}+2\,\mu\,\tau\,\theta+{\mu}^{2}\,{\tau}^{2}-{\sigma}^{2}\,\tau-{\mu}^{2}\,\tau\right)^{2} \\ + G_{(2,2)}(\theta;y_t) &=\left( y_t-\theta-\mu\,\tau\right) \,\left( + {y_t}^{2}-2\,\theta\,y_t-2\,\mu\,\tau\,y_t+{\theta}^{2}+2\,\mu\,\tau\,\theta+{\mu}^{2}\,{\tau}^{2}-{\sigma}^{2}\,\tau-{\mu}^{2}\,\tau\right). +\end{align*} + + +\subsection{Variance-covariance des paramètres} + +On obtient la variance-covariance asymptotique des paramètres à partir +de la variance-covariance associée aux conditions de moment, en +utilisant la méthode delta multivariée. Pour ce faire, on évalue +d'abord la valeur théorique du gradient $D(\theta)$. +\begin{align} + D(\theta) &= E \left[ \begin{array}{cc} + -1 & -2\,\left( Y-\theta-\mu\,\tau\right) \\ + 0 & -2\,\sigma\,\tau \\ + -\tau & -2\,\tau\,\left( Y-\theta-\mu\,\tau\right) -2\,\mu\,\tau \\ + -\nu & -2\,\mu\,\left( Y-\theta-\mu\,\tau\right) + -{\sigma}^{2}-{\mu}^{2} + \end{array}\right] \nonumber\\ + &= \left[ \begin{array}{cc} + -1 & 0 \\ + 0 & -2\,\sigma\,\tau \\ + -\tau & -2\,\mu\,\tau \\ + -\nu & -{\sigma}^{2}-{\mu}^{2} + \end{array}\right]. +\end{align} + +Ensuite, on utilise le résultat \eqref{matricevcovparamGMMnc} afin +d'obtenir la valeur de la variance-covariance $\mathcal{J}_0^{-1}$. + +De la même manière, on évalue la variance-covariance des paramètres +optimaux en utilisant les valeurs empiriques de la variance-covariance +des conditions de moment \eqref{eq:matvcovGMMemp1} et du gradient +$\hat{D}(\theta,\mathbf{y}_T)$, défini par la matrice +\eqref{eq:gradientGMM}. On a essentiellement à calculer la moyenne +empirique des gradients obtenus en chaque point $y_t$, puis à utiliser +le résultat \eqref{matricevcovparamGMMnc} avec les estimateurs +$\hat{W}_T = \hat{S}^{-1}(\theta,\mathbf{y}_T)$ et +$\hat{D}(\theta,\mathbf{y}_T)$: +\begin{align} + \hat{D}(\theta,\mathbf{y}_T) &= \frac{1}{T} \sum_{t=1}^{T} + \left[ \begin{array}{cc} + -1 & -2\,\left( y_t-\theta-\mu\,\tau\right) \\ + 0 & -2\,\sigma\,\tau \\ + -\tau & -2\,\tau\,\left( y_t-\theta-\mu\,\tau\right) -2\,\mu\,\tau \\ + -\nu & -2\,\mu\,\left( y_t-\theta-\mu\,\tau\right) + -{\sigma}^{2}-{\mu}^{2} + \end{array}\right]. +\end{align} + +\subsection{Contraintes linéaires} + +Une fois l'estimation des paramètres effectuée, on peut tester si ces +derniers pourraient en fait correspondre à un cas particulier de la +distribution de Laplace asymétrique généralisée ayant le même support, +parmi celles figurant à la table \ref{tab:casspeciauxGAL}. Pour ce +faire, on doit déterminer les paramètres de la distribution sous un +ensemble de contraintes linéaires, sous la forme +\eqref{eq:contraintelin}, correspondant au cas particulier, tel +qu'énuméré dans la table \ref{tab:contraintesGAL}. +\begin{table}[h!] + \centering + \begin{tabular}{ccc} + & \multicolumn{2}{c}{\textbf{Paramètres}} \\ + \hline + $\text{\textbf{Distribution}}$ & $R$ & $r$ \\ + \hline + Laplace symétrique & $\begin{bmatrix} + 0&0&1&0\\ + 0&0&0&1 + \end{bmatrix}$ & $\begin{bmatrix} + 0\\ + 1 + \end{bmatrix}$ \\[1cm] + Laplace asymétrique & $\begin{bmatrix} 0&0&0&1 + \end{bmatrix}$ & $\begin{bmatrix} 1 + \end{bmatrix}$ \\[1cm] + Dégénérée à $\theta$ & $\begin{bmatrix} + 0&1&0&0\\ + 0&0&1&0\\ + 0&0&0&1 + \end{bmatrix}$ & $\begin{bmatrix} + 0\\ + 0\\ + 0\\ + \end{bmatrix}$\\\hline + \end{tabular} + \caption{Contraintes linéaires pour les cas particuliers de la distribution de Laplace asymétrique généralisée $GAL(\theta,\sigma,\mu,\tau)$} + \label{tab:contraintesGAL} +\end{table} + +On doit ensuite utiliser un algorithme d'optimisation numérique afin +de maximiser le lagrangien \eqref{eq:estimateurGMMlagrange} défini par +la fonction objectif $Q_T(\theta)$ \eqref{eq:objectifGMM2} et +l'ensemble de contraintes $a(\theta) = R\theta-r$ sélectionné +précédemment. Enfin, on peut effectuer les tests statistiques de la +section \ref{sec:testwald} afin de vérifier si les paramètres de la +distribution contrainte sont significativement différents de ceux de +la distribution non contrainte. Dans cette situation, l'hypothèse +selon laquelle le cas particulier serait approprié pour décrire la +distribution des données de l'échantillon serait rejetée. + +Notons qu'on peut aussi estimer les paramètres de chacun des cas +particuliers directement par la méthode des moments généralisée en +utilisant des conditions de moment basées sur la moyenne $m_1$ et la +variance $m_2$ de la forme \eqref{eq:condmom}. Par contre, dans ces +cas, on devra utiliser des tests d'adéquation non paramétriques afin +de décider laquelle représente mieux la distribution de l'échantillon +de données. Ces tests seront présentés au chapitre suivant, à la +section \ref{sec:testnonparam}. + +\section{Méthode de l'équation d'estimation optimale} +\label{sec:CrowderGAL} + +On utilise maintenant la méthode de l'équation d'estimation optimale +afin d’estimer les paramètres de la distribution de Laplace +asymétrique généralisée. On considère une équation d'estimation de la +forme quadratique \eqref{eq:generalquad}, basée sur les conditions de +moments \eqref{eq:condmomGAL} utilisées à la section précédente. + +On évalue d'abord les expressions des dérivées premières de la moyenne +et de l'écart-type par rapport au vecteur de paramètres +\eqref{eq:derivmom}: +\begin{subequations}\label{eq:derivmomGAL} + \begin{align} + \mu^{\prime}\left( \theta \right) &= \frac{d}{d\theta}\left(\theta+\tau\,\mu\right) \nonumber\\ + &= \left[ \begin{array}{c} 1\\ + 0\\ + \tau\\ + \mu\end{array} \right]^T \label{eq:derivmom1GAL}\\ + \sigma^{\prime}\left( \theta \right) &= \frac{d}{d\theta}\sqrt{\tau\sigma^2+\tau\mu^2} \nonumber\\ + &= \frac{1}{\sqrt {\tau\,{\sigma}^{2}+\tau\,{\mu}^{2}}} + \left[ \begin{array}{c} + 0\\ + {\tau\,\sigma}\\ + {\tau\,\mu}\\ + \frac{{\sigma}^{2}+{\mu}^{2}}{2} + \end{array} \right]^T.\label{eq:derivmom2GAL} + \end{align} +\end{subequations} + +On peut alors obtenir une forme analytique en utilisant les +expressions déterminées précédemment pour $\mathbf{a}(\theta)$ et +$\mathbf{b}(\theta)$ \eqref{eq:coefficientscrowder}, les moments de la +distribution \eqref{eq:momentsGAL} ainsi que les coefficients +d'asymétrie et d'aplatissement \eqref{eq:moments56GAL}. Étant donné la +longueur des expressions obtenues, elles ne seront pas présentées dans +ce texte. Cependant, on peut facilement les évaluer à l'aide d'un +logiciel de calcul symbolique. On combine ensuite celles-ci pour +obtenir l'équation d'estimation optimale $g\left(\theta;\mathbf{y}_T +\right)$ de forme quadratique. + +On obtient cependant des expressions plus faciles à manipuler pour +$\mathbf{a}(\theta)$ et $\mathbf{b}(\theta)$ en utilisant la version +modifiée du vecteur de pondération optimal +\eqref{eq:vecteurcrowdermod}: +\begin{subequations} + \label{eq:vecteurcrowdermodGAL} + \begin{align} + \mathbf{a}_{mod}^{*}(\theta;\mathbf{y}_T) &= \begin{bmatrix} + \frac{-\hat\gamma_2\left(\mathbf{y}_T\right)-2}{\left( {\sigma}^{2}+{\mu}^{2}\right) \,\tau\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + \frac{2\,\sigma\,\hat\gamma_1\left(\mathbf{y}_T\right)}{{\left( {\sigma}^{2}+{\mu}^{2}\right) }^{3/2}\,\sqrt{\tau}\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + \frac{\frac{2\,\mu\,\sqrt{\tau}\,\hat\gamma_1\left(\mathbf{y}_T\right)}{\sqrt{{\sigma}^{2}+{\mu}^{2}}}+\tau\,\left( -\hat\gamma_2\left(\mathbf{y}_T\right)-2\right) }{\left( {\sigma}^{2}+{\mu}^{2}\right) \,\tau\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + \frac{\frac{\sqrt{{\sigma}^{2}+{\mu}^{2}}\,\hat\gamma_1\left(\mathbf{y}_T\right)}{\sqrt{\tau}}+\mu\,\left( + -\hat\gamma_2\left(\mathbf{y}_T\right)-2\right) }{\left( + {\sigma}^{2}+{\mu}^{2}\right) + \,\tau\,\hat\gamma_3\left(\mathbf{y}_T\right) } + \end{bmatrix} \\ + \mathbf{b}_{mod}^{*}(\theta;\mathbf{y}_T) &= \begin{bmatrix} + \frac{\hat\gamma_1\left(\mathbf{y}_T\right)}{{\left( {\sigma}^{2}+{\mu}^{2}\right) }^{3/2}\,{\tau}^{3/2}\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + -\frac{2\,\sigma}{{\left( {\sigma}^{2}+{\mu}^{2}\right) }^{2}\,\tau\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + \frac{\tau\,\hat\gamma_1\left(\mathbf{y}_T\right)-\frac{2\,\mu\,\sqrt{\tau}}{\sqrt{{\sigma}^{2}+{\mu}^{2}}}}{{\left( {\sigma}^{2}+{\mu}^{2}\right) }^{3/2}\,{\tau}^{3/2}\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + \frac{\mu\,\hat\gamma_1\left(\mathbf{y}_T\right)-\frac{\sqrt{{\sigma}^{2}+{\mu}^{2}}}{\sqrt{\tau}}}{{\left( + {\sigma}^{2}+{\mu}^{2}\right) + }^{3/2}\,{\tau}^{3/2}\,\hat\gamma_3\left(\mathbf{y}_T\right) } + \end{bmatrix}. + \end{align} +\end{subequations} + +On insère ensuite ces dernières dans la forme générale +\eqref{eq:generalquad} pour obtenir l'équation d'estimation optimale +modifiée $g_{mod}^{*}\left(\theta;Y \right)$: +\begin{align} + \label{eq:generalquadmod} + g_{mod}^{*}\left(\theta;Y \right) = \sum_{t=1}^n \left[ + \mathbf{a}_{mod}^{*}(\theta;\mathbf{y}_T)(y_t-\mu\left(\theta\right)) + + \mathbf{b}_{mod}^{*}(\theta;\mathbf{y}_T)\left((y_t-\mu\left(\theta\right))^2-\sigma^2\left(\theta\right) + \right)\right]. +\end{align} + +On note qu'afin d'éviter des irrégularités numériques, on suggère, +dans les deux cas, de faire l'estimation sur des données centrées et +réduites: +\begin{align} + X_t &= + \frac{Y_t-\hat{m_1}}{\sqrt{\hat{m_2}}}. \label{eq:defcentrereduite} +\end{align} + +Comme précédemment, on pourra utiliser la propriété d'invariance +\eqref{eq:GALscaletrans} de la paramétrisation en $\kappa$ afin de +retrouver les paramètres de la distribution de la variable aléatoire +$Y_t$ une fois l'estimation effectuée: +\begin{align} + Y_t &= \hat{\sigma}_{t} X_t + \hat{\mu}_{t} \sim + GAL(\hat{\sigma}_{t} \theta + + \hat{\mu}_{t},\hat{\sigma}_{t}\sigma,\kappa,\tau). \label{eq:paramnonreduit} +\end{align} + +Afin d'obtenir le vecteur de paramètres estimés $\hat\theta$, on +minimise la fonction objectif +$\Lambda\left(\theta;\mathbf{y}_T\right)$ +\eqref{eq:eqnobjectifEE}. Une fois les paramètres obtenus, on peut +ensuite évaluer la variance-covariance de ces estimateurs, à partir du +résultat \eqref{eq:Moptimalestimetheta}. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/chapitre7.aux b/memoire/chapitre7.aux new file mode 100644 index 0000000..8594ffa --- /dev/null +++ b/memoire/chapitre7.aux @@ -0,0 +1,160 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{shapiro1965analysis} +\citation{shapiro1965analysis} +\citation{shapiro1965analysis} +\citation{epps1983test} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {chapter}{\chapternumberline {7}Tests statistiques}{81}{chapter.7}} +\@writefile{toc}{\contentsline {section}{\numberline {7.1}Test de normalit\IeC {\'e}}{81}{section.7.1}} +\newlabel{sec:testnormalite}{{7.1}{81}{Test de normalité}{section.7.1}{}} +\newlabel{sec:testnormalite@cref}{{[section][1][7]7.1}{81}} +\@writefile{toc}{\contentsline {subsection}{\numberline {7.1.1}Test de Shapiro-Wilk}{81}{subsection.7.1.1}} +\newlabel{sec:test-de-shapiro}{{7.1.1}{81}{Test de Shapiro-Wilk}{subsection.7.1.1}{}} +\newlabel{sec:test-de-shapiro@cref}{{[subsection][1][7,1]7.1.1}{81}} +\newlabel{eq:shapirowilkb1}{{7.1.1}{81}{Test de Shapiro-Wilk}{equation.7.1.0.1}{}} +\newlabel{eq:shapirowilkb1@cref}{{[equation][1][7,1]7.1.1}{81}} +\newlabel{eq:shapirowilkb2}{{7.1.2}{81}{Test de Shapiro-Wilk}{equation.7.1.0.2}{}} +\newlabel{eq:shapirowilkb2@cref}{{[equation][2][7,1]7.1.2}{81}} +\citation{henze1990approximation} +\citation{henze1990approximation} +\@writefile{toc}{\contentsline {subsection}{\numberline {7.1.2}Test d\IeC {\textquoteright }Epps-Pulley}{82}{subsection.7.1.2}} +\newlabel{sec:test-de-epps}{{7.1.2}{82}{Test d’Epps-Pulley}{subsection.7.1.2}{}} +\newlabel{sec:test-de-epps@cref}{{[subsection][2][7,1]7.1.2}{82}} +\newlabel{eq:statistiqueEppsPulley}{{7.1.3}{82}{Test d’Epps-Pulley}{equation.7.1.0.3}{}} +\newlabel{eq:statistiqueEppsPulley@cref}{{[equation][3][7,1]7.1.3}{82}} +\newlabel{eq:EppsPulley}{{7.1.4}{82}{Test d’Epps-Pulley}{equation.7.1.0.4}{}} +\newlabel{eq:EppsPulley@cref}{{[equation][4][7,1]7.1.4}{82}} +\newlabel{eq:EppsPulleyMod}{{7.1.5}{82}{Test d’Epps-Pulley}{equation.7.1.0.5}{}} +\newlabel{eq:EppsPulleyMod@cref}{{[equation][5][7,1]7.1.5}{82}} +\newlabel{eq:ZEppsPulley}{{7.1.6}{82}{Test d’Epps-Pulley}{equation.7.1.0.6}{}} +\newlabel{eq:ZEppsPulley@cref}{{[equation][6][7,1]7.1.6}{82}} +\@writefile{toc}{\contentsline {section}{\numberline {7.2}Tests d'ad\IeC {\'e}quation}{82}{section.7.2}} +\newlabel{sec:testnonparam}{{7.2}{82}{Tests d'adéquation}{section.7.2}{}} +\newlabel{sec:testnonparam@cref}{{[section][2][7]7.2}{82}} +\citation{hogg1978introduction} +\newlabel{eq:hypotheseadeqcomp}{{7.2.1}{83}{Tests d'adéquation}{equation.7.2.0.1}{}} +\newlabel{eq:hypotheseadeqcomp@cref}{{[equation][1][7,2]7.2.1}{83}} +\newlabel{eq:hypotheseadeq}{{7.2.2}{83}{Tests d'adéquation}{equation.7.2.0.2}{}} +\newlabel{eq:hypotheseadeq@cref}{{[equation][2][7,2]7.2.2}{83}} +\@writefile{toc}{\contentsline {subsection}{\numberline {7.2.1}Test $\chi ^2$ de Pearson}{83}{subsection.7.2.1}} +\newlabel{sec:testchi2}{{7.2.1}{83}{Test $\chi ^2$ de Pearson}{subsection.7.2.1}{}} +\newlabel{sec:testchi2@cref}{{[subsection][1][7,2]7.2.1}{83}} +\newlabel{eq:pmultinomiale}{{7.2.3}{83}{Test $\chi ^2$ de Pearson}{equation.7.2.0.3}{}} +\newlabel{eq:pmultinomiale@cref}{{[equation][3][7,2]7.2.3}{83}} +\newlabel{eq:statchi2}{{7.2.4}{83}{Test $\chi ^2$ de Pearson}{equation.7.2.0.4}{}} +\newlabel{eq:statchi2@cref}{{[equation][4][7,2]7.2.4}{83}} +\citation{wang2003evaluating} +\newlabel{eq:hyp0-1}{{7.2.5}{84}{Test $\chi ^2$ de Pearson}{equation.7.2.0.5}{}} +\newlabel{eq:hyp0-1@cref}{{[equation][5][7,2]7.2.5}{84}} +\newlabel{eq:hyp0-2}{{7.2.6}{84}{Test $\chi ^2$ de Pearson}{equation.7.2.0.6}{}} +\newlabel{eq:hyp0-2@cref}{{[equation][6][7,2]7.2.6}{84}} +\newlabel{eq:statchi2param}{{7.2.7}{84}{Test $\chi ^2$ de Pearson}{equation.7.2.0.7}{}} +\newlabel{eq:statchi2param@cref}{{[equation][7][7,2]7.2.7}{84}} +\newlabel{eq:quantilechisq}{{7.2.8}{84}{Test $\chi ^2$ de Pearson}{equation.7.2.0.8}{}} +\newlabel{eq:quantilechisq@cref}{{[equation][8][7,2]7.2.8}{84}} +\@writefile{toc}{\contentsline {subsection}{\numberline {7.2.2}Test de Kolmogorov-Smirnov}{84}{subsection.7.2.2}} +\newlabel{sec:kolmogorovsmirnov}{{7.2.2}{84}{Test de Kolmogorov-Smirnov}{subsection.7.2.2}{}} +\newlabel{sec:kolmogorovsmirnov@cref}{{[subsection][2][7,2]7.2.2}{84}} +\newlabel{eq:statks}{{7.2.9}{84}{Test de Kolmogorov-Smirnov}{equation.7.2.0.9}{}} +\newlabel{eq:statks@cref}{{[equation][9][7,2]7.2.9}{84}} +\newlabel{eq:repartemp}{{7.2.10}{84}{Test de Kolmogorov-Smirnov}{equation.7.2.0.10}{}} +\newlabel{eq:repartemp@cref}{{[equation][10][7,2]7.2.10}{84}} +\newlabel{eq:repartkolmogorov}{{7.2.11}{84}{Test de Kolmogorov-Smirnov}{equation.7.2.0.11}{}} +\newlabel{eq:repartkolmogorov@cref}{{[equation][11][7,2]7.2.11}{84}} +\citation{luong1987minimum} +\citation{feuerverger1981efficiency} +\citation{KOUTROUVELIS01011980} +\newlabel{eq:quantilekolmogorov}{{7.2.13}{85}{Test de Kolmogorov-Smirnov}{equation.7.2.0.13}{}} +\newlabel{eq:quantilekolmogorov@cref}{{[equation][13][7,2]7.2.13}{85}} +\@writefile{toc}{\contentsline {subsection}{\numberline {7.2.3}Test de distance minimale bas\IeC {\'e} sur la fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments}{85}{subsection.7.2.3}} +\newlabel{sec:test-de-distance}{{7.2.3}{85}{Test de distance minimale basé sur la fonction génératrice des moments}{subsection.7.2.3}{}} +\newlabel{sec:test-de-distance@cref}{{[subsection][3][7,2]7.2.3}{85}} +\newlabel{eq:4}{{7.2.14}{85}{Test de distance minimale basé sur la fonction génératrice des moments}{equation.7.2.0.14}{}} +\newlabel{eq:4@cref}{{[equation][14][7,2]7.2.14}{85}} +\newlabel{eq:5}{{7.2.15}{85}{Test de distance minimale basé sur la fonction génératrice des moments}{equation.7.2.0.15}{}} +\newlabel{eq:5@cref}{{[equation][15][7,2]7.2.15}{85}} +\newlabel{eq:9}{{7.2.16}{85}{Test de distance minimale basé sur la fonction génératrice des moments}{equation.7.2.0.16}{}} +\newlabel{eq:9@cref}{{[equation][16][7,2]7.2.16}{85}} +\newlabel{eq:14}{{7.2.17}{86}{Test de distance minimale basé sur la fonction génératrice des moments}{equation.7.2.0.17}{}} +\newlabel{eq:14@cref}{{[equation][17][7,2]7.2.17}{86}} +\newlabel{eq:10}{{7.2.18}{86}{Test de distance minimale basé sur la fonction génératrice des moments}{equation.7.2.0.18}{}} +\newlabel{eq:10@cref}{{[equation][18][7,2]7.2.18}{86}} +\@setckpt{chapitre7}{ +\setcounter{page}{87} +\setcounter{equation}{18} +\setcounter{enumi}{2} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{7} +\setcounter{section}{2} +\setcounter{subsection}{3} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{105} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{0} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{26} +\setcounter{Hfootnote}{1} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{82} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/chapitre7.log b/memoire/chapitre7.log new file mode 100644 index 0000000..08ae071 --- /dev/null +++ b/memoire/chapitre7.log @@ -0,0 +1,9686 @@ +This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2013.11.30) 2 DEC 2013 21:08 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**chapitre7.tex +(./chapitre7.tex +LaTeX2e <2011/06/27> +Babel <3.9f> and hyphenation patterns for 78 languages loaded. +! Undefined control sequence. +l.1 \chapter + {Tests statistiques} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1 \chapter{T + ests statistiques} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 1--2 +[] + [] + +! Undefined control sequence. +l.3 \section + {Test de normalité} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3 \section{T + est de normalité} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 3--5 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.6 O + n utilise un test de normalité afin de vérifier si l'on peut rejeter +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no â in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no ™ in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 6--13 +[] + [] + +! Undefined control sequence. +l.14 \subsection + {Test de Shapiro-Wilk} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.14 \subsection{T + est de Shapiro-Wilk} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no k in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 14--16 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.17 L + e test de Shapiro-Wilk \citep{shapiro1965analysis} est basé sur les +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no k in font nullfont! +! Undefined control sequence. +l.17 Le test de Shapiro-Wilk \citep + {shapiro1965analysis} est basé sur les +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no s in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 19. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 19. +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 17--21 +[] + [] + + +Overfull \hbox (21.67706pt too wide) in paragraph at lines 17--21 +\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 T < + [] + + +Overfull \hbox (13.88893pt too wide) in paragraph at lines 17--21 +\OT1/cmr/m/n/10 20)$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.22 O + n considère l'échantillon ordonné +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{shapiro1965analysis} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.25 ...mph{Table 5} de \cite{shapiro1965analysis} + : +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `shapiro1965analysis' on page undefined on input line +25. + +Missing character: There is no ] in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 22--26 +[] + [] + + +Overfull \hbox (78.17471pt too wide) in paragraph at lines 22--26 +\OML/cmm/m/it/10 y[]; y[]; [] ; y[];$ + [] + + +Overfull \hbox (11.19444pt too wide) in paragraph at lines 22--26 +\OML/cmm/m/it/10 S[]$ + [] + + +Overfull \hbox (4.29166pt too wide) in paragraph at lines 22--26 +\OML/cmm/m/it/10 b$ + [] + + +Overfull \hbox (30.78989pt too wide) in paragraph at lines 22--26 +\OML/cmm/m/it/10 a[]$ + [] + + +Overfull \hbox (15.07767pt too wide) in paragraph at lines 22--26 +\OT1/cmr/m/it/10 Ta- + [] + + +Overfull \hbox (11.75546pt too wide) in paragraph at lines 22--26 +\OT1/cmr/m/it/10 ble + [] + + +Overfull \hbox (5.11108pt too wide) in paragraph at lines 22--26 +\OT1/cmr/m/it/10 5 + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 22--26 +[] + [] + +Missing character: There is no S in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.29 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no b in font nullfont! +! Misplaced alignment tab character &. +l.31 b & + = \sum_{i=1}^{k}a_{T-i+1}(y_{(T-i+1)}-y_{(i)}). +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +! Missing $ inserted. + + $ +l.31 b &= \sum + _{i=1}^{k}a_{T-i+1}(y_{(T-i+1)}-y_{(i)}). +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{enumerate} on input line 26 ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.32 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.32 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (7.23265pt too wide) in paragraph at lines 27--33 +\OML/cmm/m/it/10 T$ + [] + + +Overfull \hbox (16.07635pt too wide) in paragraph at lines 27--33 +\OML/cmm/m/it/10 k \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (8.19965pt too wide) in paragraph at lines 27--33 +[]$ + [] + + +Overfull \hbox (104.78998pt too wide) in paragraph at lines 27--33 +[][] \OML/cmm/m/it/10 a[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y[] \OMS/cmsy/m/n/10 + ^^@ + [] + + +Overfull \hbox (21.14882pt too wide) in paragraph at lines 27--33 +\OML/cmm/m/it/10 y[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no S in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.35 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no b in font nullfont! +! Misplaced alignment tab character &. +l.37 b & + = a_{T}(y_{(T)}-y_{(1)}) + \ldots + a_{k+2}(y_{(k+2)}-y_{(k)}). +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +Missing character: There is no a in font nullfont! +! Missing $ inserted. + + $ +l.37 b &= a_ + {T}(y_{(T)}-y_{(1)}) + \ldots + a_{k+2}(y_{(k+2)}-y_{(k)}). +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{enumerate} on input line 26 ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.38 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.38 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (7.23265pt too wide) in paragraph at lines 33--39 +\OML/cmm/m/it/10 T$ + [] + + +Overfull \hbox (16.07635pt too wide) in paragraph at lines 33--39 +\OML/cmm/m/it/10 k \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (18.43579pt too wide) in paragraph at lines 33--39 +[]$ + [] + + +Overfull \hbox (35.41885pt too wide) in paragraph at lines 33--39 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (27.30568pt too wide) in paragraph at lines 33--39 +\OML/cmm/m/it/10 y[]\OT1/cmr/m/n/10 ) + + [] + + +Overfull \hbox (19.44441pt too wide) in paragraph at lines 33--39 +[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (58.19586pt too wide) in paragraph at lines 33--39 +\OML/cmm/m/it/10 a[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (22.7397pt too wide) in paragraph at lines 33--39 +\OML/cmm/m/it/10 y[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{shapiro1965analysis} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.43 \emph{Table 6} de \cite{shapiro1965analysis} + . On rejette l'hypothèse +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `shapiro1965analysis' on page undefined on input line +43. + +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 41--46 +[] + [] + + +Overfull \hbox (21.38885pt too wide) in paragraph at lines 41--46 +\OML/cmm/m/it/10 W \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (11.5959pt too wide) in paragraph at lines 41--46 +[]$ + [] + + +Overfull \hbox (15.07767pt too wide) in paragraph at lines 41--46 +\OT1/cmr/m/it/10 Ta- + [] + + +Overfull \hbox (11.75546pt too wide) in paragraph at lines 41--46 +\OT1/cmr/m/it/10 ble + [] + + +Overfull \hbox (5.11108pt too wide) in paragraph at lines 41--46 +\OT1/cmr/m/it/10 6 + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 41--46 +[] + [] + + +Overfull \hbox (10.83334pt too wide) in paragraph at lines 41--46 +\OML/cmm/m/it/10 W$ + [] + +! Undefined control sequence. +l.47 \subsection + {Test d’Epps-Pulley} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no â in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no ™ in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no y in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 47--49 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no â in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no ™ in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no y in font nullfont! +! Undefined control sequence. +l.50 Le test d’Epps-Pulley \citep + {epps1983test} est basé sur le carré de la +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.57 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no E in font nullfont! +Missing character: There is no P in font nullfont! +! Missing $ inserted. + + $ +l.59 EP_ + T &= T\int_{-\infty}^{\infty} \left\vert\frac{1}{T}\sum_{j=1}^T +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.59 EP_T & + = T\int_{-\infty}^{\infty} \left\vert\frac{1}{T}\sum_{j=1}^T +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.62 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.62 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 50--63 +[] + [] + + +Overfull \hbox (9.06943pt too wide) in paragraph at lines 50--63 +[]$ + [] + + +Overfull \hbox (6.70831pt too wide) in paragraph at lines 50--63 +\OML/cmm/m/it/10 S$ + [] + + +Overfull \hbox (16.85516pt too wide) in paragraph at lines 50--63 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (233.76042pt too wide) in paragraph at lines 50--63 +\OML/cmm/m/it/10 T [][] [][] w\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/1 +0 )\OML/cmm/m/it/10 dt:$ + [] + +Missing character: There is no [ in font nullfont! +\citation{henze1990approximation} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.64 \cite{henze1990approximation} + propose une procédure simple à +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `henze1990approximation' on page undefined on input li +ne 64. + +Missing character: There is no ] in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no â in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no ™ in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.71 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no E in font nullfont! +Missing character: There is no P in font nullfont! +! Missing $ inserted. + + $ +l.73 EP_ + T &= \frac{2}{T} \sum_{1\leq\,j for immediate help. + ... + +l.77 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.77 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (25.43053pt too wide) in paragraph at lines 64--78 +[][] + [] + + +Overfull \hbox (20.6781pt too wide) in paragraph at lines 64--78 +\OML/cmm/m/it/10 EP[]$ + [] + + +Overfull \hbox (17.98384pt too wide) in paragraph at lines 64--78 +\OML/cmm/m/it/10 w\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 ^^A\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (16.85516pt too wide) in paragraph at lines 64--78 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (143.96034pt too wide) in paragraph at lines 64--78 +[] [][] [] [] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (122.3431pt too wide) in paragraph at lines 64--78 +[] [][] [] [] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (20.7334pt too wide) in paragraph at lines 64--78 +[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 64--78 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 79--80 +[] + [] + + +Overfull \hbox (17.78816pt too wide) in paragraph at lines 79--80 +\OML/cmm/m/it/10 T > + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 79--80 +\OT1/cmr/m/n/10 10$ + [] + + +Overfull \hbox (192.44495pt too wide) detected at line 83 +[]\OML/cmm/m/it/10 EP[] \OT1/cmr/m/n/10 = [] [] \OML/cmm/m/it/10 : + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.87 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no Z in font nullfont! +! Missing $ inserted. + + $ +l.89 Z_ + T &= \gamma + \delta\ln\left(\left(EP_T^{*}-\xi +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.89 Z_T & + = \gamma + \delta\ln\left(\left(EP_T^{*}-\xi +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.91 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.91 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 85--92 +[] + [] + + +Overfull \hbox (16.85516pt too wide) in paragraph at lines 85--92 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (13.51065pt too wide) in paragraph at lines 85--92 +\OML/cmm/m/it/10 ^^M \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (139.4601pt too wide) in paragraph at lines 85--92 +\OML/cmm/m/it/10 ^^N [] [] :$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{henze1990approximation} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.94 \cite{henze1990approximation} + : +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `henze1990approximation' on page undefined on input li +ne 94. + +Missing character: There is no ] in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align* undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.95 \begin{align*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.96 \gamma + &\approx 3.55295 \\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.96 \gamma & + \approx 3.55295 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.97 \delta & + \approx 1.23062 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.98 \lambda & + \approx 2.26664 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.99 \xi & + \approx -0.020682. +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.100 \end{align*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.100 \end{align*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 93--101 +[] + [] + + +Overfull \hbox (13.28882pt too wide) in paragraph at lines 93--101 +\OT1/cmr/m/it/10 4.1 + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 93--101 +[] + [] + + +Overfull \hbox (16.28836pt too wide) in paragraph at lines 93--101 +\OML/cmm/m/it/10 ^^M \OMS/cmsy/m/n/10 ^^Y + [] + + +Overfull \hbox (32.77788pt too wide) in paragraph at lines 93--101 +\OT1/cmr/m/n/10 3\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 55295[] + [] + + +Overfull \hbox (15.37842pt too wide) in paragraph at lines 93--101 +\OML/cmm/m/it/10 ^^N \OMS/cmsy/m/n/10 ^^Y + [] + + +Overfull \hbox (32.77788pt too wide) in paragraph at lines 93--101 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 23062[] + [] + + +Overfull \hbox (16.38887pt too wide) in paragraph at lines 93--101 +\OML/cmm/m/it/10 ^^U \OMS/cmsy/m/n/10 ^^Y + [] + + +Overfull \hbox (32.77788pt too wide) in paragraph at lines 93--101 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 26664[] + [] + + +Overfull \hbox (15.3906pt too wide) in paragraph at lines 93--101 +\OML/cmm/m/it/10 ^^X \OMS/cmsy/m/n/10 ^^Y + [] + + +Overfull \hbox (48.33348pt too wide) in paragraph at lines 93--101 +\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 020682\O +ML/cmm/m/it/10 :$ + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 102--105 +[] + [] + + +Overfull \hbox (22.7613pt too wide) in paragraph at lines 102--105 +\OML/cmm/m/it/10 Z[]$ + [] + + +Overfull \hbox (6.43404pt too wide) in paragraph at lines 102--105 +\OML/cmm/m/it/10 $ + [] + + +Overfull \hbox (22.82524pt too wide) in paragraph at lines 102--105 +\OML/cmm/m/it/10 Z[] > + [] + + +Overfull \hbox (22.7613pt too wide) in paragraph at lines 102--105 +\OML/cmm/m/it/10 Z[]$ + [] + +! Undefined control sequence. +l.106 \section + {Tests d'adéquation} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 106--108 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 109--111 +[] + [] + +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 119--121 +[] + [] + + +Overfull \hbox (8.02779pt too wide) in paragraph at lines 119--121 +\OML/cmm/m/it/10 Y$ + [] + +Missing character: There is no S in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (71.37418pt too wide) detected at line 128 +[]\OML/cmm/m/it/10 H[] \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 Y \OMS/cmsy/m/n/10 ^^ +X \OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 ([])\OML/cmm/m/it/10 : + [] + +Missing character: There is no S in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (75.58252pt too wide) detected at line 135 +[]\OML/cmm/m/it/10 H[] \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 Y \OMS/cmsy/m/n/10 ^^ +X \OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^R[]\OT1/cmr/m/n/10 )\ +OML/cmm/m/it/10 : + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 138--144 +[] + [] + +! Undefined control sequence. +l.145 \subsection + {Test $\chi^2$ de Pearson} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 145--147 +[] + [] + + +Overfull \hbox (10.74304pt too wide) in paragraph at lines 145--147 +\OML/cmm/m/it/10 ^^_[]$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.148 Le test $\chi^2$ de Pearson \citep + [ch. 8]{hogg1978introduction} est +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 148--157 +[] + [] + + +Overfull \hbox (10.74304pt too wide) in paragraph at lines 148--157 +\OML/cmm/m/it/10 ^^_[]$ + [] + + +Overfull \hbox (36.94238pt too wide) in paragraph at lines 148--157 +\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ; \OML/c +mm/m/it/10 ^^R\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (48.72098pt too wide) in paragraph at lines 148--157 +\OML/cmm/m/it/10 X[]; [] ; X[]$ + [] + + +Overfull \hbox (9.06943pt too wide) in paragraph at lines 148--157 +\OML/cmm/m/it/10 X$ + [] + + +Overfull \hbox (5.52084pt too wide) in paragraph at lines 148--157 +\OML/cmm/m/it/10 k$ + [] + + +Overfull \hbox (21.11057pt too wide) in paragraph at lines 148--157 +\OML/cmm/m/it/10 p[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^R\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (9.06943pt too wide) in paragraph at lines 148--157 +\OML/cmm/m/it/10 X$ + [] + + +Overfull \hbox (35.54982pt too wide) in paragraph at lines 148--157 +[]$ + [] + + +Overfull \hbox (36.94238pt too wide) in paragraph at lines 148--157 +\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ; \OML/c +mm/m/it/10 ^^R\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (9.06943pt too wide) in paragraph at lines 148--157 +\OML/cmm/m/it/10 X$ + [] + + +Overfull \hbox (186.96912pt too wide) detected at line 160 +[]\OML/cmm/m/it/10 p[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^R\OT1/cmr/m/n/10 ) = +\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c[]\OT1/cmr/m/n/10 ; \OML +/cmm/m/it/10 ^^R\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 F[]\OT1 +/cmr/m/n/10 (\OML/cmm/m/it/10 c[]\OT1/cmr/m/n/10 ; \OML/cmm/m/it/10 ^^R\OT1/cmr +/m/n/10 )\OML/cmm/m/it/10 ; i \OT1/cmr/m/n/10 = 1\OML/cmm/m/it/10 ; [] ; k: + [] + +Missing character: There is no S in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 162--169 +[] + [] + + +Overfull \hbox (11.364pt too wide) in paragraph at lines 162--169 +\OML/cmm/m/it/10 N[]$ + [] + + +Overfull \hbox (7.90555pt too wide) in paragraph at lines 162--169 +\OML/cmm/m/it/10 Q$ + [] + + +Overfull \hbox (27.93059pt too wide) in paragraph at lines 162--169 +\OML/cmm/m/it/10 ^^_[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (8.88892pt too wide) in paragraph at lines 162--169 +\OT1/cmr/m/n/10 1)$ + [] + + +Overfull \hbox (13.29865pt too wide) in paragraph at lines 162--169 +\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 162--169 +\OT1/cmr/m/n/10 1$ + [] + + +Overfull \hbox (175.00801pt too wide) detected at line 172 +[]\OML/cmm/m/it/10 Q[] \OT1/cmr/m/n/10 = [] [] \OMS/cmsy/m/n/10 ^^X \OML/cmm/m/ +it/10 ^^_[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/ +n/10 1)\OML/cmm/m/it/10 : + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.176 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathcal allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.178 \mathcal + {H}_0: \theta = \theta_0. +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no H in font nullfont! +! Missing $ inserted. + + $ +l.178 \mathcal{H}_ + 0: \theta = \theta_0. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.179 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.179 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 174--180 +[] + [] + + +Overfull \hbox (23.06186pt too wide) in paragraph at lines 174--180 +\OML/cmm/m/it/10 Q[]$ + [] + + +Overfull \hbox (10.04163pt too wide) in paragraph at lines 174--180 +[] \OT1/cmr/m/n/10 : + [] + + +Overfull \hbox (15.52774pt too wide) in paragraph at lines 174--180 +\OML/cmm/m/it/10 ^^R \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (11.95836pt too wide) in paragraph at lines 174--180 +\OML/cmm/m/it/10 ^^R[]:$ + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.184 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathcal allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.186 \mathcal + {H}_0: F \in \lbrace F_{\theta} \rbrace, \theta \in \Omega. +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no H in font nullfont! +! Missing $ inserted. + + $ +l.186 \mathcal{H}_ + 0: F \in \lbrace F_{\theta} \rbrace, \theta \in \Omega. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.187 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.187 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 181--188 +[] + [] + + +Overfull \hbox (4.8229pt too wide) in paragraph at lines 181--188 +\OML/cmm/m/it/10 q$ + [] + + +Overfull \hbox (12.49797pt too wide) in paragraph at lines 181--188 +\OML/cmm/m/it/10 f[]$ + [] + + +Overfull \hbox (21.11057pt too wide) in paragraph at lines 181--188 +\OML/cmm/m/it/10 p[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^R\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (10.04163pt too wide) in paragraph at lines 181--188 +[] \OT1/cmr/m/n/10 : + [] + + +Overfull \hbox (17.26385pt too wide) in paragraph at lines 181--188 +\OML/cmm/m/it/10 F \OMS/cmsy/m/n/10 2 + [] + + +Overfull \hbox (39.8159pt too wide) in paragraph at lines 181--188 +\OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 F[]\OMS/cmsy/m/n/10 g\OML/cmm/m/it/10 ; ^^R +\OMS/cmsy/m/n/10 2 + [] + + +Overfull \hbox (10.00002pt too wide) in paragraph at lines 181--188 +\OT1/cmr/m/n/10 +\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 189--192 +[] + [] + + +Overfull \hbox (7.90555pt too wide) in paragraph at lines 189--192 +\OML/cmm/m/it/10 Q$ + [] + + +Overfull \hbox (197.73338pt too wide) detected at line 195 +[]\OML/cmm/m/it/10 Q[] \OT1/cmr/m/n/10 = [] [] \OMS/cmsy/m/n/10 ^^X \OML/cmm/m/ +it/10 ^^_[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/ +it/10 q \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/10 1)\OML/cmm/m/it/10 : + [] + +Missing character: There is no I in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.200 \eqref + {eq:statchi2param}, qui devient alors une fonction objectif. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 197--201 +[] + [] + + +Overfull \hbox (10.74304pt too wide) in paragraph at lines 197--201 +\OML/cmm/m/it/10 ^^_[]$ + [] + + +Overfull \hbox (33.22821pt too wide) in paragraph at lines 197--201 +\OML/cmm/m/it/10 Q[]$ + [] + + +! LaTeX Error: The font size command \normalsize is not defined: + there is probably something wrong with the class file. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.201 + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +[1 +! Undefined control sequence. + \newlabel{sec:testnormalite}{{}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:testnormalite}{{}{}} +! Undefined control sequence. + ...label{sec:test-de-shapiro}{{}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:test-de-shapiro}{{}{}} +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} +! Undefined control sequence. + \newlabel{eq:shapirowilkb1}{{1}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:shapirowilkb1}{{1}{}} +! Undefined control sequence. + \newlabel{eq:shapirowilkb2}{{2}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:shapirowilkb2}{{2}{}} +! Undefined control sequence. + \newlabel{sec:test-de-epps}{{}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:test-de-epps}{{}{}} +! Undefined control sequence. + ...{eq:statistiqueEppsPulley}{{}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:statistiqueEppsPulley}{{}{}} +! Undefined control sequence. + \newlabel{eq:EppsPulley}{{}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:EppsPulley}{{}{}} +! Undefined control sequence. + \newlabel{eq:EppsPulleyMod}{{1}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:EppsPulleyMod}{{1}{}} +! Undefined control sequence. + \newlabel{eq:ZEppsPulley}{{}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:ZEppsPulley}{{}{}} +! Undefined control sequence. + \newlabel{sec:testnonparam}{{}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:testnonparam}{{}{}} +! Undefined control sequence. + ...bel{eq:hypotheseadeqcomp}{{2}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:hypotheseadeqcomp}{{2}{}} +! Undefined control sequence. + \newlabel{eq:hypotheseadeq}{{3}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:hypotheseadeq}{{3}{}} +! Undefined control sequence. + \newlabel{sec:testchi2}{{}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:testchi2}{{}{}} +! Undefined control sequence. + \newlabel{eq:pmultinomiale}{{4}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:pmultinomiale}{{4}{}} +! Undefined control sequence. + \newlabel{eq:statchi2}{{5}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:statchi2}{{5}{}} +! Undefined control sequence. + \newlabel{eq:hyp0-1}{{}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:hyp0-1}{{}{}} +! Undefined control sequence. + \newlabel{eq:hyp0-2}{{}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:hyp0-2}{{}{}} +! Undefined control sequence. + \newlabel{eq:statchi2param}{{6}{\thepage + }} +l.201 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:statchi2param}{{6}{}} +] +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 202--206 +[] + [] + + +Overfull \hbox (12.9313pt too wide) in paragraph at lines 202--206 +\OMS/cmsy/m/n/10 H[]$ + [] + + +Overfull \hbox (43.78372pt too wide) in paragraph at lines 202--206 +\OML/cmm/m/it/10 Q[] > + [] + + +Overfull \hbox (39.37936pt too wide) in paragraph at lines 202--206 +\OML/cmm/m/it/10 ^^_[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (12.60071pt too wide) in paragraph at lines 202--206 +\OML/cmm/m/it/10 q \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (8.88892pt too wide) in paragraph at lines 202--206 +\OT1/cmr/m/n/10 1)$ + [] + + +Overfull \hbox (15.37842pt too wide) in paragraph at lines 202--206 +\OML/cmm/m/it/10 q \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 202--206 +\OT1/cmr/m/n/10 0$ + [] + + +Overfull \hbox (27.93059pt too wide) in paragraph at lines 202--206 +\OML/cmm/m/it/10 ^^_[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (12.60071pt too wide) in paragraph at lines 202--206 +\OML/cmm/m/it/10 q \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (8.88892pt too wide) in paragraph at lines 202--206 +\OT1/cmr/m/n/10 1)$ + [] + +! Undefined control sequence. +l.208 \operatorname + {Pr}\left(Q_{k-q-1}\leq \chi^2_{1-\alpha}(k-q-1)\right)... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +Overfull \hbox (175.0737pt too wide) detected at line 209 +[][] [] \OT1/cmr/m/n/10 = 1 \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 : + [] + +! Undefined control sequence. +l.211 \subsection + {Test de Kolmogorov-Smirnov} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 211--213 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.215 \eqref + {eq:hypotheseadeq}. On définit la statistique $D_n$ comme étant +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 214--219 +[] + [] + + +Overfull \hbox (13.72249pt too wide) in paragraph at lines 214--219 +\OML/cmm/m/it/10 D[]$ + [] + + +Overfull \hbox (25.36696pt too wide) in paragraph at lines 214--219 +\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (27.52574pt too wide) in paragraph at lines 214--219 +\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (12.79863pt too wide) in paragraph at lines 214--219 +\OML/cmm/m/it/10 H[]$ + [] + + +Overfull \hbox (110.004pt too wide) detected at line 222 +[]\OML/cmm/m/it/10 D[] \OT1/cmr/m/n/10 = [] \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 +F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ \O +ML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\OMS/cmsy +/m/n/10 j + [] + +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! + +Overfull \hbox (93.06448pt too wide) detected at line 227 +[]\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) = [] + [] \OML/cmm/m/it/10 I[]: + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{wang2003evaluating} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.230 Kolmogorov, selon \cite{wang2003evaluating} + , dont la fonction de +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `wang2003evaluating' on page undefined on input line 2 +30. + +Missing character: There is no ] in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.232 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Undefined control sequence. +l.234 \operatorname + {Pr}(\sqrt{n}D_n\leq x)&=1-2\sum_{k=1}^\infty (-1)^{k-1... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ( in font nullfont! +! Missing $ inserted. + + $ +l.234 \operatorname{Pr}(\sqrt{ + n}D_n\leq x)&=1-2\sum_{k=1}^\infty (-1)^{k-1... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.234 \operatorname{Pr}(\sqrt{n}D_n\leq x)& + =1-2\sum_{k=1}^\infty (-1)^{k-1... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.235 & + =\frac{\sqrt{2\pi}}{x}\sum_{k=1}^\infty e^{-(2k-1)^2\pi^2/(8x^2)}. +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.236 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.236 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 229--237 +[] + [] + + +Overfull \hbox (28.0582pt too wide) in paragraph at lines 229--237 +[]\OML/cmm/m/it/10 D[]$ + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 229--237 +[] + [] + + +Overfull \hbox (38.61371pt too wide) in paragraph at lines 229--237 +[]\OML/cmm/m/it/10 D[] \OMS/cmsy/m/n/10 ^^T + [] + + +Overfull \hbox (20.15968pt too wide) in paragraph at lines 229--237 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (12.77782pt too wide) in paragraph at lines 229--237 +\OT1/cmr/m/n/10 1 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (100.13234pt too wide) in paragraph at lines 229--237 +\OT1/cmr/m/n/10 2 [][](\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1)[]\OML/cmm/m/it/10 + e[][] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 229--237 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (121.39775pt too wide) in paragraph at lines 229--237 +[] [][] \OML/cmm/m/it/10 e[]:$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 238--241 +[] + [] + + +Overfull \hbox (12.79863pt too wide) in paragraph at lines 238--241 +\OML/cmm/m/it/10 H[]$ + [] + + +Overfull \hbox (38.61371pt too wide) in paragraph at lines 238--241 +[]\OML/cmm/m/it/10 D[] > + [] + + +Overfull \hbox (24.42796pt too wide) in paragraph at lines 238--241 +\OML/cmm/m/it/10 K[]$ + [] + +! Undefined control sequence. +l.243 \operatorname + {Pr}(\sqrt{n}D_n \leq K_{1-\alpha})=1-\alpha. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +Overfull \hbox (121.51845pt too wide) detected at line 244 +[][]\OT1/cmr/m/n/10 ([]\OML/cmm/m/it/10 D[] \OMS/cmsy/m/n/10 ^^T \OML/cmm/m/it/ +10 K[]\OT1/cmr/m/n/10 ) = 1 \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 : + [] + +! Undefined control sequence. +l.246 \subsection + {Test de distance minimale basé sur la fonction génératrice +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 246--249 +[] + [] + +Missing character: There is no [ in font nullfont! +\citation{luong1987minimum} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.250 \cite{luong1987minimum} + développent un ensemble de tests d'ajustement +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `luong1987minimum' on page undefined on input line 250 +. + +Missing character: There is no ] in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.251 pour l'hypothèse simple \eqref + {eq:hypotheseadeq} basée sur une +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no â in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no ™ in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ...ce `sec:test-de-epps' on page \thepage + \space undefined\on@line . +l.255 section \ref{sec:test-de-epps} + . On retrouve aussi le test K-L de +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:test-de-epps' on page undefined on input line 25 +5. + +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{feuerverger1981efficiency} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.256 \cite{feuerverger1981efficiency} + basé sur les parties réelles et +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `feuerverger1981efficiency' on page undefined on input + line 256. + +Missing character: There is no ] in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{KOUTROUVELIS01011980} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.263 \cite{KOUTROUVELIS01011980} + , qui a développé le test K-L pour cette +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `KOUTROUVELIS01011980' on page undefined on input line + 263. + +Missing character: There is no ] in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (25.43053pt too wide) in paragraph at lines 250--266 +[][] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 250--266 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 250--266 +[] + [] + + +Overfull \hbox (29.23724pt too wide) in paragraph at lines 250--266 +\OML/cmm/m/it/10 M[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (46.61707pt too wide) in paragraph at lines 250--266 +\OML/cmm/m/it/10 D\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 ; \OML/c +mm/m/it/10 f[]\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 250--266 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.270 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no z in font nullfont! +! Missing $ inserted. + + $ +l.272 z_ + j(F) &= \int_{-\infty}^{\infty} h_j(y) dF_Y(y)\nonumber\\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.272 z_j(F) & + = \int_{-\infty}^{\infty} h_j(y) dF_Y(y)\nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.273 & + = E[h_j(y)]\nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.274 & + = M_Y(t_j),\quad j=1,\ldots,K. +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.275 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.275 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 267--276 +[] + [] + + +Overfull \hbox (42.70496pt too wide) in paragraph at lines 267--276 +\OML/cmm/m/it/10 h[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 )\OML/cm +m/m/it/10 ; j \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (36.4304pt too wide) in paragraph at lines 267--276 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; [] ; K$ + [] + + +Overfull \hbox (26.39291pt too wide) in paragraph at lines 267--276 +\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (31.26385pt too wide) in paragraph at lines 267--276 +\OT1/cmr/bx/n/10 z\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (80.46904pt too wide) in paragraph at lines 267--276 +\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F\OT1/cm +r/m/n/10 )\OML/cmm/m/it/10 ; [] ; z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F\OT1/cm +r/m/n/10 )]$ + [] + + +Overfull \hbox (30.36633pt too wide) in paragraph at lines 267--276 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (75.72343pt too wide) in paragraph at lines 267--276 +[][] \OML/cmm/m/it/10 h[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 )\O +ML/cmm/m/it/10 dF[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 )[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 267--276 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (36.52843pt too wide) in paragraph at lines 267--276 +\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 h[]\OT1/cmr/m/n/10 (\OML/cm +m/m/it/10 y\OT1/cmr/m/n/10 )][] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 267--276 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (51.91725pt too wide) in paragraph at lines 267--276 +\OML/cmm/m/it/10 M[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 )\OML/ +cmm/m/it/10 ; j \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (38.65263pt too wide) in paragraph at lines 267--276 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; [] ; K:$ + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.281 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no z in font nullfont! +! Missing $ inserted. + + $ +l.283 z_ + j(F_T) = \frac{1}{T} \sum_{t=1}^T e^{t_j y_t}. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.284 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.284 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 277--285 +[] + [] + + +Overfull \hbox (33.57005pt too wide) in paragraph at lines 277--285 +\OML/cmm/m/it/10 h[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (16.21971pt too wide) in paragraph at lines 277--285 +\OML/cmm/m/it/10 e[]$ + [] + + +Overfull \hbox (24.46132pt too wide) in paragraph at lines 277--285 +\OML/cmm/m/it/10 z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (25.7696pt too wide) in paragraph at lines 277--285 +\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (36.1746pt too wide) in paragraph at lines 277--285 +\OT1/cmr/bx/n/10 z\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (90.29054pt too wide) in paragraph at lines 277--285 +\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[]\OT1/ +cmr/m/n/10 )\OML/cmm/m/it/10 ; [] ; z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[]\OT +1/cmr/m/n/10 )]$ + [] + + +Overfull \hbox (35.27708pt too wide) in paragraph at lines 277--285 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (57.79903pt too wide) in paragraph at lines 277--285 +[] [][] \OML/cmm/m/it/10 e[]:$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.287 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no T in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no F in font nullfont! +! Missing $ inserted. + + $ +l.289 Td(F_ + n,F_\theta) &= T\left\{\mathbf{z}(F_n)-\mathbf{z}(F_\theta) +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.289 Td(F_n,F_\theta) & + = T\left\{\mathbf{z}(F_n)-\mathbf{z}(F_\theta) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.292 & + = \mathbf{v}_n^{\prime }\mathbf{Q}(F_\theta)\mathbf{v}_n. +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.293 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.293 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 286--294 +[] + [] + + +Overfull \hbox (35.28699pt too wide) in paragraph at lines 286--294 +[]\OML/cmm/m/it/10 ; F[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (185.7335pt too wide) in paragraph at lines 286--294 +\OML/cmm/m/it/10 T [][] \OT1/cmr/bx/n/10 Q\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[] +\OT1/cmr/m/n/10 ) [] [] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 286--294 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (53.17474pt too wide) in paragraph at lines 286--294 +\OT1/cmr/bx/n/10 v[]Q\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 )\OT1 +/cmr/bx/n/10 v[]\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 295--308 +[] + [] + + +Overfull \hbox (27.37148pt too wide) in paragraph at lines 295--308 +\OT1/cmr/bx/n/10 Q\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (47.48845pt too wide) in paragraph at lines 295--308 +\OML/cmm/m/it/10 Td\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[]; F[]\OT1/cmr/m/n/10 )$ + + [] + + +Overfull \hbox (10.74304pt too wide) in paragraph at lines 295--308 +\OML/cmm/m/it/10 ^^_[]$ + [] + + +Overfull \hbox (22.06825pt too wide) in paragraph at lines 295--308 +\OT1/cmr/bx/n/10 v[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (88.0613pt too wide) in paragraph at lines 295--308 +[] []$ + [] + + +Overfull \hbox (8.30551pt too wide) in paragraph at lines 295--308 +\OT1/cmr/bx/n/10 ^^F$ + [] + + +Overfull \hbox (8.30551pt too wide) in paragraph at lines 295--308 +\OT1/cmr/bx/n/10 ^^F$ + [] + + +Overfull \hbox (19.19435pt too wide) in paragraph at lines 295--308 +\OT1/cmr/bx/n/10 Q \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (19.04166pt too wide) in paragraph at lines 295--308 +\OT1/cmr/bx/n/10 ^^F[]$ + [] + + +Overfull \hbox (50.39696pt too wide) in paragraph at lines 295--308 +\OT1/cmr/bx/n/10 v[]Q\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 )\OT1 +/cmr/bx/n/10 v[]$ + [] + + +Overfull \hbox (10.74304pt too wide) in paragraph at lines 295--308 +\OML/cmm/m/it/10 ^^_[]$ + [] + + +Overfull \hbox (9.20833pt too wide) in paragraph at lines 295--308 +\OML/cmm/m/it/10 K$ + [] + + +Overfull \hbox (8.30551pt too wide) in paragraph at lines 295--308 +\OT1/cmr/bx/n/10 ^^F$ + [] + + +Overfull \hbox (23.01454pt too wide) in paragraph at lines 295--308 +\OML/cmm/m/it/10 h[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (23.72115pt too wide) in paragraph at lines 295--308 +\OML/cmm/m/it/10 h[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (41.38144pt too wide) in paragraph at lines 295--308 +\OML/cmm/m/it/10 z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (60.24957pt too wide) in paragraph at lines 295--308 +\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 h[]\OT1/cmr/m/n/10 (\OML/cm +m/m/it/10 y\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 h[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/ +10 y\OT1/cmr/m/n/10 )]$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.311 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathbf allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.313 \mathbf{\Sigma} + &= \begin{bmatrix} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +! Missing $ inserted. + + $ +l.313 \mathbf{\Sigma} + &= \begin{bmatrix} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Extra }, or forgotten $. +\select@group ...pandafter \non@alpherr \fi #1{#4} + } +l.313 \mathbf{\Sigma} + &= \begin{bmatrix} +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\select@group ...andafter \non@alpherr \fi #1{#4}} + +l.313 \mathbf{\Sigma} + &= \begin{bmatrix} +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Misplaced alignment tab character &. +l.313 \mathbf{\Sigma} & + = \begin{bmatrix} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.313 \mathbf{\Sigma} &= \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.314 \sigma_{1,1} & + \cdots & \sigma_{1,k} \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.314 \sigma_{1,1} & \cdots & + \sigma_{1,k} \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +! Misplaced alignment tab character &. +l.315 \vdots & + \ddots & \vdots \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +! Misplaced alignment tab character &. +l.315 \vdots & \ddots & + \vdots \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +! Misplaced alignment tab character &. +l.316 \sigma_{j,1} & + \cdots & \sigma_{j,k} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.316 \sigma_{j,1} & \cdots & + \sigma_{j,k} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.317 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.318 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.318 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.318 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.318 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.320 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.322 \sigma + _{j,k} &= z_{j,k}(F) - z_j(F)z_k(F)\nonumber\\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.322 \sigma_{j,k} & + = z_{j,k}(F) - z_j(F)z_k(F)\nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.323 & + = M_Y(t_j+t_k) - M_Y(t_j)M_Y(t_k),\quad j,k=1,\ldots,K. +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.324 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.324 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 309--325 +[] + [] + + +Overfull \hbox (8.30551pt too wide) in paragraph at lines 309--325 +\OT1/cmr/bx/n/10 ^^F$ + [] + + +Overfull \hbox (17.77774pt too wide) in paragraph at lines 309--325 +\OT1/cmr/m/n/10 ^^F = + [] + + +Overfull \hbox (48.55667pt too wide) in paragraph at lines 309--325 +\OML/cmm/m/it/10 ^^[[] [] ^^[[][] + [] + + +Overfull \hbox (6.6665pt too wide) in paragraph at lines 309--325 +[] [] [][] + [] + + +Overfull \hbox (47.15042pt too wide) in paragraph at lines 309--325 +\OML/cmm/m/it/10 ^^[[] [] ^^[[]$ + [] + + +Overfull \hbox (26.84781pt too wide) in paragraph at lines 309--325 +\OML/cmm/m/it/10 ^^[[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (38.60373pt too wide) in paragraph at lines 309--325 +\OML/cmm/m/it/10 z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 ) \OMS/c +msy/m/n/10 ^^@ + [] + + +Overfull \hbox (49.62924pt too wide) in paragraph at lines 309--325 +\OML/cmm/m/it/10 z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 )\OML/cm +m/m/it/10 z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 )[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 309--325 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (36.11574pt too wide) in paragraph at lines 309--325 +\OML/cmm/m/it/10 M[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t[] \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (20.19798pt too wide) in paragraph at lines 309--325 +\OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (94.26039pt too wide) in paragraph at lines 309--325 +\OML/cmm/m/it/10 M[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 )\OML/ +cmm/m/it/10 M[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 )\OML/cmm/m +/it/10 ; j; k \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (38.65263pt too wide) in paragraph at lines 309--325 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; [] ; K:$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 326--331 +[] + [] + + +Overfull \hbox (47.48845pt too wide) in paragraph at lines 326--331 +\OML/cmm/m/it/10 Td\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[]; F[]\OT1/cmr/m/n/10 )$ + + [] + + +Overfull \hbox (12.79863pt too wide) in paragraph at lines 326--331 +\OML/cmm/m/it/10 H[]$ + [] + + +Overfull \hbox (6.43404pt too wide) in paragraph at lines 326--331 +\OML/cmm/m/it/10 $ + [] + + +Overfull \hbox (10.74304pt too wide) in paragraph at lines 326--331 +\OML/cmm/m/it/10 ^^_[]$ + [] + + +Overfull \hbox (9.20833pt too wide) in paragraph at lines 326--331 +\OML/cmm/m/it/10 K$ + [] + + +Overfull \hbox (58.04396pt too wide) in paragraph at lines 326--331 +\OML/cmm/m/it/10 Td\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F[]; F[]\OT1/cmr/m/n/10 ) +\OML/cmm/m/it/10 > + [] + + +Overfull \hbox (39.17795pt too wide) in paragraph at lines 326--331 +\OML/cmm/m/it/10 ^^_[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ + [] + +) +! Emergency stop. +<*> chapitre7.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 34 strings out of 493307 + 497 string characters out of 6139904 + 61685 words of memory out of 5000000 + 3584 multiletter control sequences out of 15000+600000 + 4913 words of font info for 18 fonts, out of 8000000 for 9000 + 957 hyphenation exceptions out of 8191 + 18i,6n,21p,129b,121s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/memoire/chapitre7.tex b/memoire/chapitre7.tex new file mode 100644 index 0000000..61204b5 --- /dev/null +++ b/memoire/chapitre7.tex @@ -0,0 +1,336 @@ +\chapter{Tests statistiques} + +\section{Test de normalité} +\label{sec:testnormalite} + +On utilise un test de normalité afin de vérifier si l'on peut rejeter +l'hypothèse de normalité de la distribution des rendements. Comme +présenté au premier chapitre, c'est cette observation qui est à la +base de la recherche de modèles alternatifs, étant donné que cette +hypothèse est rejetée pour la plupart des séries financières. Il +existe plusieurs tests disponibles pour ce faire, dont ceux de +Shapiro-Wilk et d’Epps-Pulley. + +\subsection{Test de Shapiro-Wilk} +\label{sec:test-de-shapiro} + +Le test de Shapiro-Wilk \citep{shapiro1965analysis} est basé sur les +statistiques d'ordre de la distribution normale. Ce test est +particulièrement efficace même pour de petits échantillons $(T<20)$, +selon les auteurs. + +On considère l'échantillon ordonné +$y_{(1)},y_{(2)},\ldots,y_{(T)},$. On évalue la variance $S^2$ de +l'échantillon puis la statistique $b$, où les valeurs $a_{T-i+1}$ +proviennent de la \emph{Table 5} de \cite{shapiro1965analysis}: +\begin{enumerate} +\item Si la taille de l'échantillon $T$ est paire, alors + $k=\frac{T}{2}$ et + \begin{align} + \label{eq:shapirowilkb1} + b &= \sum_{i=1}^{k}a_{T-i+1}(y_{(T-i+1)}-y_{(i)}). + \end{align} +\item Si la taille de l'échantillon $T$ est impaire, alors + $k=\frac{T-1}{2}$ et + \begin{align} + \label{eq:shapirowilkb2} + b &= a_{T}(y_{(T)}-y_{(1)}) + \ldots + a_{k+2}(y_{(k+2)}-y_{(k)}). + \end{align} +\end{enumerate} + +La statistique de Shapiro-Wilk $W = \frac{b}{S^2}$ suit une +distribution particulière dont on retrouve différents quantiles à la +\emph{Table 6} de \cite{shapiro1965analysis}. On rejette l'hypothèse +de la normalité lorsque la statistique $W$ est inférieure au seuil +critique déterminé à partir de cette dernière table. + +\subsection{Test d’Epps-Pulley} +\label{sec:test-de-epps} + +Le test d’Epps-Pulley \citep{epps1983test} est basé sur le carré de la +différence entre les fonctions caractéristiques empirique et théorique +de la distribution normale. Ce test est considéré comme un des plus +puissants pour un seuil de tolérance donné, pour de larges +échantillons. La statistique de test prend la forme suivante, où +$\overline{X}$ et $S$ sont respectivement la moyenne et l'écart-type +échantillonnal: +\begin{align} + \label{eq:statistiqueEppsPulley} + EP_T &= T\int_{-\infty}^{\infty} \left\vert\frac{1}{T}\sum_{j=1}^T + \exp\left[it\frac{\left(X_j-\bar{X} \right)}{S} \right] - \exp + \left[-\frac{t^2}{2} \right]\right\vert^2w(t)dt. +\end{align} + +\cite{henze1990approximation} propose une procédure simple à +implémenter pour effectuer un test basé sur la statistique +d’Epps-Pulley $EP_T$, où la pondération de la différence quadratique +$w(\cdot)$ est remplacée par la densité de la loi normale centrée +réduite, ce qui permet d'accorder davantage d'importance aux +observations près de l'origine et aussi d'obtenir une forme +intégrable: +\begin{align} + \label{eq:EppsPulley} + EP_T &= \frac{2}{T} \sum_{1\leq\,j10$, on calcule une version modifiée de la statistique: +\begin{equation} + \label{eq:EppsPulleyMod} + EP_T^{*} = \left(EP_T-\frac{0.365}{T}+\frac{1.34}{T^2}\right)\left(1+\frac{1.3}{T}\right). +\end{equation} + +On obtient ensuite une approximation d'un quantile de la loi normale +centrée réduite: +\begin{align} + \label{eq:ZEppsPulley} + Z_T &= \gamma + \delta\ln\left(\left(EP_T^{*}-\xi + \right)\left(\xi+\lambda-EP_T^{*} \right) \right). +\end{align} + +Les constantes sont dérivées à l'équation \emph{4.1} de +\cite{henze1990approximation}: +\begin{align*} + \gamma &\approx 3.55295 \\ + \delta &\approx 1.23062 \\ + \lambda &\approx 2.26664 \\ + \xi &\approx -0.020682. +\end{align*} + +Puis, on compare cette statistique au quantile $Z_{1-\alpha}$ correspondant +au seuil de tolérance $\alpha$. Si $Z_n>Z_{1-\alpha}$ +alors on rejette la normalité des données. + +\section{Tests d'adéquation} +\label{sec:testnonparam} + +Les tests d'adéquation vérifient l'ajustement à l'échantillon de la +fonction de répartition estimée soit: +\begin{itemize} +\item à partir de l'échantillon ayant servi à l'estimation du vecteur + de paramètres +\item à partir d'un échantillon séparé, afin de vérifier, par exemple, + si un modèle estimé à partir d'anciennes données s'applique toujours + à de nouvelles informations. +\end{itemize} + +On cherche à rejeter ou non une hypothèse concernant l'échantillon $Y$ +et le vecteur de paramètres estimés: +\begin{itemize} +\item Si on considère le vecteur de paramètres comme étant estimé à + partir du même échantillon sur lequel on effectue le test, alors on + pose l'hypothèse composée suivante: + \begin{equation} + \label{eq:hypotheseadeqcomp} + H_0: Y \sim F_Y(\hat\theta). + \end{equation} +\item Si on considère le vecteur de paramètres comme étant connu ou + estimé à partir d'un autre échantillon, alors on pose l'hypothèse + simple suivante: + \begin{equation} + \label{eq:hypotheseadeq} + H_0: Y \sim F_Y(\theta_0). + \end{equation} +\end{itemize} + +La différence entre ces deux hypothèses est le nombre de degrés de +liberté de la statistique de test. Les deux premiers tests présentés +sont des classiques, largement utilisés, mais dont l'application est +basée sur la fonction de répartition. Étant donné qu'on ne dispose que +d'une approximation de celle-ci, on utilisera alors un test basé sur +la fonction génératrice des moments, qui a une forme analytique. + +\subsection{Test $\chi^2$ de Pearson} +\label{sec:testchi2} + +Le test $\chi^2$ de Pearson \citep[ch. 8]{hogg1978introduction} est +basé sur une approximation multinomiale de la fonction de répartition +$F_X(x;\theta)$ pour laquelle on veut vérifier l'ajustement des +données échantillonnales $X_1,\ldots,X_n$. Pour ce faire, on divise le +domaine de la variable aléatoire $X$ en $k$ intervalles, appelés +classes. On associe la probabilité $p_i(\theta)$ que $X$ prend une +valeur dans l'intervalle $\left[c_{i-1},c_{i}\right]$. Cette +probabilité est évaluée à l'aide de la fonction de répartition +$F_X(x;\theta)$ de la variable aléatoire $X$: +\begin{equation} + \label{eq:pmultinomiale} + p_i(\theta) = F_X(c_{i};\theta) - F_X(c_{i-1};\theta),\qquad i = 1,\ldots,k. +\end{equation} + +Soit $N_i$, le nombre de données de l'échantillon observées dans +l'intervalle défini précédemment. On définit alors la statistique $Q$ +comme étant la somme, pour chaque intervalle, du carré de la +différence entre le nombre d'observations obtenues et espérées, +pondérée par cette dernière valeur. Cette quantité a approximativement +une distribution asymptotique $\chi^2(k-1)$ avec $k-1$ degrés de +liberté: +\begin{equation} + \label{eq:statchi2} + Q_{k-1} = \sum_{i=1}^{k} \frac{\left(N_i-np_i(\theta_0)\right)^2}{np_i(\theta_0)} \sim \chi^2(k-1). +\end{equation} + +La statistique $Q_{k-1}$ peut être utilisée pour vérifier une +hypothèse simple, c'est-à-dire lorsque les paramètres sont connus: +\begin{align} + \label{eq:hyp0-1} + \mathcal{H}_0: \theta = \theta_0. +\end{align} + +Cependant, comme les $q$ paramètres de la distribution $f_X$ doivent +être estimés pour évaluer les probabilités $p_i(\theta)$, on retrouve +alors une hypothèse composée: +\begin{align} + \label{eq:hyp0-2} + \mathcal{H}_0: F \in \lbrace F_{\theta} \rbrace, \theta \in \Omega. +\end{align} + +On doit retrancher le même nombre de degrés de liberté à la +distribution asymptotique approximative de la statistique $Q$ qui +devient: +\begin{equation} + \label{eq:statchi2param} + Q_{k-q-1} = \sum_{i=1}^{k} \frac{\left(N_i-n{p}_i(\hat\theta)\right)^2}{n{p}_i(\hat\theta)} \sim \chi^2(k-q-1). +\end{equation} + +Il est important de noter que ce test est approximatif, car pour +obtenir asymptotiquement la distribution $\chi^2$, les paramètres +doivent être estimés en minimisant la statistique $Q_{k-q-1}$ +\eqref{eq:statchi2param}, qui devient alors une fonction objectif. + +On rejette l'hypothèse composée $\mathcal{H}_0$ lorsque la condition +$Q_{k-q-1} > \chi^2_{1-\alpha}(k-q-1)$ est respectée. On effectue le +même test pour l'hypothèse simple, en posant $q=0$. Le point critique +de la distribution $\chi^2(k-q-1)$ est sélectionné selon le critère +\begin{equation} + \label{eq:quantilechisq} + \operatorname{Pr}\left(Q_{k-q-1}\leq \chi^2_{1-\alpha}(k-q-1)\right)=1-\alpha. +\end{equation} + +\subsection{Test de Kolmogorov-Smirnov} +\label{sec:kolmogorovsmirnov} + +Le test de Kolmogorov-Smirnov vérifie l'hypothèse simple +\eqref{eq:hypotheseadeq}. On définit la statistique $D_n$ comme étant +la valeur maximale de la distance entre les fonctions de répartition +empiriques $F_n(x)$ et celle $F_X(x)$ de la distribution exacte +spécifiée par l'hypothèse nulle $H_0$: +\begin{equation} + \label{eq:statks} + D_n=\sup_x |F_n(x)-F_X(x)| +\end{equation} +où +\begin{equation} + \label{eq:repartemp} + F_n(x)={1 \over n}\sum_{i=1}^n I_{X_i\leq x}. +\end{equation} + +La statistique $\sqrt{n}D_n$ suit asymptotiquement une distribution de +Kolmogorov, selon \cite{wang2003evaluating}, dont la fonction de +répartition est définie comme suit: +\begin{align} + \label{eq:repartkolmogorov} + \operatorname{Pr}(\sqrt{n}D_n\leq x)&=1-2\sum_{k=1}^\infty (-1)^{k-1} e^{-2k^2 x^2}\\ + &=\frac{\sqrt{2\pi}}{x}\sum_{k=1}^\infty e^{-(2k-1)^2\pi^2/(8x^2)}. +\end{align} + +On rejette l'hypothèse nulle $H_0$ lorsque la condition +$\sqrt{n}D_n>K_{1-\alpha}$ est respectée. On sélectionne le quantile +de cette distribution selon le critère suivant: +\begin{equation} + \label{eq:quantilekolmogorov} + \operatorname{Pr}(\sqrt{n}D_n \leq K_{1-\alpha})=1-\alpha. +\end{equation} + +\subsection{Test de distance minimale basé sur la fonction génératrice + des moments} +\label{sec:test-de-distance} + +\cite{luong1987minimum} développent un ensemble de tests d'ajustement +pour l'hypothèse simple \eqref{eq:hypotheseadeq} basée sur une +transformation de la fonction de densité ou de répartition. Parmi +cette classe de tests, on retrouve celui d’Epps-Pulley, développé pour +vérifier l'hypothèse de normalité des données, tel que présenté à la +section \ref{sec:test-de-epps}. On retrouve aussi le test K-L de +\cite{feuerverger1981efficiency} basé sur les parties réelles et +imaginaires de la fonction caractéristique. Cependant, pour la +distribution de Laplace asymétrique généralisée, on ne peut effectuer +cette séparation. On préfèrera donc utiliser la fonction génératrice +des moments $M_Y(\xi)$ afin de construire la statistique de test +$D(F_T;f_{\theta_0})$. Comme on a déjà estimé les paramètres, on veut donc +vérifier une hypothèse simple. On se réfère à +\cite{KOUTROUVELIS01011980}, qui a développé le test K-L pour cette +situation. Cependant, on remplacera la fonction caractéristique par la +fonction génératrice des moments. + +On considère un ensemble de fonctions $h_j(y), j=1,\ldots,K$. La +transformée de la fonction de répartition $F_Y(y)$ est donnée par le +vecteur $\mathbf{z}(F)=[z_1(F),\ldots,z_K(F)]$ : +\begin{align} + \label{eq:4} + z_j(F) &= \int_{-\infty}^{\infty} h_j(y) dF_Y(y)\nonumber\\ + &= E[h_j(y)]\nonumber\\ + &= M_Y(t_j),\quad j=1,\ldots,K. +\end{align} + +En posant $h_j(y)=e^{t_j y}$, on définit $z_j(F)$ comme étant la +fonction génératrice des moments. On définit aussi la transformée de +la fonction de répartition empirique $F_T(y)$ par le vecteur +$\mathbf{z}(F_T)=[z_1(F_T),\ldots,z_K(F_T)]$, où +\begin{align} + \label{eq:5} + z_j(F_T) = \frac{1}{T} \sum_{t=1}^T e^{t_j y_t}. +\end{align} + +La statistique de distance quadratique prend alors la forme suivante: +\begin{align} + \label{eq:9} + Td(F_n,F_\theta) &= T\left\{\mathbf{z}(F_n)-\mathbf{z}(F_\theta) + \right\}^{\prime} \mathbf{Q}(F_\theta) + \left\{\mathbf{z}(F_n)-\mathbf{z}(F_\theta) \right\} \nonumber\\ + &= \mathbf{v}_n^{\prime }\mathbf{Q}(F_\theta)\mathbf{v}_n. +\end{align} + +On doit maintenant sélectionner une matrice $\mathbf{Q}(F_\theta)$ de +sorte que la distribution de la statistique $Td(F_n,F_\theta)$ soit +$\chi^2$. Comme $\mathbf{v}_n = +\sqrt{T}\left\{\mathbf{z}(F_n)-\mathbf{z}(F_\theta) \right\}$ suit +asymptotiquement une distribution normale multivariée de moyenne 0 et +de variance-covariance $\mathbf{\Sigma}$, on a, si la matrice +$\mathbf{\Sigma}$ est inversible et que l'on pose +$\mathbf{Q}=\mathbf{\Sigma}^{-1}$, que $\mathbf{v}_n^{\prime +}\mathbf{Q}(F_\theta)\mathbf{v}_n$ suit une distribution $\chi^2$ avec +$K$ degrés de liberté, dans le cadre d'une hypothèse simple. Afin +d'évaluer la variance-covariance $\mathbf{\Sigma}$, on définit +l'espérance du produit des deux fonctions $h_j(y)$ et $h_k(y)$ +:$z_{j,k}(F) = E[h_j(y)h_k(y)]$. + +On décrit la matrice de variance-covariance $\mathbf{\Sigma}$ comme +suit: +\begin{align} + \label{eq:14} + \mathbf{\Sigma} &= \begin{bmatrix} + \sigma_{1,1} & \cdots & \sigma_{1,k} \\ + \vdots & \ddots & \vdots \\ + \sigma_{j,1} & \cdots & \sigma_{j,k} + \end{bmatrix} +\end{align} +où +\begin{align} + \label{eq:10} + \sigma_{j,k} &= z_{j,k}(F) - z_j(F)z_k(F)\nonumber\\ + &= M_Y(t_j+t_k) - M_Y(t_j)M_Y(t_k),\quad j,k=1,\ldots,K. +\end{align} + +On peut maintenant évaluer la statistique $Td(F_n,F_\theta)$ et +effectuer le test. On rejette l'hypothèse $H_0$ lorsque la valeur de +celle-ci est supérieure au quantile correspondant au seuil $\alpha$ de +la distribution $\chi^2$ avec $K$ degrés de liberté: +$Td(F_n,F_\theta)>\chi^2_{1-\alpha}(K)$. + + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/chapitre8.aux b/memoire/chapitre8.aux new file mode 100644 index 0000000..729f866 --- /dev/null +++ b/memoire/chapitre8.aux @@ -0,0 +1,188 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{bingham2004risk} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {chapter}{\chapternumberline {8}\IeC {\'E}valuation d'options}{87}{chapter.8}} +\newlabel{chap:options}{{8}{87}{Évaluation d'options}{chapter.8}{}} +\newlabel{chap:options@cref}{{[chapter][8][]8}{87}} +\@writefile{toc}{\contentsline {section}{\numberline {8.1}D\IeC {\'e}finitions}{87}{section.8.1}} +\newlabel{sec:options}{{8.1}{87}{Définitions}{section.8.1}{}} +\newlabel{sec:options@cref}{{[section][1][8]8.1}{87}} +\newlabel{eq:callpayoff}{{8.1.1}{87}{Définitions}{equation.8.1.0.1}{}} +\newlabel{eq:callpayoff@cref}{{[equation][1][8,1]8.1.1}{87}} +\citation{black1973pricing} +\newlabel{eq:reclamationcall}{{8.1.2}{88}{Définitions}{equation.8.1.0.2}{}} +\newlabel{eq:reclamationcall@cref}{{[equation][2][8,1]8.1.2}{88}} +\newlabel{eq:reclamationcallintlog}{{8.1.3}{88}{Définitions}{equation.8.1.0.3}{}} +\newlabel{eq:reclamationcallintlog@cref}{{[equation][3][8,1]8.1.3}{88}} +\newlabel{eq:pariteputcall}{{8.1.4}{88}{Définitions}{equation.8.1.0.4}{}} +\newlabel{eq:pariteputcall@cref}{{[equation][4][8,1]8.1.4}{88}} +\@writefile{toc}{\contentsline {subsection}{\numberline {8.1.1}\IeC {\'E}quation martingale}{88}{subsection.8.1.1}} +\newlabel{sec:equationmartingale}{{8.1.1}{88}{Équation martingale}{subsection.8.1.1}{}} +\newlabel{sec:equationmartingale@cref}{{[subsection][1][8,1]8.1.1}{88}} +\newlabel{eq:equationmartingale}{{8.1.5}{89}{Équation martingale}{equation.8.1.0.5}{}} +\newlabel{eq:equationmartingale@cref}{{[equation][5][8,1]8.1.5}{89}} +\@writefile{toc}{\contentsline {subsection}{\numberline {8.1.2}Param\IeC {\`e}tres neutres au risque}{89}{subsection.8.1.2}} +\newlabel{sec:GALrn}{{8.1.2}{89}{Paramètres neutres au risque}{subsection.8.1.2}{}} +\newlabel{sec:GALrn@cref}{{[subsection][2][8,1]8.1.2}{89}} +\newlabel{eq:martingaleGAL}{{8.1.6}{89}{Paramètres neutres au risque}{equation.8.1.0.6}{}} +\newlabel{eq:martingaleGAL@cref}{{[equation][6][8,1]8.1.6}{89}} +\newlabel{eq:omegaGAL}{{8.1.7}{89}{Paramètres neutres au risque}{equation.8.1.0.7}{}} +\newlabel{eq:omegaGAL@cref}{{[equation][7][8,1]8.1.7}{89}} +\newlabel{eq:inegaliteparamrn}{{8.1.8}{89}{Paramètres neutres au risque}{equation.8.1.0.8}{}} +\newlabel{eq:inegaliteparamrn@cref}{{[equation][8][8,1]8.1.8}{89}} +\newlabel{eq:fncaractGALrn}{{8.1.9}{89}{Paramètres neutres au risque}{equation.8.1.0.9}{}} +\newlabel{eq:fncaractGALrn@cref}{{[equation][9][8,1]8.1.9}{89}} +\citation{heston1993closed} +\@writefile{toc}{\contentsline {section}{\numberline {8.2}Aper\IeC {\c c}u du mod\IeC {\`e}le de Black-Scholes}{90}{section.8.2}} +\newlabel{sec:blackscholes}{{8.2}{90}{Aperçu du modèle de Black-Scholes}{section.8.2}{}} +\newlabel{sec:blackscholes@cref}{{[section][2][8]8.2}{90}} +\newlabel{eq:blackscholes}{{8.2.1}{90}{Aperçu du modèle de Black-Scholes}{equation.8.2.0.1}{}} +\newlabel{eq:blackscholes@cref}{{[equation][1][8,2]8.2.1}{90}} +\newlabel{eq:putblackscholes}{{8.2.2}{90}{Aperçu du modèle de Black-Scholes}{equation.8.2.0.2}{}} +\newlabel{eq:putblackscholes@cref}{{[equation][2][8,2]8.2.2}{90}} +\@writefile{toc}{\contentsline {section}{\numberline {8.3}M\IeC {\'e}thodes d'\IeC {\'e}valuation pour options europ\IeC {\'e}ennes}{90}{section.8.3}} +\newlabel{sec:methodesoptions}{{8.3}{90}{Méthodes d'évaluation pour options européennes}{section.8.3}{}} +\newlabel{sec:methodesoptions@cref}{{[section][3][8]8.3}{90}} +\citation{carr1999option} +\citation{madan1998variance} +\citation{epps2007pricing} +\@writefile{toc}{\contentsline {subsection}{\numberline {8.3.1}M\IeC {\'e}thode de Heston}{91}{subsection.8.3.1}} +\newlabel{sec:heston1993}{{8.3.1}{91}{Méthode de Heston}{subsection.8.3.1}{}} +\newlabel{sec:heston1993@cref}{{[subsection][1][8,3]8.3.1}{91}} +\newlabel{eq:PutHeston93-1}{{8.3.1}{91}{Méthode de Heston}{equation.8.3.0.1}{}} +\newlabel{eq:PutHeston93-1@cref}{{[equation][1][8,3]8.3.1}{91}} +\newlabel{eq:PutHeston93}{{8.3.2}{91}{Méthode de Heston}{equation.8.3.0.2}{}} +\newlabel{eq:PutHeston93@cref}{{[equation][2][8,3]8.3.2}{91}} +\newlabel{eq:mesureGHeston}{{8.3.3}{91}{Méthode de Heston}{equation.8.3.0.3}{}} +\newlabel{eq:mesureGHeston@cref}{{[equation][3][8,3]8.3.3}{91}} +\@writefile{toc}{\contentsline {subsection}{\numberline {8.3.2}M\IeC {\'e}thode de Carr et Madan}{91}{subsection.8.3.2}} +\newlabel{sec:carrmadanfftoptions}{{8.3.2}{91}{Méthode de Carr et Madan}{subsection.8.3.2}{}} +\newlabel{sec:carrmadanfftoptions@cref}{{[subsection][2][8,3]8.3.2}{91}} +\newlabel{eq:callamortiCarr}{{8.3.4}{91}{Méthode de Carr et Madan}{equation.8.3.0.4}{}} +\newlabel{eq:callamortiCarr@cref}{{[equation][4][8,3]8.3.4}{91}} +\citation{carr1999option} +\newlabel{eq:fouriercallamortiCarr}{{8.3.6}{92}{Méthode de Carr et Madan}{equation.8.3.0.6}{}} +\newlabel{eq:fouriercallamortiCarr@cref}{{[equation][6][8,3]8.3.6}{92}} +\newlabel{eq:prixoptionCarr}{{8.3.7}{92}{Méthode de Carr et Madan}{equation.8.3.0.7}{}} +\newlabel{eq:prixoptionCarr@cref}{{[equation][7][8,3]8.3.7}{92}} +\newlabel{eq:prixoptionFFT}{{8.3.10}{92}{Méthode de Carr et Madan}{equation.8.3.0.10}{}} +\newlabel{eq:prixoptionFFT@cref}{{[equation][10][8,3]8.3.10}{92}} +\newlabel{eq:prixoptionFFTsimpson}{{8.3.11}{92}{Méthode de Carr et Madan}{equation.8.3.0.11}{}} +\newlabel{eq:prixoptionFFTsimpson@cref}{{[equation][11][8,3]8.3.11}{92}} +\citation{carr1999option} +\citation{itkin2005pricing} +\citation{epps2007pricing} +\@writefile{toc}{\contentsline {subsection}{\numberline {8.3.3}Prix d'exercice hors du cours}{93}{subsection.8.3.3}} +\newlabel{sec:outofmoneyfft}{{8.3.3}{93}{Prix d'exercice hors du cours}{subsection.8.3.3}{}} +\newlabel{sec:outofmoneyfft@cref}{{[subsection][3][8,3]8.3.3}{93}} +\newlabel{eq:prixoptionCarrOOM}{{8.3.12}{93}{Prix d'exercice hors du cours}{equation.8.3.0.12}{}} +\newlabel{eq:prixoptionCarrOOM@cref}{{[equation][12][8,3]8.3.12}{93}} +\newlabel{eq:prixoptionFFTsimpsonOOM}{{8.3.14}{93}{Prix d'exercice hors du cours}{equation.8.3.0.14}{}} +\newlabel{eq:prixoptionFFTsimpsonOOM@cref}{{[equation][14][8,3]8.3.14}{93}} +\@writefile{toc}{\contentsline {subsection}{\numberline {8.3.4}Critique de la m\IeC {\'e}thode de Carr-Madan}{93}{subsection.8.3.4}} +\newlabel{sec:critiquecarrmadanfft}{{8.3.4}{93}{Critique de la méthode de Carr-Madan}{subsection.8.3.4}{}} +\newlabel{sec:critiquecarrmadanfft@cref}{{[subsection][4][8,3]8.3.4}{93}} +\@writefile{toc}{\contentsline {subsection}{\numberline {8.3.5}M\IeC {\'e}thode d\IeC {\textquoteright }Epps}{93}{subsection.8.3.5}} +\newlabel{sec:epps2007}{{8.3.5}{93}{Méthode d’Epps}{subsection.8.3.5}{}} +\newlabel{sec:epps2007@cref}{{[subsection][5][8,3]8.3.5}{93}} +\newlabel{eq:putepps-1}{{8.3.15}{94}{Méthode d’Epps}{equation.8.3.0.15}{}} +\newlabel{eq:putepps-1@cref}{{[equation][15][8,3]8.3.15}{94}} +\newlabel{eq:putepps-resultat1}{{8.3.16}{94}{Méthode d’Epps}{equation.8.3.0.16}{}} +\newlabel{eq:putepps-resultat1@cref}{{[equation][16][8,3]8.3.16}{94}} +\newlabel{eq:fubini-integrale-a-EPPS}{{8.3.17}{94}{Méthode d’Epps}{equation.8.3.0.17}{}} +\newlabel{eq:fubini-integrale-a-EPPS@cref}{{[equation][17][8,3]8.3.17}{94}} +\newlabel{eq:PutEpps8.37}{{8.3.18}{94}{Méthode d’Epps}{equation.8.3.0.18}{}} +\newlabel{eq:PutEpps8.37@cref}{{[equation][18][8,3]8.3.18}{94}} +\citation{black1976pricing} +\@writefile{toc}{\contentsline {section}{\numberline {8.4}Particularit\IeC {\'e}s}{95}{section.8.4}} +\newlabel{sec:monnaiescontrats}{{8.4}{95}{Particularités}{section.8.4}{}} +\newlabel{sec:monnaiescontrats@cref}{{[section][4][8]8.4}{95}} +\@writefile{toc}{\contentsline {subsection}{\numberline {8.4.1}Option sur actions avec dividendes}{95}{subsection.8.4.1}} +\newlabel{sec:dividentoptions}{{8.4.1}{95}{Option sur actions avec dividendes}{subsection.8.4.1}{}} +\newlabel{sec:dividentoptions@cref}{{[subsection][1][8,4]8.4.1}{95}} +\newlabel{eq:prixinitialdividende}{{8.4.1}{95}{Option sur actions avec dividendes}{equation.8.4.0.1}{}} +\newlabel{eq:prixinitialdividende@cref}{{[equation][1][8,4]8.4.1}{95}} +\newlabel{eq:pricinitialdivcontinu}{{8.4.2}{95}{Option sur actions avec dividendes}{equation.8.4.0.2}{}} +\newlabel{eq:pricinitialdivcontinu@cref}{{[equation][2][8,4]8.4.2}{95}} +\@writefile{toc}{\contentsline {subsection}{\numberline {8.4.2}Options sur contrats \IeC {\`a} terme et taux de change}{95}{subsection.8.4.2}} +\newlabel{sec:futureoptions}{{8.4.2}{95}{Options sur contrats à terme et taux de change}{subsection.8.4.2}{}} +\newlabel{sec:futureoptions@cref}{{[subsection][2][8,4]8.4.2}{95}} +\@setckpt{chapitre8}{ +\setcounter{page}{96} +\setcounter{equation}{2} +\setcounter{enumi}{4} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{2} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{8} +\setcounter{section}{4} +\setcounter{subsection}{2} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{114} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{0} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{30} +\setcounter{Hfootnote}{3} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{96} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/chapitre8.log b/memoire/chapitre8.log new file mode 100644 index 0000000..a39380e --- /dev/null +++ b/memoire/chapitre8.log @@ -0,0 +1,15774 @@ +This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2013.11.30) 2 DEC 2013 21:08 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**chapitre8.tex +(./chapitre8.tex +LaTeX2e <2011/06/27> +Babel <3.9f> and hyphenation patterns for 78 languages loaded. +! Undefined control sequence. +l.1 \chapter + {Évaluation d'options} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1 \chapter{à + ‰valuation d'options} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ® in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 1--6 +[] + [] + +! Undefined control sequence. +l.7 \section + {Définitions} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.7 \section{D + éfinitions} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no D in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{bingham2004risk} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.9 En se référant à \cite{bingham2004risk} + , on définit: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `bingham2004risk' on page undefined on input line 9. + +Missing character: There is no ] in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 7--10 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 7--10 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.11 \begin{itemize} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 11--11 +[] + [] + +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 13. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 13. +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.70824pt too wide) in paragraph at lines 12--16 +\OT1/cmr/bx/n/10 pro- + [] + + +Overfull \hbox (20.44434pt too wide) in paragraph at lines 12--16 +\OT1/cmr/bx/n/10 duit + [] + + +Overfull \hbox (20.3888pt too wide) in paragraph at lines 12--16 +\OT1/cmr/bx/n/10 driv + [] + + +Overfull \hbox (7.23265pt too wide) in paragraph at lines 12--16 +\OML/cmm/m/it/10 T$ + [] + + +Overfull \hbox (7.23265pt too wide) in paragraph at lines 12--16 +\OML/cmm/m/it/10 T$ + [] + + +Overfull \hbox (22.23265pt too wide) in paragraph at lines 12--16 +[]$ + [] + +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (15.97214pt too wide) in paragraph at lines 16--19 +\OT1/cmr/bx/n/10 op- + [] + + +Overfull \hbox (19.80545pt too wide) in paragraph at lines 16--19 +\OT1/cmr/bx/n/10 tion + [] + +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (15.97214pt too wide) in paragraph at lines 19--22 +\OT1/cmr/bx/n/10 op- + [] + + +Overfull \hbox (19.80545pt too wide) in paragraph at lines 19--22 +\OT1/cmr/bx/n/10 tion + [] + + +Overfull \hbox (36.41643pt too wide) in paragraph at lines 19--22 +\OT1/cmr/bx/n/10 d'achat + [] + + +Overfull \hbox (16.13187pt too wide) in paragraph at lines 19--22 +\OT1/cmr/bx/n/10 (de + [] + + +Overfull \hbox (31.3054pt too wide) in paragraph at lines 19--22 +\OT1/cmr/bx/n/10 vente) + [] + + +Overfull \hbox (15.49298pt too wide) in paragraph at lines 19--22 +\OT1/cmr/bx/n/10 eu- + [] + + +Overfull \hbox (40.5137pt too wide) in paragraph at lines 19--22 +\OT1/cmr/bx/n/10 ropenne + [] + + +Overfull \hbox (24.54155pt too wide) in paragraph at lines 19--22 +\OT1/cmr/bx/n/10 droit + [] + + +Overfull \hbox (20.3888pt too wide) in paragraph at lines 19--22 +\OT1/cmr/bx/n/10 prix + [] + + +Overfull \hbox (49.61783pt too wide) in paragraph at lines 19--22 +\OT1/cmr/bx/n/10 d'exercice + [] + + +Overfull \hbox (9.20833pt too wide) in paragraph at lines 19--22 +\OML/cmm/m/it/10 K$ + [] + + +Overfull \hbox (7.23265pt too wide) in paragraph at lines 19--22 +\OML/cmm/m/it/10 T$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ; in font nullfont! + +Overfull \hbox (32.54164pt too wide) in paragraph at lines 25--27 +\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/ +m/n/10 ) \OML/cmm/m/it/10 > + [] + + +Overfull \hbox (13.09723pt too wide) in paragraph at lines 25--27 +\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (22.904pt too wide) in paragraph at lines 25--27 +\OT1/cmr/bx/n/10 dans + [] + + +Overfull \hbox (8.46524pt too wide) in paragraph at lines 25--27 +\OT1/cmr/bx/n/10 le + [] + + +Overfull \hbox (26.52208pt too wide) in paragraph at lines 25--27 +\OT1/cmr/bx/n/10 cours + [] + +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ; in font nullfont! + +Overfull \hbox (32.54164pt too wide) in paragraph at lines 27--28 +\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/ +m/n/10 ) = + [] + + +Overfull \hbox (13.09723pt too wide) in paragraph at lines 27--28 +\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (11.97908pt too wide) in paragraph at lines 27--28 +\OT1/cmr/bx/n/10 au + [] + + +Overfull \hbox (26.52208pt too wide) in paragraph at lines 27--28 +\OT1/cmr/bx/n/10 cours + [] + +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (32.54164pt too wide) in paragraph at lines 28--30 +\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/ +m/n/10 ) \OML/cmm/m/it/10 < + [] + + +Overfull \hbox (13.09723pt too wide) in paragraph at lines 28--30 +\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (21.411pt too wide) in paragraph at lines 28--30 +\OT1/cmr/bx/n/10 hors + [] + + +Overfull \hbox (12.77771pt too wide) in paragraph at lines 28--30 +\OT1/cmr/bx/n/10 du + [] + + +Overfull \hbox (26.52208pt too wide) in paragraph at lines 28--30 +\OT1/cmr/bx/n/10 cours + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.35 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no C in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no = in font nullfont! +! Missing $ inserted. + + $ +l.37 C(T) = \left + \{ \begin{array}{cc} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.41 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.41 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 33--42 +[] + [] + + +Overfull \hbox (19.2514pt too wide) in paragraph at lines 33--42 +\OML/cmm/m/it/10 C\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (100.55397pt too wide) in paragraph at lines 33--42 +[]$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.48 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no C in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +! Misplaced alignment tab character &. +l.49 C(t) & + = B(t,T) E \left[C(T) \right] \nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no E in font nullfont! +! Missing $ inserted. + + $ +l.49 C(t) &= B(t,T) E \left + [C(T) \right] \nonumber\\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. + & + +l.50 & + = B(t,T) \int_{K}^{\infty} (S(T)-K) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.52 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.52 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 43--56 +[] + [] + + +Overfull \hbox (32.20816pt too wide) in paragraph at lines 43--56 +\OT1/cmr/bx/n/10 crance + [] + + +Overfull \hbox (44.88173pt too wide) in paragraph at lines 43--56 +\OT1/cmr/bx/n/10 ventuelle + [] + + +Overfull \hbox (30.09514pt too wide) in paragraph at lines 43--56 +[] [] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 43--56 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (81.3438pt too wide) in paragraph at lines 43--56 +\OML/cmm/m/it/10 B\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t; T\OT1/cmr/m/n/10 ) [][]( +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 T\OT1/cmr/m/n/10 ) \OMS/cms +y/m/n/10 ^^@ + [] + + +Overfull \hbox (45.3532pt too wide) in paragraph at lines 43--56 +\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 d[][]:$ + [] + + +Overfull \hbox (31.15279pt too wide) in paragraph at lines 43--56 +\OML/cmm/m/it/10 B\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t; T\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (3.61111pt too wide) in paragraph at lines 43--56 +\OML/cmm/m/it/10 t$ + [] + + +Overfull \hbox (4.78937pt too wide) in paragraph at lines 43--56 +\OML/cmm/m/it/10 r$ + [] + + +Overfull \hbox (7.23265pt too wide) in paragraph at lines 43--56 +\OML/cmm/m/it/10 T$ + [] + + +Overfull \hbox (39.44797pt too wide) in paragraph at lines 43--56 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 T\OT1/cm +r/m/n/10 ))$ + [] + + +Overfull \hbox (21.71877pt too wide) in paragraph at lines 43--56 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 T\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 57--58 +[] + [] + +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! + +Overfull \hbox (18.09723pt too wide) in paragraph at lines 59--60 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! + +Overfull \hbox (9.20833pt too wide) in paragraph at lines 60--61 +\OML/cmm/m/it/10 K$ + [] + +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! + +Overfull \hbox (6.0729pt too wide) in paragraph at lines 61--62 +\OML/cmm/m/it/10 ^^[$ + [] + +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! + +Overfull \hbox (15.01045pt too wide) in paragraph at lines 62--63 +\OML/cmm/m/it/10 T \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (3.61111pt too wide) in paragraph at lines 62--63 +\OML/cmm/m/it/10 t$ + [] + +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (4.78937pt too wide) in paragraph at lines 63--64 +\OML/cmm/m/it/10 r$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.66 On peut aussi récrire \eqref + {eq:reclamationcall} avec le logarithme du +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.70 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no C in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +! Missing $ inserted. + + $ +l.71 C(t) = B(t,T) \int + _{k}^{\infty} (e^{s(T)}-e^{k}) d{F}_{s(T)}. \label{... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.72 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.72 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 66--73 +[] + [] + + +Overfull \hbox (16.07635pt too wide) in paragraph at lines 66--73 +\OML/cmm/m/it/10 k \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (19.20831pt too wide) in paragraph at lines 66--73 +[] \OML/cmm/m/it/10 K$ + [] + + +Overfull \hbox (26.63193pt too wide) in paragraph at lines 66--73 +\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (31.71875pt too wide) in paragraph at lines 66--73 +[] []$ + [] + + +Overfull \hbox (46.34947pt too wide) in paragraph at lines 66--73 +\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (70.13417pt too wide) in paragraph at lines 66--73 +\OML/cmm/m/it/10 Pr []$ + [] + + +Overfull \hbox (47.78589pt too wide) in paragraph at lines 66--73 +[][]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (44.20255pt too wide) in paragraph at lines 66--73 +\OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 dF[]:[]$ + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 74--77 +[] + [] + + +Overfull \hbox (19.19795pt too wide) in paragraph at lines 74--77 +\OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (19.2514pt too wide) in paragraph at lines 74--77 +\OML/cmm/m/it/10 C\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +Overfull \hbox (31.15279pt too wide) in paragraph at lines 80--81 +\OML/cmm/m/it/10 B\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t; T\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (18.09723pt too wide) in paragraph at lines 81--82 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.84 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no P in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no + in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no + in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.87 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 86--88 +[] + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 89--91 +[] + [] + +! Undefined control sequence. +l.92 \subsection + {Équation martingale} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +! Undefined control sequence. + \mathbb + +l.95 ...abilités neutre au risque, noté $\mathbb + {Q}$, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{black1973pricing} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.98 \cite{black1973pricing} + . Dans cette perspective, les rendements +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `black1973pricing' on page undefined on input line 98. + + +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.100 rendements cumulés \eqref + {eq:rendementcumL} répond à la propriété +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.107 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no E in font nullfont! +! Missing $ inserted. + + $ +l.109 E\left + [\exp(L_t) \right] = e^{rt}. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.110 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.110 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 92--111 +[] + [] + + +Overfull \hbox (7.90555pt too wide) in paragraph at lines 92--111 +\OML/cmm/m/it/10 Q$ + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 92--111 +[] + [] + + +Overfull \hbox (47.965pt too wide) in paragraph at lines 92--111 +\OT1/cmr/bx/n/10 L'quation + [] + + +Overfull \hbox (23.74292pt too wide) in paragraph at lines 92--111 +\OT1/cmr/bx/n/10 mar- + [] + + +Overfull \hbox (17.8888pt too wide) in paragraph at lines 92--111 +\OT1/cmr/bx/n/10 tin- + [] + + +Overfull \hbox (19.80544pt too wide) in paragraph at lines 92--111 +\OT1/cmr/bx/n/10 gale + [] + + +Overfull \hbox (3.61111pt too wide) in paragraph at lines 92--111 +\OML/cmm/m/it/10 t$ + [] + + +Overfull \hbox (49.49312pt too wide) in paragraph at lines 92--111 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (14.88315pt too wide) in paragraph at lines 92--111 +\OML/cmm/m/it/10 e[]:$ + [] + +! Undefined control sequence. +l.112 \subsection + {Paramètres neutres au risque} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 112--114 +[] + [] + +Missing character: There is no A in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.120 ... on utilise l'équation martingale \eqref + {eq:equationmartingale} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.121 ...fonction génératrice des moments \eqref + {eq:fgmGAL}. On +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.124 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no e in font nullfont! +! Missing $ inserted. + + $ +l.125 e^ + {rt} &= E\left[\exp(L_t) \right] +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.125 e^{rt} & + = E\left[\exp(L_t) \right] +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.127 & + = M_{R_1+\ldots+R_t}(1) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.129 & + = \left(\frac{e^{\theta_{RN}}}{(1-\mu-\sigma^2/2)^{\tau}} \right)^t... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.130 r & + = \ln \left(\frac{e^{\theta_{RN}}}{(1-\mu-\sigma^2/2)^{\tau}} \ri... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.131 & + = \theta_{RN} - \tau\ln(1-\mu-\frac{\sigma^2}{2}) \nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.132 \theta_{RN} & + = r + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.134 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.134 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 115--135 +[] + [] + + +Overfull \hbox (10.32642pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 L[]$ + [] + + +Overfull \hbox (4.78937pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 r$ + [] + + +Overfull \hbox (18.33926pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 ^^R[]$ + [] + + +Overfull \hbox (18.00461pt too wide) in paragraph at lines 115--135 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (59.11806pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 E [] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (99.2064pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 E [] [] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 115--135 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (72.20723pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 M[]\OT1/cmr/m/n/10 (1) = + [] + + +Overfull \hbox (40.33795pt too wide) in paragraph at lines 115--135 +\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 M[]\OT1/cmr/m/n/10 (1))[][] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 115--135 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (68.48961pt too wide) in paragraph at lines 115--135 +[][] [] + [] + + +Overfull \hbox (15.34488pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 r \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (74.96875pt too wide) in paragraph at lines 115--135 +[] [] [] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 115--135 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (26.11707pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 ^^R[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (32.17018pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 ^^\ []\OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (13.80328pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 ^^V \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (15.07683pt too wide) in paragraph at lines 115--135 +[]\OT1/cmr/m/n/10 )[] + [] + + +Overfull \hbox (28.89478pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 ^^R[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (12.56717pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 r \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (32.17018pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 ^^\ []\OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (13.80328pt too wide) in paragraph at lines 115--135 +\OML/cmm/m/it/10 ^^V \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (17.85461pt too wide) in paragraph at lines 115--135 +[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 :[]$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.137 partir de l'équation martingale \eqref + {eq:martingaleGAL}: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.138 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.140 \omega + = \tau\ln(1-\mu-\frac{\sigma^2}{2}). +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.141 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.141 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 136--142 +[] + [] + + +Overfull \hbox (6.58331pt too wide) in paragraph at lines 136--142 +\OML/cmm/m/it/10 !$ + [] + + +Overfull \hbox (17.13882pt too wide) in paragraph at lines 136--142 +\OML/cmm/m/it/10 ! \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (32.17018pt too wide) in paragraph at lines 136--142 +\OML/cmm/m/it/10 ^^\ []\OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (13.80328pt too wide) in paragraph at lines 136--142 +\OML/cmm/m/it/10 ^^V \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (17.85461pt too wide) in paragraph at lines 136--142 +[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.145 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.147 \mu + +\frac{\sigma^2}{2} < 1. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.148 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.148 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 143--149 +[] + [] + + +Overfull \hbox (6.02548pt too wide) in paragraph at lines 143--149 +\OML/cmm/m/it/10 ^^V$ + [] + + +Overfull \hbox (6.0729pt too wide) in paragraph at lines 143--149 +\OML/cmm/m/it/10 ^^[$ + [] + + +Overfull \hbox (13.80328pt too wide) in paragraph at lines 143--149 +\OML/cmm/m/it/10 ^^V \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (21.74344pt too wide) in paragraph at lines 143--149 +[] \OML/cmm/m/it/10 < + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 143--149 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + \mathbb + +l.151 risque $\mathbb + {Q}$ pour $L_T$ en remplaçant $\theta$ par +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no § in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.153 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.155 \phi + (\xi,L_T) = \frac{\exp\left(i\xi*\left(\ln\left(S_{t}\right)+\le... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.156 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.156 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 150--157 +[] + [] + + +Overfull \hbox (7.90555pt too wide) in paragraph at lines 150--157 +\OML/cmm/m/it/10 Q$ + [] + + +Overfull \hbox (13.10522pt too wide) in paragraph at lines 150--157 +\OML/cmm/m/it/10 L[]$ + [] + + +Overfull \hbox (4.97223pt too wide) in paragraph at lines 150--157 +\OML/cmm/m/it/10 ^^R$ + [] + + +Overfull \hbox (18.33926pt too wide) in paragraph at lines 150--157 +\OML/cmm/m/it/10 ^^R[]$ + [] + + +Overfull \hbox (46.67638pt too wide) in paragraph at lines 150--157 +\OML/cmm/m/it/10 ^^^\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X; L[]\OT1/cmr/m/n/10 ) + = + [] + + +Overfull \hbox (110.8806pt too wide) in paragraph at lines 150--157 +[]\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 158--160 +[] + [] + + +Overfull \hbox (21.66925pt too wide) in paragraph at lines 158--160 +\OML/cmm/m/it/10 R[] \OMS/cmsy/m/n/10 ^^X + [] + + +Overfull \hbox (38.62415pt too wide) in paragraph at lines 158--160 +\OML/cmm/m/it/10 GAL\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (32.17018pt too wide) in paragraph at lines 158--160 +\OML/cmm/m/it/10 ^^\ []\OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (13.80328pt too wide) in paragraph at lines 158--160 +\OML/cmm/m/it/10 ^^V \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (58.71642pt too wide) in paragraph at lines 158--160 +\OML/cmm/m/it/10 ^^[[]=\OT1/cmr/m/n/10 2)\OML/cmm/m/it/10 ; ^^[; ^^V; ^^\\OT1/c +mr/m/n/10 )$ + [] + + +! LaTeX Error: The font size command \normalsize is not defined: + there is probably something wrong with the class file. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.160 + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +[1 +! Undefined control sequence. + \newlabel{chap:options}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{chap:options}{{}{}} +! Undefined control sequence. + \newlabel{sec:options}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:options}{{}{}} +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} +! Undefined control sequence. + \newlabel{eq:callpayoff}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:callpayoff}{{}{}} +! Undefined control sequence. + \newlabel{eq:reclamationcall}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:reclamationcall}{{}{}} +! Undefined control sequence. + ...{eq:reclamationcallintlog}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:reclamationcallintlog}{{}{}} +! Undefined control sequence. + \newlabel{eq:pariteputcall}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:pariteputcall}{{}{}} +! Undefined control sequence. + ...el{sec:equationmartingale}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:equationmartingale}{{}{}} +! Undefined control sequence. + ...bel{eq:equationmartingale}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:equationmartingale}{{}{}} +! Undefined control sequence. + \newlabel{sec:GALrn}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:GALrn}{{}{}} +! Undefined control sequence. + \newlabel{eq:martingaleGAL}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:martingaleGAL}{{}{}} +! Undefined control sequence. + \newlabel{eq:omegaGAL}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:omegaGAL}{{}{}} +! Undefined control sequence. + ...label{eq:inegaliteparamrn}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:inegaliteparamrn}{{}{}} +! Undefined control sequence. + \newlabel{eq:fncaractGALrn}{{}{\thepage + }} +l.160 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:fncaractGALrn}{{}{}} +] +! Undefined control sequence. +l.161 \section + {Aperçu du modèle de Black-Scholes} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no A in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no § in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 161--163 +[] + [] + +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 164--167 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (6.02548pt too wide) in paragraph at lines 168--172 +\OML/cmm/m/it/10 ^^V$ + [] + + +Overfull \hbox (10.55904pt too wide) in paragraph at lines 168--172 +\OML/cmm/m/it/10 ^^[[]$ + [] + + +Overfull \hbox (28.09726pt too wide) in paragraph at lines 168--172 +\OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr +/m/n/10 )\OMS/cmsy/m/n/10 g$ + [] + + +Overfull \hbox (6.02548pt too wide) in paragraph at lines 168--172 +\OML/cmm/m/it/10 ^^V$ + [] + + +Overfull \hbox (6.0729pt too wide) in paragraph at lines 168--172 +\OML/cmm/m/it/10 ^^[$ + [] + +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (4.78937pt too wide) in paragraph at lines 172--174 +\OML/cmm/m/it/10 r$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.180 En utilisant l'équation martingale \eqref + {eq:equationmartingale}, on +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 180--184 +[] + [] + + +Overfull \hbox (12.56717pt too wide) in paragraph at lines 180--184 +\OML/cmm/m/it/10 r \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (11.18793pt too wide) in paragraph at lines 180--184 +[]$ + [] + + +Overfull \hbox (10.55904pt too wide) in paragraph at lines 180--184 +\OML/cmm/m/it/10 ^^[[]$ + [] + +Missing character: There is no à in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.187 \eqref + {eq:pariteputcall} européenne: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.188 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no C in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +! Misplaced alignment tab character &. +l.190 C(S(t),K,T) & + = \Phi(\Delta_1) ~ S(t) - \Phi(\Delta_2) ~ K B(t,T) \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +! Missing $ inserted. + + $ +l.190 C(S(t),K,T) &= \Phi + (\Delta_1) ~ S(t) - \Phi(\Delta_2) ~ K B(t,T) \\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.191 \Delta_1 & + = \frac{\ln\left(\frac{S(t)}{K}\right) + \left(r + \frac{\... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.192 \Delta_2 & + = \Delta_1 - \sigma\sqrt{T - t} \nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.193 P(S(t),K,T) & + = \Phi(-\delta_2)K B(t,T) - +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.195 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.195 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 185--196 +[] + [] + + +Overfull \hbox (53.69455pt too wide) in paragraph at lines 185--196 +\OT1/cmr/m/n/10 ^^H(^^A[]) []\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/1 +0 t\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (68.18063pt too wide) in paragraph at lines 185--196 +\OT1/cmr/m/n/10 ^^H(^^A[]) []\OML/cmm/m/it/10 KB\OT1/cmr/m/n/10 (\OML/cmm/m/it/ +10 t; T\OT1/cmr/m/n/10 )[] + [] + + +Overfull \hbox (23.375pt too wide) in paragraph at lines 185--196 +\OT1/cmr/m/n/10 ^^A[] = + [] + + +Overfull \hbox (90.66849pt too wide) in paragraph at lines 185--196 +[][] + [] + + +Overfull \hbox (23.375pt too wide) in paragraph at lines 185--196 +\OT1/cmr/m/n/10 ^^A[] = + [] + + +Overfull \hbox (20.59729pt too wide) in paragraph at lines 185--196 +\OT1/cmr/m/n/10 ^^A[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (37.47217pt too wide) in paragraph at lines 185--196 +\OML/cmm/m/it/10 ^^[[][] + [] + + +Overfull \hbox (69.01382pt too wide) in paragraph at lines 185--196 +\OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/ +m/it/10 t\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; K; T\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (79.84732pt too wide) in paragraph at lines 185--196 +\OT1/cmr/m/n/10 ^^H(\OMS/cmsy/m/n/10 ^^@\OML/cmm/m/it/10 ^^N[]\OT1/cmr/m/n/10 ) +\OML/cmm/m/it/10 KB\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t; T\OT1/cmr/m/n/10 ) \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (52.58342pt too wide) in paragraph at lines 185--196 +\OT1/cmr/m/n/10 ^^H(\OMS/cmsy/m/n/10 ^^@\OML/cmm/m/it/10 ^^N[]\OT1/cmr/m/n/10 ) +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )\OML/cmm/ +m/it/10 :[]$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 197--206 +[] + [] + +! Undefined control sequence. +l.207 \section + {Méthodes d'évaluation pour options européennes} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no M in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 207--209 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no Ž in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 210--216 +[] + [] + +! Undefined control sequence. +l.217 \subsection + {Méthode de Heston} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no M in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no H in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 217--219 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{heston1993closed} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.220 ...che utilisée par \cite{heston1993closed} + est de calculer +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `heston1993closed' on page undefined on input line 220 +. + +Missing character: There is no ] in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + \mathbb + +l.223 $\mathbb + {Q}$: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.224 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no P in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +! Misplaced alignment tab character &. +l.225 P(S(t),K,T) & + = B(t,T) \int_{0}^{\infty} \max\left(K-S(T),0\right)\cd... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +! Missing $ inserted. + + $ +l.225 P(S(t),K,T) &= B(t,T) \int + _{0}^{\infty} \max\left(K-S(T),0\right)\cd... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. + & + +l.226 & + = B(t,T)K \int_{0}^{K} dF_{S(T)} - B(t,T)\int_{0}^{K} S(T)\cdot d{F... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.227 & + = B(t,T)K F_{S(t)}(K) - S(t) G_{S(t)}(K). \label{eq:PutHeston93} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.228 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.228 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 220--229 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 220--229 +[] + [] + + +Overfull \hbox (62.34721pt too wide) in paragraph at lines 220--229 +\OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/ +m/it/10 t\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; K; T \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (7.50002pt too wide) in paragraph at lines 220--229 +\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (7.90555pt too wide) in paragraph at lines 220--229 +\OML/cmm/m/it/10 Q$ + [] + + +Overfull \hbox (100.23259pt too wide) in paragraph at lines 220--229 +[][] [] [] \OMS/cmsy/m/n/10 ^^A + [] + + +Overfull \hbox (29.47818pt too wide) in paragraph at lines 220--229 +\OML/cmm/m/it/10 dF[][][] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 220--229 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (95.32678pt too wide) in paragraph at lines 220--229 +\OML/cmm/m/it/10 B\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t; T\OT1/cmr/m/n/10 )\OML/c +mm/m/it/10 K [][] dF[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (73.35902pt too wide) in paragraph at lines 220--229 +\OML/cmm/m/it/10 B\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t; T\OT1/cmr/m/n/10 ) [][] +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 T\OT1/cmr/m/n/10 ) \OMS/cms +y/m/n/10 ^^A + [] + + +Overfull \hbox (29.47818pt too wide) in paragraph at lines 220--229 +\OML/cmm/m/it/10 dF[][] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 220--229 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (86.61955pt too wide) in paragraph at lines 220--229 +\OML/cmm/m/it/10 B\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t; T\OT1/cmr/m/n/10 )\OML/c +mm/m/it/10 KF[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ) \OMS/cmsy/m +/n/10 ^^@ + [] + + +Overfull \hbox (60.78758pt too wide) in paragraph at lines 220--229 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )\OML/cmm/ +m/it/10 G[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 + :[]$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + \mathbb + +l.231 ...ire $S(t) \leq K$ sous la mesure $\mathbb + {Q}$. On définit aussi +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +! Undefined control sequence. + \mathbb + +l.233 équivalant à $\mathbb + {Q}$. Soit $\phi_F(\xi)$ la fonction +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. + \mathbb + +l.234 caractéristique de $S(T)$ sous $\mathbb + {Q}$, celle de cette mesure est +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.236 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.238 \phi + _G(\xi) = \frac{\phi_F(\xi-i)}{\phi_F(-i)}. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.239 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.239 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +Missing character: There is no C in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + \mathbb + +l.241 $\mathbb + {Q}$. On remarquera la forme de \eqref{eq:PutHeston93} qui est +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.241 ...hbb{Q}$. On remarquera la forme de \eqref + {eq:PutHeston93} qui est +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no H in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.243 Black-Scholes \eqref + {eq:putblackscholes}. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 230--244 +[] + [] + + +Overfull \hbox (38.48064pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (28.65274pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OMS/cms +y/m/n/10 ^^T + [] + + +Overfull \hbox (9.20833pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 K$ + [] + + +Overfull \hbox (7.90555pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 Q$ + [] + + +Overfull \hbox (39.91257pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 G[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (7.90555pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 Q$ + [] + + +Overfull \hbox (25.21918pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (21.71877pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 T\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (7.90555pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 Q$ + [] + + +Overfull \hbox (35.85272pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (34.89798pt too wide) in paragraph at lines 230--244 +[]\OML/cmm/m/it/10 :$ + [] + + +Overfull \hbox (16.3171pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 h \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 230--244 +\OT1/cmr/m/n/10 1$ + [] + + +Overfull \hbox (7.90555pt too wide) in paragraph at lines 230--244 +\OML/cmm/m/it/10 Q$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.246 peuvent être évaluées à partir de \eqref + {eq:approxinvfncaract} ou de +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 245--248 +[] + [] + + +Overfull \hbox (38.48064pt too wide) in paragraph at lines 245--248 +\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (39.91257pt too wide) in paragraph at lines 245--248 +\OML/cmm/m/it/10 G[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no [ in font nullfont! +\citation{carr1999option} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.249 \cite{carr1999option} + notent qu'on ne peut pas inverser +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `carr1999option' on page undefined on input line 249. + +Missing character: There is no ] in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.250 l'approximation \eqref + {eq:approxinvfncaract} en utilisant la méthode +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.255 \eqref + {eq:saddlepointGAL}. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (25.43053pt too wide) in paragraph at lines 249--256 +[][] + [] + +! Undefined control sequence. +l.257 \subsection + {Méthode de Carr et Madan} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no M in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 257--259 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no H in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{madan1998variance} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.262 caractéristiques. \cite{madan1998variance} + ont développé une méthode +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `madan1998variance' on page undefined on input line 26 +2. + +Missing character: There is no ] in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{epps2007pricing} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.264 \cite{epps2007pricing} + . Cette méthode nécessite cependant la sélection +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `epps2007pricing' on page undefined on input line 264. + + +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¢ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +\@footnotetext ...tins {\reset@font \footnotesize + \interlinepenalty \interfo... +l.267 ...\footnote{traduction de l'anglais damped} + de +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +\@footnotetext ...}\color@begingroup \@makefntext + {\rule \z@ \footnotesep \i... +l.267 ...\footnote{traduction de l'anglais damped} + de +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +Overfull \hbox (45.58345pt too wide) in paragraph at lines 267--267 +[][]\OT1/cmr/m/n/10 traduction + [] + + +Overfull \hbox (10.00002pt too wide) in paragraph at lines 267--267 +\OT1/cmr/m/n/10 de + [] + + +Overfull \hbox (35.6112pt too wide) in paragraph at lines 267--267 +\OT1/cmr/m/n/10 l'anglais + [] + + +Overfull \hbox (34.72232pt too wide) in paragraph at lines 267--267 +\OT1/cmr/m/n/10 damped| + [] + +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.269 \eqref + {eq:reclamationcallintlog}: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.270 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no C in font nullfont! +! Missing $ inserted. + + $ +l.272 C_ + {\theta_D}(S(t),k,T) &= e^{\theta_{D}k} C(S(t),k,T) \\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.272 C_{\theta_D}(S(t),k,T) & + = e^{\theta_{D}k} C(S(t),k,T) \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.273 & + = e^{\theta_{D}k} B(t,T) \int_{K}^{\infty} (e^{s(T)}-e^{k}) +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.275 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.275 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 260--276 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 260--276 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 260--276 +[] + [] + + +Overfull \hbox (11.94821pt too wide) in paragraph at lines 260--276 +\OML/cmm/m/it/10 ^^R[]$ + [] + + +Overfull \hbox (26.19427pt too wide) in paragraph at lines 260--276 +\OT1/cmr/m/it/10 amor- + [] + + +Overfull \hbox (10.9888pt too wide) in paragraph at lines 260--276 +\OT1/cmr/m/it/10 tie + [] + + +Overfull \hbox (4.48613pt too wide) in paragraph at lines 260--276 +[] + [] + + +Overfull \hbox (68.60486pt too wide) in paragraph at lines 260--276 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cm +r/m/n/10 )\OML/cmm/m/it/10 ; k; T\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (74.98827pt too wide) in paragraph at lines 260--276 +\OML/cmm/m/it/10 e[]C\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/c +mm/m/it/10 t\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; k; T\OT1/cmr/m/n/10 )[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 260--276 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (100.21373pt too wide) in paragraph at lines 260--276 +\OML/cmm/m/it/10 e[]B\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t; T\OT1/cmr/m/n/10 ) [] +[](\OML/cmm/m/it/10 e[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (44.20255pt too wide) in paragraph at lines 260--276 +\OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 dF[]:$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.278 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathcal allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.280 \mathcal + {F}C_{\theta_D}(\nu) = B(t,T) \frac{\phi\left( \nu - i(1+\th... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no F in font nullfont! +Missing character: There is no C in font nullfont! +! Missing $ inserted. + + $ +l.280 \mathcal{F}C_ + {\theta_D}(\nu) = B(t,T) \frac{\phi\left( \nu - i(1+\th... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.281 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.281 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 277--282 +[] + [] + + +Overfull \hbox (34.4417pt too wide) in paragraph at lines 277--282 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^W\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (105.90752pt too wide) in paragraph at lines 277--282 +\OML/cmm/m/it/10 B\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t; T\OT1/cmr/m/n/10 )[]\OML +/cmm/m/it/10 :$ + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font cmti10! +Missing character: There is no © in font cmti10! +! Undefined control sequence. +\@footnotetext ...tins {\reset@font \footnotesize + \interlinepenalty \interfo... +l.284 l'anglais amplified} + pour retrouver le prix de l'option: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +\@footnotetext ...}\color@begingroup \@makefntext + {\rule \z@ \footnotesep \i... +l.284 l'anglais amplified} + pour retrouver le prix de l'option: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +Overfull \hbox (45.58345pt too wide) in paragraph at lines 284--284 +[][]\OT1/cmr/m/n/10 traduction + [] + + +Overfull \hbox (10.00002pt too wide) in paragraph at lines 284--284 +\OT1/cmr/m/n/10 de + [] + + +Overfull \hbox (35.6112pt too wide) in paragraph at lines 284--284 +\OT1/cmr/m/n/10 l'anglais + [] + + +Overfull \hbox (16.6667pt too wide) in paragraph at lines 284--284 +\OT1/cmr/m/n/10 am- + [] + + +Overfull \hbox (14.44447pt too wide) in paragraph at lines 284--284 +\OT1/cmr/m/n/10 pli- + [] + + +Overfull \hbox (15.55559pt too wide) in paragraph at lines 284--284 +\OT1/cmr/m/n/10 fied| + [] + +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.285 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no C in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +! Misplaced alignment tab character &. +l.286 C(S(t),k,T) & + = e^{-\theta_{D}k} \mathcal{F}^{-1}\left(\mathcal{F}C_{... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +Missing character: There is no e in font nullfont! +! Missing $ inserted. + + $ +l.286 C(S(t),k,T) &= e^ + {-\theta_{D}k} \mathcal{F}^{-1}\left(\mathcal{F}C_{... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. + & + +l.287 & + = \frac{e^{-\theta_{D}k}}{2\pi} \int_{-\infty}^{\infty} e^{-i\nu k} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.290 & + = \frac{e^{-\theta_{D}k}}{\pi} \int_{0}^{\infty} e^{-i\nu k} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no ( in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ) in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.293 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.293 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 283--294 +[] + [] + + +Overfull \hbox (16.86656pt too wide) in paragraph at lines 283--294 +\OT1/cmr/m/it/10 am- + [] + + +Overfull \hbox (16.35547pt too wide) in paragraph at lines 283--294 +\OT1/cmr/m/it/10 plifi + [] + + +Overfull \hbox (4.48613pt too wide) in paragraph at lines 283--294 +[] + [] + + +Overfull \hbox (90.44383pt too wide) in paragraph at lines 283--294 +[]\OMS/cmsy/m/n/10 F[] [] [] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 283--294 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (121.26955pt too wide) in paragraph at lines 283--294 +[] [][] \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 F\OML/cmm/m/it/10 + C[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 ^^W\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^ +A + [] + + +Overfull \hbox (10.78125pt too wide) in paragraph at lines 283--294 +\OML/cmm/m/it/10 d^^W[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 283--294 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (116.96397pt too wide) in paragraph at lines 283--294 +[] [][] \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 F\OML/cmm/m/it/10 + C[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 ^^W\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^ +A + [] + + +Overfull \hbox (13.55904pt too wide) in paragraph at lines 283--294 +\OML/cmm/m/it/10 d^^W []:[]$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ...ence `sec:methodeFFT' on page \thepage + \space undefined\on@line . +l.296 (section \ref{sec:methodeFFT} + ) pour retrouver les prix d'options en se +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:methodeFFT' on page undefined on input line 296. + + +Missing character: There is no ) in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{carr1999option} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.297 référant à \cite{carr1999option} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `carr1999option' on page undefined on input line 297. + +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 295--298 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 295--298 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 295--298 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +! Undefined control sequence. +l.299 On doit donc discrétiser \eqref + {eq:prixoptionCarr} et effectuer des +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.300 ...de variables pour obtenir la forme \eqref + {eq:sommefft}. On +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.304 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.305 \lambda + = \frac{2\pi}{N\eta}. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.306 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.306 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 299--307 +[] + [] + + +Overfull \hbox (9.12497pt too wide) in paragraph at lines 299--307 +\OML/cmm/m/it/10 N$ + [] + + +Overfull \hbox (5.32408pt too wide) in paragraph at lines 299--307 +\OML/cmm/m/it/10 ^^Q$ + [] + + +Overfull \hbox (5.83336pt too wide) in paragraph at lines 299--307 +\OML/cmm/m/it/10 ^^U$ + [] + + +Overfull \hbox (10.4341pt too wide) in paragraph at lines 299--307 +\OML/cmm/m/it/10 k[]$ + [] + + +Overfull \hbox (16.28009pt too wide) in paragraph at lines 299--307 +\OML/cmm/m/it/10 u \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (36.34705pt too wide) in paragraph at lines 299--307 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; [] ; N$ + [] + + +Overfull \hbox (16.38887pt too wide) in paragraph at lines 299--307 +\OML/cmm/m/it/10 ^^U \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (16.63385pt too wide) in paragraph at lines 299--307 +[]\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.309 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no b in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no N in font nullfont! +! Missing $ inserted. + + $ +l.310 b=\frac{N\lambda}{2} + . +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing $ inserted. + + $ +l.310 b=\frac{N\lambda}{2} + . +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.310 b=\frac{N\lambda}{2} + . +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Extra }, or forgotten $. +\frac #1#2->{\begingroup #1\endgroup \over #2} + +l.310 b=\frac{N\lambda}{2} + . +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.311 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.311 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.311 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 308--312 +[] + [] + + +Overfull \hbox (4.29166pt too wide) in paragraph at lines 308--312 +\OML/cmm/m/it/10 b$ + [] + + +Overfull \hbox (14.59448pt too wide) in paragraph at lines 308--312 +\OML/cmm/m/it/10 ^^U$$[]$ + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.317 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no C in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no k in font nullfont! +! Missing $ inserted. + + $ +l.318 C(S(t),k_ + u,T) &\approx \frac{e^{-\theta_D\,k}}{\pi} \sum_{j=1}^N e^{... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.318 C(S(t),k_u,T) & + \approx \frac{e^{-\theta_D\,k}}{\pi} \sum_{j=1}^N e^{... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.319 & + \approx \frac{e^{-\theta_D\,k}}{\pi} \sum_{j=1}^N +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.322 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.322 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 313--323 +[] + [] + + +Overfull \hbox (19.7089pt too wide) in paragraph at lines 313--323 +\OML/cmm/m/it/10 ^^W[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (21.68127pt too wide) in paragraph at lines 313--323 +\OML/cmm/m/it/10 ^^Q\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 j \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (8.88892pt too wide) in paragraph at lines 313--323 +\OT1/cmr/m/n/10 1)$ + [] + + +Overfull \hbox (31.34953pt too wide) in paragraph at lines 313--323 +[]\OML/cmm/m/it/10 ; T\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^Y + [] + + +Overfull \hbox (133.38046pt too wide) in paragraph at lines 313--323 +[] [][] \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 F\OML/cmm/m/it/10 + C[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 ^^W[]\OT1/cmr/m/n/10 )[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 313--323 +\OMS/cmsy/m/n/10 ^^Y + [] + + +Overfull \hbox (197.72702pt too wide) in paragraph at lines 313--323 +[] [][] \OML/cmm/m/it/10 e[]e[]\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 F\OML/cmm/m/it +/10 C[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 ^^W[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/1 +0 ^^Q:[]$ + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.328 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no C in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no k in font nullfont! +! Missing $ inserted. + + $ +l.329 C(S(t),k_ + u,T) &\approx \frac{e^{-\theta_D\,k}}{\pi} \sum_{j=1}^N +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.329 C(S(t),k_u,T) & + \approx \frac{e^{-\theta_D\,k}}{\pi} \sum_{j=1}^N +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.333 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.333 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 324--334 +[] + [] + + +Overfull \hbox (31.34953pt too wide) in paragraph at lines 324--334 +[]\OML/cmm/m/it/10 ; T\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^Y + [] + + +Overfull \hbox (284.78186pt too wide) in paragraph at lines 324--334 +[] [][] \OML/cmm/m/it/10 e[]e[]\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 F\OML/cmm/m/it +/10 C[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 ^^W[]\OT1/cmr/m/n/10 )[] [] \OML/cmm/ +m/it/10 :[]$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 335--338 +[] + [] + +! Undefined control sequence. +l.339 \subsection + {Prix d'exercice hors du cours} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 339--341 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.342 L'intégration numérique de \eqref + {eq:prixoptionCarr} pose problème +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{carr1999option} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.345 \cite{carr1999option} + développent une formule alternative à l'équation +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `carr1999option' on page undefined on input line 345. + +Missing character: There is no ] in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.346 \eqref + {eq:prixoptionCarr} pour évaluer le prix de l'option d'achat : +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.347 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no C in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +! Misplaced alignment tab character &. +l.348 C(S(t),k,T) & + = \frac{1}{\pi\sinh(\theta^{*}_D k)} \int_{0}^{\infty} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +Missing character: There is no 1 in font nullfont! +! Missing $ inserted. + + $ +l.348 ...,T) &= \frac{1}{\pi\sinh(\theta^{*}_D k)} + \int_{0}^{\infty} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Extra }, or forgotten $. +\frac #1#2->{\begingroup #1\endgroup \over #2} + +l.348 ...,T) &= \frac{1}{\pi\sinh(\theta^{*}_D k)} + \int_{0}^{\infty} +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.351 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.351 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.351 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.354 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathcal allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.355 \mathcal + {F}C^{*}_{\theta^{*}_D} &= \frac{\zeta_T(\nu-i\theta^{*}_D) ... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no F in font nullfont! +Missing character: There is no C in font nullfont! +! Missing $ inserted. + + $ +l.355 \mathcal{F}C^ + {*}_{\theta^{*}_D} &= \frac{\zeta_T(\nu-i\theta^{*}_D) ... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.355 \mathcal{F}C^{*}_{\theta^{*}_D} & + = \frac{\zeta_T(\nu-i\theta^{*}_D) ... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.356 \zeta_T(\nu) & + = e^{-r(T-t)}\bigg[\frac{1}{1+i\nu} - +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.359 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.359 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 342--360 +[] + [] + + +Overfull \hbox (15.01045pt too wide) in paragraph at lines 342--360 +\OML/cmm/m/it/10 T \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (3.61111pt too wide) in paragraph at lines 342--360 +\OML/cmm/m/it/10 t$ + [] + + +Overfull \hbox (9.20833pt too wide) in paragraph at lines 342--360 +\OML/cmm/m/it/10 K$ + [] + + +Overfull \hbox (16.07635pt too wide) in paragraph at lines 342--360 +\OML/cmm/m/it/10 k > + [] + + +Overfull \hbox (35.59729pt too wide) in paragraph at lines 342--360 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (0))$ + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 342--360 +[] + [] + + +Overfull \hbox (132.74002pt too wide) in paragraph at lines 342--360 +[]$ + [] + + +Overfull \hbox (21.08751pt too wide) in paragraph at lines 342--360 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (86.54999pt too wide) in paragraph at lines 342--360 +[][] + [] + + +Overfull \hbox (34.58437pt too wide) in paragraph at lines 342--360 +\OML/cmm/m/it/10 ^^P[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^W\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (69.54073pt too wide) in paragraph at lines 342--360 +\OML/cmm/m/it/10 e[][][] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (36.6834pt too wide) in paragraph at lines 342--360 +[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (35.06897pt too wide) in paragraph at lines 342--360 +[][]\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 361--363 +[] + [] + + +Overfull \hbox (11.94821pt too wide) in paragraph at lines 361--363 +\OML/cmm/m/it/10 ^^R[]$ + [] + + +Overfull \hbox (11.94821pt too wide) in paragraph at lines 361--363 +\OML/cmm/m/it/10 ^^R[]$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.366 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no C in font nullfont! +! Missing $ inserted. + + $ +l.367 C^ + {*}(S(t),k_u,T) &\approx \frac{1}{\pi\sinh(\theta^{*}_D k)} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.367 C^{*}(S(t),k_u,T) & + \approx \frac{1}{\pi\sinh(\theta^{*}_D k)} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.371 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.371 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.374 le cours \eqref + {eq:prixoptionFFTsimpson}. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 364--375 +[] + [] + + +Overfull \hbox (67.58334pt too wide) in paragraph at lines 364--375 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cm +r/m/n/10 )\OML/cmm/m/it/10 ; k[]; T\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^Y + [] + + +Overfull \hbox (301.62492pt too wide) in paragraph at lines 364--375 +[] [][] \OML/cmm/m/it/10 e[]e[]\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 F\OML/cmm/m/it +/10 C[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 ^^W[]\OT1/cmr/m/n/10 )[] [] \OML/cmm/ +m/it/10 :[]$ + [] + + +Overfull \hbox (28.61568pt too wide) in paragraph at lines 364--375 +\OML/cmm/m/it/10 N; ^^Q; ^^U$ + [] + + +Overfull \hbox (9.15338pt too wide) in paragraph at lines 364--375 +\OML/cmm/m/it/10 ^^W[]$ + [] + +! Undefined control sequence. +l.376 \subsection + {Critique de la méthode de Carr-Madan} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no C in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 376--378 +[] + [] + +Missing character: There is no U in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no H in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no â in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no ™ in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{itkin2005pricing} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.382 européennes. \cite{itkin2005pricing} + démontre que pour certaines +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `itkin2005pricing' on page undefined on input line 382 +. + +Missing character: There is no ] in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.384 \eqref + {eq:prixoptionCarr} a un comportement très irrégulier. De plus, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 379--387 +[] + [] + + +Overfull \hbox (11.94821pt too wide) in paragraph at lines 379--387 +\OML/cmm/m/it/10 ^^R[]$ + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 379--387 +[] + [] + + +Overfull \hbox (11.94821pt too wide) in paragraph at lines 379--387 +\OML/cmm/m/it/10 ^^R[]$ + [] + +! Undefined control sequence. +l.388 \subsection + {Méthode d’Epps} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no M in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no â in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no ™ in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 388--390 +[] + [] + +Missing character: There is no [ in font nullfont! +\citation{epps2007pricing} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.391 \cite{epps2007pricing} + propose une méthode qui, contrairement à celle +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `epps2007pricing' on page undefined on input line 391. + + +Missing character: There is no ] in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.395 ... vente développée précédemment \eqref + {eq:PutHeston93-1}, qui +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no H in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.397 \eqref + {eq:reclamationcallintlog}. Après l'utilisation de la formule +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.398 d'inversion \eqref + {eq:gilpelaez2}, on obtient le résultat suivant: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.399 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no E in font nullfont! +! Missing $ inserted. + + $ +l.401 E\left + [max(K-S_T;0) \right] &= \int_{-\infty}^{k} {F}_{S(T)}(s)e^s\c... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.401 E\left[max(K-S_T;0) \right] & + = \int_{-\infty}^{k} {F}_{S(T)}(s)e^s\c... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.402 & + = \frac{K}{2}-\frac{1}{2}\int_{-\infty}^{k} \lim_{c\to\infty} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.405 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.405 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (25.43053pt too wide) in paragraph at lines 391--406 +[][] + [] + + +Overfull \hbox (86.97667pt too wide) in paragraph at lines 391--406 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (69.5581pt too wide) in paragraph at lines 391--406 +[][] \OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 )\O +ML/cmm/m/it/10 e[] \OMS/cmsy/m/n/10 ^^A + [] + + +Overfull \hbox (9.89236pt too wide) in paragraph at lines 391--406 +\OML/cmm/m/it/10 ds[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 391--406 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (17.38756pt too wide) in paragraph at lines 391--406 +[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (160.53036pt too wide) in paragraph at lines 391--406 +[] [][] [][] [] \OML/cmm/m/it/10 e[]ds:$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.410 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no | in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +! Missing $ inserted. + + $ +l.412 |sup_ + s\left[1-2{F}_{S(t)}(s)-a(c_{\epsilon},s) \right]|<\epsilon. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.413 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.413 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 407--414 +[] + [] + + +Overfull \hbox (79.87466pt too wide) in paragraph at lines 407--414 +\OMS/cmsy/m/n/10 j [][] \OML/cmm/m/it/10 a\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c; +s\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 j \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (15.5556pt too wide) in paragraph at lines 407--414 +\OMS/cmsy/m/n/10 j\OT1/cmr/m/n/10 1 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (52.29312pt too wide) in paragraph at lines 407--414 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 F[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 s\OT1/cm +r/m/n/10 )\OMS/cmsy/m/n/10 j ^^T + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 407--414 +\OT1/cmr/m/n/10 1$ + [] + + +Overfull \hbox (14.61455pt too wide) in paragraph at lines 407--414 +\OML/cmm/m/it/10 ^^O > + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 407--414 +\OT1/cmr/m/n/10 0$ + [] + + +Overfull \hbox (8.16055pt too wide) in paragraph at lines 407--414 +\OML/cmm/m/it/10 c[]$ + [] + + +Overfull \hbox (128.03452pt too wide) in paragraph at lines 407--414 +[] [] \OMS/cmsy/m/n/10 j \OML/cmm/m/it/10 < + [] + + +Overfull \hbox (6.83682pt too wide) in paragraph at lines 407--414 +\OML/cmm/m/it/10 ^^O:$ + [] + +Missing character: There is no D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ...e `sec:theor-de-conv' on page \thepage + \space undefined\on@line . +l.418 ...Lebesgue (section \ref{sec:theor-de-conv} + ) implique que +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:theor-de-conv' on page undefined on input line 4 +18. + +Missing character: There is no ) in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.420 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.421 \int + _{-\infty}^{k} \lim_{c\to\infty} a(c,s)e^sds &= \lim_{c\to\infty... +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.421 ...nfty}^{k} \lim_{c\to\infty} a(c,s)e^sds & + = \lim_{c\to\infty}\int_{-... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.422 & + = \lim_{c\to\infty} \int_{-\infty}^{k}\int_{-c}^{c} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.424 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.424 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 415--425 +[] + [] + + +Overfull \hbox (51.56483pt too wide) in paragraph at lines 415--425 +\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 a\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c; s\OT1/ +cmr/m/n/10 )\OML/cmm/m/it/10 e[]\OMS/cmsy/m/n/10 j ^^T + [] + + +Overfull \hbox (25.5973pt too wide) in paragraph at lines 415--425 +\OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (1 + + [] + + +Overfull \hbox (7.94794pt too wide) in paragraph at lines 415--425 +\OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (4.32756pt too wide) in paragraph at lines 415--425 +\OML/cmm/m/it/10 c$ + [] + + +Overfull \hbox (8.93059pt too wide) in paragraph at lines 415--425 +\OML/cmm/m/it/10 e[]$ + [] + + +Overfull \hbox (35.52087pt too wide) in paragraph at lines 415--425 +\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 ^^@1\OML/cmm/m/it/10 ; k\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 415--425 +[] + [] + + +Overfull \hbox (112.58653pt too wide) in paragraph at lines 415--425 +[][] [][] \OML/cmm/m/it/10 a\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c; s\OT1/cmr/m/n/ +10 )\OML/cmm/m/it/10 e[]ds \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (102.03102pt too wide) in paragraph at lines 415--425 +[][] [][] \OML/cmm/m/it/10 a\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c; s\OT1/cmr/m/n/ +10 )\OML/cmm/m/it/10 e[]ds[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 415--425 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (147.02365pt too wide) in paragraph at lines 415--425 +[][] [][] [][] []\OML/cmm/m/it/10 ^^^\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^W\OT1/ +cmr/m/n/10 )\OML/cmm/m/it/10 d^^Wds:$ + [] + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ...c:theoreme-de-fubini' on page \thepage + \space undefined\on@line . +l.429 \ref{sec:theoreme-de-fubini} + ) permet d'inverser l'ordre d'intégration: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:theoreme-de-fubini' on page undefined on input l +ine 429. + +Missing character: There is no ) in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.430 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.431 \lim + _{c\to\infty} \int_{-\infty}^{k}\int_{-c}^{c} \frac{e^{-i\nu +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +l.432 s}}{\pi i\nu}\phi(\nu)d\nu ds & + = \lim_{c\to\infty} \int_{-c}^{c}... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.433 & + = \frac{K}{\pi} \lim_{c\to\infty} \int_{-c}^{c} +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.435 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.435 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 426--436 +[] + [] + + +Overfull \hbox (16.98613pt too wide) in paragraph at lines 426--436 +\OML/cmm/m/it/10 K \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (91.04611pt too wide) in paragraph at lines 426--436 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 E []$ + [] + + +Overfull \hbox (35.63889pt too wide) in paragraph at lines 426--436 +[]$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 426--436 +[] + [] + + +Overfull \hbox (145.98183pt too wide) in paragraph at lines 426--436 +[][] [][] [][] []\OML/cmm/m/it/10 ^^^\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^W\OT1/ +cmr/m/n/10 )\OML/cmm/m/it/10 d^^Wds \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (144.72699pt too wide) in paragraph at lines 426--436 +[][] [][] [][] \OML/cmm/m/it/10 e[]ds[]d^^W[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 426--436 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (120.10127pt too wide) in paragraph at lines 426--436 +[] [][] [][] []\OML/cmm/m/it/10 ^^^\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^W\OT1/cm +r/m/n/10 )\OML/cmm/m/it/10 d^^W:[]$ + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no § in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.437 En remplaçant le résultat \eqref + {eq:fubini-integrale-a-EPPS} dans +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.438 l'équation de départ \eqref + {eq:putepps-1}, on obtient ainsi une +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.440 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no P in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +! Misplaced alignment tab character &. +l.441 P(S(t),K,T) & + = B(t,T)K\left[\frac{1}{2}-\frac{1}{2\pi} \int_{-c}^c +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no = in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no K in font nullfont! +! Missing $ inserted. + + $ +l.441 P(S(t),K,T) &= B(t,T)K\left + [\frac{1}{2}-\frac{1}{2\pi} \int_{-c}^c +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.444 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.444 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 437--445 +[] + [] + + +Overfull \hbox (122.80424pt too wide) in paragraph at lines 437--445 +[] \OML/cmm/m/it/10 :[]$ + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 446--449 +[] + [] + +! Undefined control sequence. +l.450 \section + {Particularités} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 450--452 +[] + [] + +! Undefined control sequence. +l.453 \subsection + {Option sur actions avec dividendes} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no O in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 453--455 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 456--462 +[] + [] + + +Overfull \hbox (22.69446pt too wide) in paragraph at lines 456--462 +\OML/cmm/m/it/10 S[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.468 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no S in font nullfont! +! Missing $ inserted. + + $ +l.470 S(t) = S^ + {*}(t) (1-\delta)^{n(T)}. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.471 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.471 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 463--472 +[] + [] + + +Overfull \hbox (4.8229pt too wide) in paragraph at lines 463--472 +\OML/cmm/m/it/10 ^^N$ + [] + + +Overfull \hbox (21.0128pt too wide) in paragraph at lines 463--472 +\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 T\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (20.84375pt too wide) in paragraph at lines 463--472 +[]$ + [] + + +Overfull \hbox (18.09723pt too wide) in paragraph at lines 463--472 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (32.65286pt too wide) in paragraph at lines 463--472 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )(1 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (28.9826pt too wide) in paragraph at lines 463--472 +\OML/cmm/m/it/10 ^^N\OT1/cmr/m/n/10 )[]\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.476 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no S in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no S in font nullfont! +! Missing $ inserted. + + $ +l.478 S(t) = S^ + {*}(t) e^{-q(T-t)}. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.479 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.479 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 473--480 +[] + [] + + +Overfull \hbox (4.8229pt too wide) in paragraph at lines 473--480 +\OML/cmm/m/it/10 q$ + [] + + +Overfull \hbox (18.09723pt too wide) in paragraph at lines 473--480 +\OML/cmm/m/it/10 S\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (55.40709pt too wide) in paragraph at lines 473--480 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 e[]:$ + [] + +! Undefined control sequence. +l.481 \subsection + {Options sur contrats à terme et taux de change} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no O in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 481--483 +[] + [] + +Missing character: There is no [ in font nullfont! +\citation{black1976pricing} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.484 \cite{black1976pricing} + (p.177) démontre que le prix d'une option sur +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `black1976pricing' on page undefined on input line 484 +. + +Missing character: There is no ] in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.487 résultats \eqref + {eq:prixinitialdividende} et +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.488 \eqref + {eq:pricinitialdivcontinu} en posant $q=r$. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (25.43053pt too wide) in paragraph at lines 484--489 +[][] + [] + + +Overfull \hbox (15.37842pt too wide) in paragraph at lines 484--489 +\OML/cmm/m/it/10 q \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (4.78937pt too wide) in paragraph at lines 484--489 +\OML/cmm/m/it/10 r$ + [] + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no R in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 490--496 +[] + [] + + +Overfull \hbox (9.69566pt too wide) in paragraph at lines 490--496 +\OML/cmm/m/it/10 r[]$ + [] + + +Overfull \hbox (15.37842pt too wide) in paragraph at lines 490--496 +\OML/cmm/m/it/10 q \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (9.69566pt too wide) in paragraph at lines 490--496 +\OML/cmm/m/it/10 r[]$ + [] + +) +! Emergency stop. +<*> chapitre8.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 33 strings out of 493307 + 501 string characters out of 6139904 + 68685 words of memory out of 5000000 + 3585 multiletter control sequences out of 15000+600000 + 4294 words of font info for 16 fonts, out of 8000000 for 9000 + 957 hyphenation exceptions out of 8191 + 20i,10n,26p,214b,148s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/memoire/chapitre8.tex b/memoire/chapitre8.tex new file mode 100644 index 0000000..2471048 --- /dev/null +++ b/memoire/chapitre8.tex @@ -0,0 +1,506 @@ +\chapter{Évaluation d'options} +\label{chap:options} +Un des principaux intérêts de connaître la distribution des rendements +d'un actif est de pouvoir évaluer la valeur de différents produits +dérivés. + +\section{Définitions} +\label{sec:options} +En se référant à \cite{bingham2004risk}, on définit: + +\begin{itemize} +\item Un \textbf{produit dérivé} est un contrat financier dont la + valeur à la date d'échéance $T$ est déterminée par le prix de + l'actif sous-jacent au temps $T$ ou par celles prises au cours de + l'intervalle $\left[0,T \right]$. +\item Une \textbf{option} est un instrument financier qui donne le + droit (et non l'obligation) d'effectuer une transaction avant ou à + une date et pour un prix spécifiés. +\item Une \textbf{option d'achat (de vente) européenne} donne le + \textbf{droit} d'acheter (de vendre) un actif au \textbf{prix + d'exercice} $K$ au temps $T$. +\item Lorsque la valeur actuelle du titre est, par rapport au prix + d'exercice: + \begin{itemize} + \item supérieure $(S(t)>K)$, l'option est dite \textbf{dans le + cours}; + \item égale $(S(t)=K)$, l'option est dite \textbf{au cours}; + \item inférieure $(S(t) K \\ + 0 & S(T) \leq K. + \end{array} \right. +\end{align} + +La valeur du prix à l'échéance est appelée une \textbf{créance + éventuelle}. L'évaluation d'un produit dérivé équivaut à calculer la +valeur espérée actualisée de la réclamation contingente définie par le +contrat. Pour l'option d'achat européenne, on obtient la formule +suivante: +\begin{align} + C(t) &= B(t,T) E \left[C(T) \right] \nonumber\\ + &= B(t,T) \int_{K}^{\infty} (S(T)-K) + d\hat{F}_{S(T)} \label{eq:reclamationcall}. +\end{align} +$B(t,T)$ est la valeur au temps $t$ d'une obligation zéro-coupon au taux sans +risque $r$ d'échéance $T$. $\hat{F}_t(S(T))$ est la fonction de +répartition de la distribution neutre au risque de $S(T)$. + +Les déterminants de la valeur d'une option sont: +\begin{itemize} +\item le prix courant de l'actif $S(t)$ +\item le prix d'exercice $K$ +\item la volatilité de l'actif $\sigma$ +\item le temps d'ici l'échéance $T-t$ +\item le taux d'intérêt sans risque $r$. +\end{itemize} + +On peut aussi récrire \eqref{eq:reclamationcall} avec le logarithme du +prix $k=\ln{K}$. On utilise donc $s(t)=\ln{S(T)}$, dont la fonction de +répartition est définie par ${F}_{s(T)}(y) = Pr\left[\ln{S(T)} < y +\right]$: +\begin{align} + C(t) = B(t,T) \int_{k}^{\infty} (e^{s(T)}-e^{k}) d{F}_{s(T)}. \label{eq:reclamationcallintlog} +\end{align} + +Enfin, une relation fondamentale, appelée parité vente-achat, permet, +dans un scénario sans arbitrage, de calculer le prix d'une option de +vente $P(t)$ (d'achat $C(t)$) en connaissant la valeur: +\begin{itemize} +\item de l'autre lorsqu'elles sont de mêmes échéance et prix + d'exercice +\item de l'obligation $B(t,T)$ +\item initiale du titre sous-jacent $S(t)$ commun aux deux options. +\end{itemize} + +\begin{align} + \label{eq:pariteputcall} + P(t)+S(t) = C(t)+ B(t,T) K. +\end{align} + +Cette relation permettra d'évaluer les deux types d'option à l'aide +d'un seul calcul. + +\subsection{Équation martingale} +\label{sec:equationmartingale} +L'évaluation d'options se fait traditionnellement dans un univers ou +un espace de probabilités neutre au risque, noté $\mathbb{Q}$, +c'est-à-dire un point de vue selon lequel les investisseurs n'exigent +pas une prime de risque. Cette approche a été introduite par +\cite{black1973pricing}. Dans cette perspective, les rendements +espérés futurs sont escomptés au taux sans risque. La distribution des +rendements cumulés \eqref{eq:rendementcumL} répond à la propriété +martingale, selon laquelle la valeur actuelle d'un titre financier +reflète l'ensemble de l'information connue sur ce dernier. Cette +propriété permet de fixer un seul prix pour les +options. \textbf{L'équation martingale} établit l'égalité entre la +valeur espérée du titre au temps $t$ et celle d'un investissement de +même valeur dans un compte en banque créditant le taux sans risque: +\begin{align} + \label{eq:equationmartingale} + E\left[\exp(L_t) \right] = e^{rt}. +\end{align} + +\subsection{Paramètres neutres au risque} +\label{sec:GALrn} + +Afin de pouvoir utiliser les résultats de l'estimation paramétrique +des chapitres précédents pour évaluer le prix de produits dérivés, on +doit tout d'abord identifier les paramètres neutres au risque de la +distribution de Laplace asymétrique généralisée associant le +rendements cumulé $L_t$ au taux d'intérêt sans risque $r$. Pour ce +faire, on utilise l'équation martingale \eqref{eq:equationmartingale} +ainsi que la fonction génératrice des moments \eqref{eq:fgmGAL}. On +obtient alors une expression pour le paramètre de dérive neutre au +risque $\theta_{RN}$: +\begin{align} + e^{rt} &= E\left[\exp(L_t) \right] + = E\left[\exp(R_1+\ldots+R_t) \right] \nonumber\\ + &= M_{R_1+\ldots+R_t}(1) + = (M_{R}(1))^t \nonumber\\ + &= \left(\frac{e^{\theta_{RN}}}{(1-\mu-\sigma^2/2)^{\tau}} \right)^t \nonumber\\ + r &= \ln \left(\frac{e^{\theta_{RN}}}{(1-\mu-\sigma^2/2)^{\tau}} \right) \nonumber\\ + &= \theta_{RN} - \tau\ln(1-\mu-\frac{\sigma^2}{2}) \nonumber\\ + \theta_{RN} &= r + + \tau\ln(1-\mu-\frac{\sigma^2}{2}). \label{eq:martingaleGAL} +\end{align} + +On obtient ainsi le paramètre de correction de la dérive $\omega$ à +partir de l'équation martingale \eqref{eq:martingaleGAL}: +\begin{align} + \label{eq:omegaGAL} + \omega = \tau\ln(1-\mu-\frac{\sigma^2}{2}). +\end{align} + +Cette égalité impose une contrainte aux paramètres $\mu$ et $\sigma$ +qui doit respecter la condition +\begin{align} + \label{eq:inegaliteparamrn} + \mu+\frac{\sigma^2}{2} < 1. +\end{align} + +On obtient ensuite la fonction caractéristique de la mesure neutre au +risque $\mathbb{Q}$ pour $L_T$ en remplaçant $\theta$ par +$\theta_{RN}$: +\begin{align} + \label{eq:fncaractGALrn} + \phi(\xi,L_T) = \frac{\exp\left(i\xi*\left(\ln\left(S_{t}\right)+\left(T-t\right)\left(r+\omega\right)\right)\right)}{\left(1-i\mu\,\xi+\left(\sigma^2\xi^2\right)/2\right)^{\tau\left(T-t\right)}}. +\end{align} + +La distribution neutre au risque des rendements est donc $R_t \sim +GAL(r+\tau\ln(1-\mu-\sigma^2/2),\sigma,\mu,\tau)$. + +\section{Aperçu du modèle de Black-Scholes} +\label{sec:blackscholes} + +Étant donné l'importance du modèle de Black-Scholes en finance, on se +doit de le présenter comme outil de référence lorsque l'on veut le +remplacer. Ce modèle est basé sur les quatre hypothèses suivantes: +\begin{enumerate} +\item Les rendements ont une distribution normale de moyenne $\mu$ et + variance $\sigma^2$, ou, de manière équivalente, le processus de + prix $\lbrace S(t) \rbrace$ est un mouvement brownien géométrique de + dérive $\mu$ et de volatilité $\sigma$. +\item Une obligation zéro-coupon au taux d'intérêt sans risque $r$ + existe sur le marché. +\item Le marché est complet et sans friction. Une position prise sur + ce marché peut être couverte de manière continue. +\item La vente à découvert est autorisée, de plus tous les actifs sont + infiniment divisibles. +\end{enumerate} + +En utilisant l'équation martingale \eqref{eq:equationmartingale}, on +obtient la distribution neutre au risque des rendements, qui est +normale avec une moyenne $r-\frac{\sigma^2}{2}$ et une variance +$\sigma^2$. + +À partir de cette information, on peut dériver la formule de +Black-Scholes pour le prix d'une option d'achat ou de vente +\eqref{eq:pariteputcall} européenne: +\begin{align} + \label{eq:blackscholes} + C(S(t),K,T) &= \Phi(\Delta_1) ~ S(t) - \Phi(\Delta_2) ~ K B(t,T) \\ + \Delta_1 &= \frac{\ln\left(\frac{S(t)}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)}{\sigma\sqrt{T - t}} \nonumber\\ + \Delta_2 &= \Delta_1 - \sigma\sqrt{T - t} \nonumber\\ + P(S(t),K,T) &= \Phi(-\delta_2)K B(t,T) - + \Phi(-\delta_1)S(t). \label{eq:putblackscholes} +\end{align} + +L'avantage de ce modèle est que toutes les données requises pour +évaluer le prix d'options sur des actifs cotés sur les marchés +boursiers, à l'exception de la volatilité, sont accessibles auprès de +fournisseurs d'informations financières. Comme les options sont +négociées sur les marchés financiers, on a aussi accès à leur prix, ce +qui permet d'extrapoler la valeur de la volatilité implicite. C'est +pour cette raison que le modèle, bien qu'il soit basé sur des +hypothèses très restrictives, est toujours utilisé comme point de +référence. + +\section{Méthodes d'évaluation pour options européennes} +\label{sec:methodesoptions} + +Le modèle de Black-Scholes est un des seuls qui présentent une forme +analytique simple pour le prix des options. Bien qu'on puisse en +dériver une pour plusieurs modèles en utilisant les équations +différentielles stochastiques et le calcul d'ÃŽto, les résultats sont +souvent difficiles à utiliser. On préfèrera alors utiliser des +méthodes numériques pour calculer le prix des options. + +\subsection{Méthode de Heston} +\label{sec:heston1993} + +L'approche utilisée par \cite{heston1993closed} est de calculer +directement l'espérance de la créance éventuelle associée à l'option +de vente européenne $P(S(t),K,T-t)$ sous la mesure neutre au risque +$\mathbb{Q}$: +\begin{align} + P(S(t),K,T) &= B(t,T) \int_{0}^{\infty} \max\left(K-S(T),0\right)\cdot d{F}_{S(T)} \label{eq:PutHeston93-1}\\ + &= B(t,T)K \int_{0}^{K} dF_{S(T)} - B(t,T)\int_{0}^{K} S(T)\cdot d{F}_{S(T)} \nonumber\\ + &= B(t,T)K F_{S(t)}(K) - S(t) G_{S(t)}(K). \label{eq:PutHeston93} +\end{align} + +On définit la fonction de répartition $F_{S(t)}(K)$ de la variable +aléatoire $S(t) \leq K$ sous la mesure $\mathbb{Q}$. On définit aussi +la fonction de répartition $G_{S(t)}(K)$ sous une autre mesure +équivalant à $\mathbb{Q}$. Soit $\phi_F(\xi)$ la fonction +caractéristique de $S(T)$ sous $\mathbb{Q}$, celle de cette mesure est +alors définie comme suit: +\begin{align} + \label{eq:mesureGHeston} + \phi_G(\xi) = \frac{\phi_F(\xi-i)}{\phi_F(-i)}. +\end{align} +C'est une transformée d'Esscher de paramètre $h=1$ de la mesure +$\mathbb{Q}$. On remarquera la forme de \eqref{eq:PutHeston93} qui est +très similaire à la formule du prix de l'option de vente du modèle de +Black-Scholes \eqref{eq:putblackscholes}. + +Les fonctions de répartition $F_{S(t)}(K)$ et $G_{S(t)}(K)$ +peuvent être évaluées à partir de \eqref{eq:approxinvfncaract} ou de +la méthode du point de selle. + +\cite{carr1999option} notent qu'on ne peut pas inverser +l'approximation \eqref{eq:approxinvfncaract} en utilisant la méthode +de la transformée de Fourier rapide. Cependant, l'approche de la +méthode du point de selle n'a pas été considérée, bien qu'elle puisse +fournir une solution analytique dans plusieurs situations, en +particulier pour la distribution de Laplace asymétrique généralisée +\eqref{eq:saddlepointGAL}. + +\subsection{Méthode de Carr et Madan} +\label{sec:carrmadanfftoptions} + +La méthode de Heston nécessite l'évaluation de deux probabilités, donc +deux inversions de fonctions +caractéristiques. \cite{madan1998variance} ont développé une méthode +qui ne nécessite qu'une seule inversion, décrite par +\cite{epps2007pricing}. Cette méthode nécessite cependant la sélection +d'un paramètre d'amortissement $\theta_{D}$ par tâtonnement afin +d'obtenir de bons résultats. Elle utilise l'expression de la créance +éventuelle \emph{amortie} \footnote{traduction de l'anglais damped} de +l'option d'achat, construite à partir de l'expression +\eqref{eq:reclamationcallintlog}: +\begin{align} + \label{eq:callamortiCarr} + C_{\theta_D}(S(t),k,T) &= e^{\theta_{D}k} C(S(t),k,T) \\ + &= e^{\theta_{D}k} B(t,T) \int_{K}^{\infty} (e^{s(T)}-e^{k}) + d{F}_{s(T)}. +\end{align} + +On applique la transformée de Fourier: +\begin{align} + \label{eq:fouriercallamortiCarr} + \mathcal{F}C_{\theta_D}(\nu) = B(t,T) \frac{\phi\left( \nu - i(1+\theta_D) \ \right)}{(\theta_D+i\nu)(1+\theta_D+i\nu)}. +\end{align} + +Ce résultat est inversé et \emph{amplifié} \footnote{traduction de + l'anglais amplified} pour retrouver le prix de l'option: +\begin{align} + C(S(t),k,T) &= e^{-\theta_{D}k} \mathcal{F}^{-1}\left(\mathcal{F}C_{\theta_D}(\nu) \right) \nonumber\\ + &= \frac{e^{-\theta_{D}k}}{2\pi} \int_{-\infty}^{\infty} e^{-i\nu k} + (\mathcal{F}C_{\theta_D})(\nu)\cdot + d\nu \nonumber\\ + &= \frac{e^{-\theta_{D}k}}{\pi} \int_{0}^{\infty} e^{-i\nu k} + (\mathcal{F}C_{\theta_D})(\nu)\cdot d\nu \qquad \mbox{(par + symétrie)}. \label{eq:prixoptionCarr} +\end{align} + +On peut utiliser la méthode de la transformée de Fourier rapide +(section \ref{sec:methodeFFT}) pour retrouver les prix d'options en se +référant à \cite{carr1999option}. + +On doit donc discrétiser \eqref{eq:prixoptionCarr} et effectuer des +changements de variables pour obtenir la forme \eqref{eq:sommefft}. On +fixe le nombre de points $N$ (idéalement une puissance de deux) et +$\eta$ le pas de discrétisation. On définit l'incrément $\lambda$ pour le +vecteur $k_u$, $u=1,\ldots,N$ des logarithmes des prix d'exercice: +\begin{align} + \lambda = \frac{2\pi}{N\eta}. +\end{align} + +La valeur maximale $b$ de ce prix d'exercice sera: +\begin{align} + b=\frac{N\lambda}{2}. +\end{align} + +Enfin, on obtient la suite des points d'évaluation de la fonction +caractéristique $\nu_j = \eta (j-1)$. L'ensemble de ces substitutions +permet d'obtenir une expression qui est de la forme requise pour +appliquer la méthode de la transformée de Fourier rapide. +\begin{align} + C(S(t),k_u,T) &\approx \frac{e^{-\theta_D\,k}}{\pi} \sum_{j=1}^N e^{-i\nu_j\,k} (\mathcal{F}C_{\theta_D})(\nu_j) \nonumber\\ + &\approx \frac{e^{-\theta_D\,k}}{\pi} \sum_{j=1}^N + e^{-i\lambda\eta(j-1)(u-1)} e^{ib\nu_j} + (\mathcal{F}C_{\theta_D})(\nu_j) \eta. \label{eq:prixoptionFFT} +\end{align} + +C'est essentiellement une intégration par la méthode du trapèze. On +suggère d'implémenter la correction de Simpson, qui permet une +intégration plus précise tout en conservant le même nombre de points +de discrétisation. On obtiendra alors la formule suivante: +\begin{align} + C(S(t),k_u,T) &\approx \frac{e^{-\theta_D\,k}}{\pi} \sum_{j=1}^N + e^{-i\lambda\eta(j-1)(u-1)} e^{ib\nu_j} + (\mathcal{F}C_{\theta_D})(\nu_j) \frac{\eta}{3} \left(3 + (-1)^j + + \delta_{j-1}\right). \label{eq:prixoptionFFTsimpson} +\end{align} + +On peut par la suite calculer un prix d'option pour chaque prix +d'exercice en utilisant une méthode d'interpolation (splines cubiques +par exemple). + +\subsection{Prix d'exercice hors du cours} +\label{sec:outofmoneyfft} + +L'intégration numérique de \eqref{eq:prixoptionCarr} pose problème +quand l'échéance $T-t$ est petite, ou encore le prix d'exercice $K$ +est hors du cours ($k > \ln(S(0))$). Dans ce contexte particulier, +\cite{carr1999option} développent une formule alternative à l'équation +\eqref{eq:prixoptionCarr} pour évaluer le prix de l'option d'achat : +\begin{align} + C(S(t),k,T) &= \frac{1}{\pi\sinh(\theta^{*}_D k)} \int_{0}^{\infty} + e^{-i\nu k} (\mathcal{F}C^{*}_{\theta^{*}_D})(\nu)\cdot + d\nu. \label{eq:prixoptionCarrOOM} +\end{align} +La transformée de Fourier de l'expression du prix de l'option d'achat +européenne est exprimée sous la forme +\begin{align} + \mathcal{F}C^{*}_{\theta^{*}_D} &= \frac{\zeta_T(\nu-i\theta^{*}_D) - \zeta_T(\nu+i\theta^{*}_D)}{2}\\ + \zeta_T(\nu) &= e^{-r(T-t)}\bigg[\frac{1}{1+i\nu} - + \frac{e^{r(T-t)}}{iv} - \frac{\phi(\nu-i)}{\nu(\nu-i)} \bigg]. + \nonumber +\end{align} + +On note que ce paramètre d'amortissement $\theta^{*}_D$ peut être +différent du paramètre $\theta_D$ qui a été utilisé précédemment. + +L'expression à utiliser pour la méthode de la transformée de Fourier +rapide devient +\begin{align} + C^{*}(S(t),k_u,T) &\approx \frac{1}{\pi\sinh(\theta^{*}_D k)} + \sum_{j=1}^N e^{-i\lambda\eta(j-1)(u-1)} e^{ib\nu_j} + (\mathcal{F}C^{*}_{\theta^{*}_D})(\nu_j) \frac{\eta}{3} \left(3 + + (-1)^j + \delta_{j-1}\right). \label{eq:prixoptionFFTsimpsonOOM} +\end{align} +Les paramètres $N ,\eta, \lambda$ et $\nu_j$ prennent les valeurs +utilisées pour évaluer le prix dans la situation où le titre est dans +le cours \eqref{eq:prixoptionFFTsimpson}. + +\subsection{Critique de la méthode de Carr-Madan} +\label{sec:critiquecarrmadanfft} + +Un paramètre d'amortissement $\theta_{D}$ inférieur à la valeur +optimale (lorsque comparé avec la méthode de Heston ou d’Epps) aura +tendance à surestimer le prix des options d'achat +européennes. \cite{itkin2005pricing} démontre que pour certaines +régions de l'espace des paramètres, l'intégrale +\eqref{eq:prixoptionCarr} a un comportement très irrégulier. De plus, +il décrit plusieurs restrictions pour $\theta_{D}$ qui, dans plusieurs +cas, ne permettent pas d'estimation convergente. + +\subsection{Méthode d’Epps} +\label{sec:epps2007} + +\cite{epps2007pricing} propose une méthode qui, contrairement à celle +de Carr et Madan, ne nécessite pas de paramètre d'amortissement et qui +ne requiert aussi qu'une seule inversion de la fonction +caractéristique. Pour ce faire, on se base sur l'expression de +l'option de vente développée précédemment \eqref{eq:PutHeston93-1}, qui +est ensuite exprimée sous la forme du logarithme, à la manière de +\eqref{eq:reclamationcallintlog}. Après l'utilisation de la formule +d'inversion \eqref{eq:gilpelaez2}, on obtient le résultat suivant: +\begin{align} + \label{eq:putepps-1} + E\left[max(K-S_T;0) \right] &= \int_{-\infty}^{k} {F}_{S(T)}(s)e^s\cdot\,ds \nonumber\\ + &= \frac{K}{2}-\frac{1}{2}\int_{-\infty}^{k} \lim_{c\to\infty} + \underbrace{\int_{-c}^{c} \frac{e^{-i\nu s}}{\pi + i\nu}\phi(\nu)d\nu}_{a(c,s)}e^sds. +\end{align} + +La formule d'inversion implique que lorsque $|\lim_{c\to\infty} +a(c,s)| = |1-2{F}_{S(t)}(s)|\leq 1$, pour toute valeur $\epsilon>0$, +$c_{\epsilon}$ existe telle que l'on a le résultat suivant: +\begin{align} + \label{eq:putepps-resultat1} + |sup_s\left[1-2{F}_{S(t)}(s)-a(c_{\epsilon},s) \right]|<\epsilon. +\end{align} + +Dans cette situation, $|a(c,s)e^s|\leq e^s(1+\epsilon)$ lorsque la +constante $c$ est suffisamment grande. Comme la fonction $e^s$ est +intégrable sur le support $(-\infty,k)$, le théorème de convergence +dominée de Lebesgue (section \ref{sec:theor-de-conv}) implique que +l'on peut poser l'égalité suivante: +\begin{align} + \int_{-\infty}^{k} \lim_{c\to\infty} a(c,s)e^sds &= \lim_{c\to\infty}\int_{-\infty}^{k} a(c,s)e^sds \nonumber\\ + &= \lim_{c\to\infty} \int_{-\infty}^{k}\int_{-c}^{c} + \frac{e^{(1-i\nu) s}}{\pi i\nu}\phi(\nu)d\nu ds. \nonumber +\end{align} + +De plus, comme la limite de la double intégrale précédente est égale à +$K-2E\left[max(K-S_T;0) \right]$ qui appartient à l'intervalle +$\left[-K,K\right]$, le théorème de Fubini (section +\ref{sec:theoreme-de-fubini}) permet d'inverser l'ordre d'intégration: +\begin{align} + \lim_{c\to\infty} \int_{-\infty}^{k}\int_{-c}^{c} \frac{e^{-i\nu + s}}{\pi i\nu}\phi(\nu)d\nu ds &= \lim_{c\to\infty} \int_{-c}^{c} \int_{-\infty}^{k}e^{(1-i\nu)s} ds \frac{\phi(\nu)}{\pi i \nu} d\nu \nonumber\\ + &= \frac{K}{\pi} \lim_{c\to\infty} \int_{-c}^{c} + \frac{K^{-i\nu}}{i\nu+\nu^2}\phi(\nu)d\nu. \label{eq:fubini-integrale-a-EPPS} +\end{align} + +En remplaçant le résultat \eqref{eq:fubini-integrale-a-EPPS} dans +l'équation de départ \eqref{eq:putepps-1}, on obtient ainsi une +expression particulièrement simple pour le prix de l'option d'achat: +\begin{align} + P(S(t),K,T) &= B(t,T)K\left[\frac{1}{2}-\frac{1}{2\pi} \int_{-c}^c + K^{-i\nu} \frac{\phi(\nu)}{\nu(i+\nu)} + d\nu\right]. \label{eq:PutEpps8.37} +\end{align} + +Cependant, puisque la forme ne se prête pas à l'utilisation de +l'algorithme de la transformée de Fourier rapide, on a recours à une +procédure d'intégration numérique. + +\section{Particularités} +\label{sec:monnaiescontrats} + +\subsection{Option sur actions avec dividendes} +\label{sec:dividentoptions} + +Lorsqu'une option a, pour titre sous-jacent, une action qui verse des +dividendes, on doit en tenir compte dans l'évaluation de son prix. Si +l'on considère que la valeur au marché de l'action $S^{*}(t)$ a été +évaluée avec la méthode de l'actualisation des flux financiers futurs, +on doit tenir compte de la valeur actualisée des dividendes qui seront +versés avant l'échéance de l'option et la soustraire de ce prix. + +Parfois, les dividendes ne sont pas fixes, mais proportionnels à la +valeur de l'action à la date ex-dividende avec un taux $\delta$. Soit +$n(T)$ le nombre de dates ex-dividende dans l'intervalle $\left] t,T +\right]$, le prix initial $S(t)$ considéré pour le calcul de la valeur +de l'option est alors défini comme suit: +\begin{align} + \label{eq:prixinitialdividende} + S(t) = S^{*}(t) (1-\delta)^{n(T)}. +\end{align} + +Lorsque l'on considère un indice composé de plusieurs titres, on peut +prendre un dividende versé de manière continue à un taux $q$. Dans ce +cas, le prix initial $S(t)$ sera +\begin{align} + \label{eq:pricinitialdivcontinu} + S(t) = S^{*}(t) e^{-q(T-t)}. +\end{align} + +\subsection{Options sur contrats à terme et taux de change} +\label{sec:futureoptions} + +\cite{black1976pricing} (p.177) démontre que le prix d'une option sur +un contrat à terme a le même prix que sur une action dont le taux de +dividende équivaut au taux sans risque. On peut donc utiliser les +résultats \eqref{eq:prixinitialdividende} et +\eqref{eq:pricinitialdivcontinu} en posant $q=r$. + +De même, on peut évaluer le prix d'une option sur une monnaie +étrangère en considérant le taux sans risque étranger $r_f$ comme un +taux de dividende $q=r_f$. Selon la Banque des Règlements +Internationaux, c'est le type d'options le plus transigé sur les +marchés non réglementés en date de 2005, même s'il est moins étudié +que les autres. + + + + + + + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/chapitre9.aux b/memoire/chapitre9.aux new file mode 100644 index 0000000..6cb8b8c --- /dev/null +++ b/memoire/chapitre9.aux @@ -0,0 +1,186 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{buckle1995bayesian} +\@writefile{lof}{\addvspace {10pt}} +\@writefile{lot}{\addvspace {10pt}} +\@writefile{toc}{\contentsline {chapter}{\chapternumberline {9}Exemple d'application}{97}{chapter.9}} +\newlabel{chap:application}{{9}{97}{Exemple d'application}{chapter.9}{}} +\newlabel{chap:application@cref}{{[chapter][9][]9}{97}} +\@writefile{toc}{\contentsline {section}{\numberline {9.1}Description des donn\IeC {\'e}es}{97}{section.9.1}} +\newlabel{sec:analysepA}{{9.1}{97}{Description des données}{section.9.1}{}} +\newlabel{sec:analysepA@cref}{{[section][1][9]9.1}{97}} +\@writefile{lot}{\contentsline {table}{\numberline {9.1}{\ignorespaces Statistiques d'ordre de l'\IeC {\'e}chantillon $R_1$\relax }}{97}{table.caption.29}} +\newlabel{tab:statordreR1}{{9.1}{97}{Statistiques d'ordre de l'échantillon $R_1$\relax }{table.caption.29}{}} +\newlabel{tab:statordreR1@cref}{{[table][1][9]9.1}{97}} +\@writefile{lof}{\contentsline {figure}{\numberline {9.1}{\ignorespaces Repr\IeC {\'e}sentation en s\IeC {\'e}rie chronologique de l'\IeC {\'e}chantillon $R_1$\relax }}{98}{figure.caption.28}} +\newlabel{fig:seriechronoR1}{{9.1}{98}{Représentation en série chronologique de l'échantillon $R_1$\relax }{figure.caption.28}{}} +\newlabel{fig:seriechronoR1@cref}{{[figure][1][9]9.1}{98}} +\@writefile{lot}{\contentsline {table}{\numberline {9.2}{\ignorespaces Valeurs relatives aux premiers moments de l'\IeC {\'e}chantillon $R_1$\relax }}{98}{table.caption.30}} +\newlabel{tab:statmomentsR1}{{9.2}{98}{Valeurs relatives aux premiers moments de l'échantillon $R_1$\relax }{table.caption.30}{}} +\newlabel{tab:statmomentsR1@cref}{{[table][2][9]9.2}{98}} +\@writefile{lot}{\contentsline {table}{\numberline {9.3}{\ignorespaces Test de normalit\IeC {\'e} d'Epps-Pulley pour $R_1$\relax }}{98}{table.caption.32}} +\newlabel{tab:eppspulleyR1}{{9.3}{98}{Test de normalité d'Epps-Pulley pour $R_1$\relax }{table.caption.32}{}} +\newlabel{tab:eppspulleyR1@cref}{{[table][3][9]9.3}{98}} +\@writefile{lof}{\contentsline {figure}{\numberline {9.2}{\ignorespaces Distribution de la variable al\IeC {\'e}atoire $R_1$\relax }}{99}{figure.caption.31}} +\newlabel{fig:distributionR1}{{9.2}{99}{Distribution de la variable aléatoire $R_1$\relax }{figure.caption.31}{}} +\newlabel{fig:distributionR1@cref}{{[figure][2][9]9.2}{99}} +\@writefile{toc}{\contentsline {section}{\numberline {9.2}Estimation}{99}{section.9.2}} +\newlabel{sec:estimation}{{9.2}{99}{Estimation}{section.9.2}{}} +\newlabel{sec:estimation@cref}{{[section][2][9]9.2}{99}} +\@writefile{lof}{\contentsline {figure}{\numberline {9.3}{\ignorespaces Graphique Quantile-Quantile\relax }}{100}{figure.caption.33}} +\newlabel{fig:qqplotR1}{{9.3}{100}{Graphique Quantile-Quantile\relax }{figure.caption.33}{}} +\newlabel{fig:qqplotR1@cref}{{[figure][3][9]9.3}{100}} +\@writefile{lot}{\contentsline {table}{\numberline {9.4}{\ignorespaces Param\IeC {\`e}tres $\theta _1$ de la premi\IeC {\`e}re optimisation\relax }}{100}{table.caption.34}} +\newlabel{tab:premiereoptimR1}{{9.4}{100}{Paramètres $\theta _1$ de la première optimisation\relax }{table.caption.34}{}} +\newlabel{tab:premiereoptimR1@cref}{{[table][4][9]9.4}{100}} +\newlabel{eq:vcov1gaussR1}{{9.2.1}{100}{Estimation}{equation.9.2.0.1}{}} +\newlabel{eq:vcov1gaussR1@cref}{{[equation][1][9,2]9.2.1}{100}} +\newlabel{eq:vcov1eeR1}{{9.2.2}{100}{Estimation}{equation.9.2.0.2}{}} +\newlabel{eq:vcov1eeR1@cref}{{[equation][2][9,2]9.2.2}{100}} +\newlabel{eq:vcov1eemodR1}{{9.2.3}{101}{Estimation}{equation.9.2.0.3}{}} +\newlabel{eq:vcov1eemodR1@cref}{{[equation][3][9,2]9.2.3}{101}} +\@writefile{lot}{\contentsline {table}{\numberline {9.5}{\ignorespaces Param\IeC {\`e}tres $\theta _1$ de la premi\IeC {\`e}re optimisation\relax }}{102}{table.caption.35}} +\newlabel{tab:secondeoptimR1}{{9.5}{102}{Paramètres $\theta _1$ de la première optimisation\relax }{table.caption.35}{}} +\newlabel{tab:secondeoptimR1@cref}{{[table][5][9]9.5}{102}} +\newlabel{eq:vcov1gmmR1}{{9.2.4}{102}{Estimation}{equation.9.2.0.4}{}} +\newlabel{eq:vcov1gmmR1@cref}{{[equation][4][9,2]9.2.4}{102}} +\newlabel{eq:matvcov2R1-gauss}{{9.2.5}{102}{Estimation}{equation.9.2.0.5}{}} +\newlabel{eq:matvcov2R1-gauss@cref}{{[equation][5][9,2]9.2.5}{102}} +\newlabel{eq:matvcov2R1-ee}{{9.2.6}{102}{Estimation}{equation.9.2.0.6}{}} +\newlabel{eq:matvcov2R1-ee@cref}{{[equation][6][9,2]9.2.6}{102}} +\newlabel{eq:matvcov2R1-eemod}{{9.2.7}{102}{Estimation}{equation.9.2.0.7}{}} +\newlabel{eq:matvcov2R1-eemod@cref}{{[equation][7][9,2]9.2.7}{102}} +\@writefile{lot}{\contentsline {table}{\numberline {9.6}{\ignorespaces Param\IeC {\`e}tres des donn\IeC {\'e}es $R_1$\relax }}{103}{table.caption.36}} +\newlabel{tab:parametresdonneesorigineR1}{{9.6}{103}{Paramètres des données $R_1$\relax }{table.caption.36}{}} +\newlabel{tab:parametresdonneesorigineR1@cref}{{[table][6][9]9.6}{103}} +\@writefile{toc}{\contentsline {section}{\numberline {9.3}Approximation}{104}{section.9.3}} +\newlabel{sec:approximationR1}{{9.3}{104}{Approximation}{section.9.3}{}} +\newlabel{sec:approximationR1@cref}{{[section][3][9]9.3}{104}} +\@writefile{lof}{\contentsline {figure}{\numberline {9.4}{\ignorespaces \IeC {\'E}quation du point de selle pour $r=0.01$\relax }}{104}{figure.caption.37}} +\newlabel{fig:equationptselle0.01R1}{{9.4}{104}{Équation du point de selle pour $r=0.01$\relax }{figure.caption.37}{}} +\newlabel{fig:equationptselle0.01R1@cref}{{[figure][4][9]9.4}{104}} +\@writefile{lot}{\contentsline {table}{\numberline {9.7}{\ignorespaces Approximation de la densit\IeC {\'e} de $R_1$\relax }}{104}{table.caption.38}} +\newlabel{tab:approximationdensiteR1}{{9.7}{104}{Approximation de la densité de $R_1$\relax }{table.caption.38}{}} +\newlabel{tab:approximationdensiteR1@cref}{{[table][7][9]9.7}{104}} +\@writefile{lot}{\contentsline {table}{\numberline {9.8}{\ignorespaces Approximation de la fonction de r\IeC {\'e}partition de $R_1$\relax }}{105}{table.caption.39}} +\newlabel{tab:approximationrepartR1}{{9.8}{105}{Approximation de la fonction de répartition de $R_1$\relax }{table.caption.39}{}} +\newlabel{tab:approximationrepartR1@cref}{{[table][8][9]9.8}{105}} +\@writefile{toc}{\contentsline {section}{\numberline {9.4}Graphiques}{105}{section.9.4}} +\newlabel{sec:graphiques}{{9.4}{105}{Graphiques}{section.9.4}{}} +\newlabel{sec:graphiques@cref}{{[section][4][9]9.4}{105}} +\@writefile{lot}{\contentsline {table}{\numberline {9.9}{\ignorespaces Courbes de densit\IeC {\'e}\relax }}{105}{table.caption.40}} +\newlabel{tab:courbesdensite}{{9.9}{105}{Courbes de densité\relax }{table.caption.40}{}} +\newlabel{tab:courbesdensite@cref}{{[table][9][9]9.9}{105}} +\@writefile{lot}{\contentsline {table}{\numberline {9.10}{\ignorespaces Valeur de l'int\IeC {\'e}grale de l'approximation de la densit\IeC {\'e} par la m\IeC {\'e}thode du point de selle \relax }}{105}{table.caption.41}} +\newlabel{tab:intapproxpointselleR1}{{9.10}{105}{Valeur de l'intégrale de l'approximation de la densité par la méthode du point de selle \relax }{table.caption.41}{}} +\newlabel{tab:intapproxpointselleR1@cref}{{[table][10][9]9.10}{105}} +\@writefile{lof}{\contentsline {figure}{\numberline {9.5}{\ignorespaces Densit\IeC {\'e} de $R_1^{*}$ selon la m\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{106}{figure.caption.42}} +\newlabel{fig:densite1R1}{{9.5}{106}{Densité de $R_1^{*}$ selon la méthode des moments généralisée\relax }{figure.caption.42}{}} +\newlabel{fig:densite1R1@cref}{{[figure][5][9]9.5}{106}} +\@writefile{lof}{\contentsline {figure}{\numberline {9.6}{\ignorespaces Densit\IeC {\'e} de $R_1^{*}$ selon la m\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale\relax }}{107}{figure.caption.43}} +\newlabel{fig:densite3R1}{{9.6}{107}{Densité de $R_1^{*}$ selon la méthode de l'équation d'estimation optimale\relax }{figure.caption.43}{}} +\newlabel{fig:densite3R1@cref}{{[figure][6][9]9.6}{107}} +\@writefile{toc}{\contentsline {section}{\numberline {9.5}Tests statistiques}{108}{section.9.5}} +\newlabel{sec:tests-statistiques}{{9.5}{108}{Tests statistiques}{section.9.5}{}} +\newlabel{sec:tests-statistiques@cref}{{[section][5][9]9.5}{108}} +\@writefile{lot}{\contentsline {table}{\numberline {9.11}{\ignorespaces Test du $\chi ^2$\relax }}{108}{table.caption.44}} +\newlabel{tab:testchi2R1}{{9.11}{108}{Test du $\chi ^2$\relax }{table.caption.44}{}} +\newlabel{tab:testchi2R1@cref}{{[table][11][9]9.11}{108}} +\@writefile{lot}{\contentsline {table}{\numberline {9.12}{\ignorespaces Test de Kolmogorov-Smirnov\relax }}{108}{table.caption.45}} +\newlabel{tab:testKSR1}{{9.12}{108}{Test de Kolmogorov-Smirnov\relax }{table.caption.45}{}} +\newlabel{tab:testKSR1@cref}{{[table][12][9]9.12}{108}} +\@writefile{lot}{\contentsline {table}{\numberline {9.13}{\ignorespaces Test de distance minimale bas\IeC {\'e} sur la fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments\relax }}{109}{table.caption.46}} +\newlabel{tab:testDMR1}{{9.13}{109}{Test de distance minimale basé sur la fonction génératrice des moments\relax }{table.caption.46}{}} +\newlabel{tab:testDMR1@cref}{{[table][13][9]9.13}{109}} +\@writefile{toc}{\contentsline {section}{\numberline {9.6}\IeC {\'E}valuation d'options}{109}{section.9.6}} +\newlabel{sec:evaluation-doptions}{{9.6}{109}{Évaluation d'options}{section.9.6}{}} +\newlabel{sec:evaluation-doptions@cref}{{[section][6][9]9.6}{109}} +\@writefile{lot}{\contentsline {table}{\numberline {9.14}{\ignorespaces Caract\IeC {\'e}ristiques de l'option\relax }}{109}{table.caption.47}} +\newlabel{tab:caracteristiqueoptionR1}{{9.14}{109}{Caractéristiques de l'option\relax }{table.caption.47}{}} +\newlabel{tab:caracteristiqueoptionR1@cref}{{[table][14][9]9.14}{109}} +\@writefile{lot}{\contentsline {table}{\numberline {9.15}{\ignorespaces Param\IeC {\`e}tres neutres au risque\relax }}{109}{table.caption.48}} +\newlabel{tab:paramrisqueneutreR1}{{9.15}{109}{Paramètres neutres au risque\relax }{table.caption.48}{}} +\newlabel{tab:paramrisqueneutreR1@cref}{{[table][15][9]9.15}{109}} +\@writefile{lof}{\contentsline {figure}{\numberline {9.7}{\ignorespaces Prix de l'option selon les param\IeC {\`e}tres estim\IeC {\'e}s avec la m\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{110}{figure.caption.49}} +\newlabel{fig:prix1R1-1}{{9.7}{110}{Prix de l'option selon les paramètres estimés avec la méthode des moments généralisée\relax }{figure.caption.49}{}} +\newlabel{fig:prix1R1-1@cref}{{[figure][7][9]9.7}{110}} +\@writefile{lof}{\contentsline {figure}{\numberline {9.8}{\ignorespaces Prix de l'option selon les param\IeC {\`e}tres estim\IeC {\'e}s avec la m\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale\relax }}{111}{figure.caption.50}} +\newlabel{fig:prix1R1-3}{{9.8}{111}{Prix de l'option selon les paramètres estimés avec la méthode de l'équation d'estimation optimale\relax }{figure.caption.50}{}} +\newlabel{fig:prix1R1-3@cref}{{[figure][8][9]9.8}{111}} +\@setckpt{chapitre9}{ +\setcounter{page}{112} +\setcounter{equation}{0} +\setcounter{enumi}{4} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{9} +\setcounter{section}{6} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{130} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{8} +\setcounter{lofdepth}{1} +\setcounter{table}{15} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{30} +\setcounter{Hfootnote}{3} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{103} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/chapitre9.log b/memoire/chapitre9.log new file mode 100644 index 0000000..7db205e --- /dev/null +++ b/memoire/chapitre9.log @@ -0,0 +1,17870 @@ +This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2013.11.30) 2 DEC 2013 21:04 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**chapitre9.tex +(./chapitre9.tex +LaTeX2e <2011/06/27> +Babel <3.9f> and hyphenation patterns for 78 languages loaded. +! Undefined control sequence. +l.1 \chapter + {Exemple d'application} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1 \chapter{E + xemple d'application} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no E in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 1--3 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.4 O + n considère un échantillon $S_1$ formé de l'ensemble des prix +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 4. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 4. +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...eference `prixabbeyn' on page \thepage + \space undefined\on@line . +l.6 le 8 octobre 1991. La table \ref{prixabbeyn} + présente l'ensemble des 50 +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `prixabbeyn' on page undefined on input line 6. + +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{buckle1995bayesian} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.8 \cite{buckle1995bayesian} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `buckle1995bayesian' on page undefined on input line 8 +. + +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 4--9 +[] + [] + + +Overfull \hbox (10.61807pt too wide) in paragraph at lines 4--9 +\OML/cmm/m/it/10 S[]$ + [] + + +Overfull \hbox (22.00699pt too wide) in paragraph at lines 4--9 +\OML/cmm/m/it/10 S[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 4--9 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 4--9 +[] + [] + +! Undefined control sequence. +l.10 \section + {Description des données} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.10 \section{D + escription des données} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 10--12 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.13 O + n évalue tout d'abord les rendements quotidiens $R_1$ à l'aide de +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.14 l'équation \eqref + {eq:rendementlogprix}. On obtient alors 49 +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...e `fig:seriechronoR1' on page \thepage + \space undefined\on@line . +l.16 la figure \ref{fig:seriechronoR1} + sous forme de série chronologique. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `fig:seriechronoR1' on page undefined on input line 1 +6. + +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 13--17 +[] + [] + + +Overfull \hbox (12.07903pt too wide) in paragraph at lines 13--17 +\OML/cmm/m/it/10 R[]$ + [] + + +Overfull \hbox (19.05907pt too wide) in paragraph at lines 13--17 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 13--17 +[] + [] + + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.18 \begin{figure} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.18 \begin{figure}[ + !ht] +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.20 \includegraphics + [height=4in, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no Y in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no f in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.22 \caption + {Représentation en série chronologique de l'échantillon +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no R in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.25 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 18--26 +[] + [] + + +Overfull \hbox (12.07903pt too wide) in paragraph at lines 18--26 +\OML/cmm/m/it/10 R[]$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.27 O + n énumère d'abord quelques propriétés de cet échantillon qui pour... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...nce `tab:statordreR1' on page \thepage + \space undefined\on@line . +l.28 ...analyse. À la table \ref{tab:statordreR1} + , on présente +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:statordreR1' on page undefined on input line 28. + + +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...e `tab:statmomentsR1' on page \thepage + \space undefined\on@line . +l.30 \ref{tab:statmomentsR1} + , on retrouve quelques valeurs relatives aux +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:statmomentsR1' on page undefined on input line 3 +0. + +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 27--32 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 27--32 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 27--32 +[] + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.33 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.33 \begin{table}[ + !ht] +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no Q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.47 \caption + {Statistiques d'ordre de l'échantillon $R_1$} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.49 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 33--50 +[] + [] + + +Overfull \hbox (127.2646pt too wide) in paragraph at lines 33--50 +[] + [] + + +Overfull \hbox (12.07903pt too wide) in paragraph at lines 33--50 +\OML/cmm/m/it/10 R[]$ + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.51 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.51 \begin{table}[ + !ht] +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.63 \caption + {Valeurs relatives aux premiers moments de l'échantillon $R_1$} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no V in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.65 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 51--66 +[] + [] + + +Overfull \hbox (86.96617pt too wide) in paragraph at lines 51--66 +[] + [] + + +Overfull \hbox (12.07903pt too wide) in paragraph at lines 51--66 +\OML/cmm/m/it/10 R[]$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.67 O + n présente maintenant la distribution des rendements sous la forme +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ... `fig:distributionR1' on page \thepage + \space undefined\on@line . +l.68 ...ité à la figure \ref{fig:distributionR1} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `fig:distributionR1' on page undefined on input line +68. + +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 67--69 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 67--69 +[] + [] + + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.70 \begin{figure} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.70 \begin{figure}[ + !ht] +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.72 \includegraphics + [height=4in,width=4in]{../contenus/r/code/ABBEYN-hist... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no Y in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no f in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.73 \caption + {Distribution de la variable aléatoire $R_1$} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.75 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 70--76 +[] + [] + + +Overfull \hbox (12.07903pt too wide) in paragraph at lines 70--76 +\OML/cmm/m/it/10 R[]$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.77 à + € l'aide du test de normalité d'Epps-Pulley (Section +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no à in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ...ce `sec:test-de-epps' on page \thepage + \space undefined\on@line . +l.78 \ref{sec:test-de-epps} + ), on vérifie si la distribution des rendements +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:test-de-epps' on page undefined on input line 78 +. + +Missing character: There is no ) in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 77--81 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 77--81 +[] + [] + + +Overfull \hbox (12.07903pt too wide) in paragraph at lines 77--81 +\OML/cmm/m/it/10 R[]$ + [] + + +Overfull \hbox (16.98955pt too wide) in paragraph at lines 77--81 +\OML/cmm/m/it/10 \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (13.33337pt too wide) in paragraph at lines 77--81 +\OT1/cmr/m/n/10 5%$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.82 O + n évalue d'abord la statistique $EP_T$ +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.83 \eqref + {eq:statistiqueEppsPulley}, puis la statistique modifiée +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.84 $EP_T^{*}$ \eqref + {eq:EppsPulleyMod} étant donné que $T>10$. La table +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...ce `tab:eppspulleyR1' on page \thepage + \space undefined\on@line . +l.85 \ref{tab:eppspulleyR1} + présente les résultats. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:eppspulleyR1' on page undefined on input line 85 +. + +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 82--86 +[] + [] + + +Overfull \hbox (20.6781pt too wide) in paragraph at lines 82--86 +\OML/cmm/m/it/10 EP[]$ + [] + + +Overfull \hbox (20.6781pt too wide) in paragraph at lines 82--86 +\OML/cmm/m/it/10 EP[]$ + [] + + +Overfull \hbox (17.78816pt too wide) in paragraph at lines 82--86 +\OML/cmm/m/it/10 T > + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 82--86 +\OT1/cmr/m/n/10 10$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 82--86 +[] + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.87 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.87 \begin{table}[ + !ht] +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.99 \caption + {Test de normalité d'Epps-Pulley pour $R_1$} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.101 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 87--102 +[] + [] + + +Overfull \hbox (86.9662pt too wide) in paragraph at lines 87--102 +[] + [] + + +Overfull \hbox (12.07903pt too wide) in paragraph at lines 87--102 +\OML/cmm/m/it/10 R[]$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.103 à + ‰tant donné que la valeur $p$ est inférieure au seuil $\alpha$, on +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...erence `fig:qqplotR1' on page \thepage + \space undefined\on@line . +l.107 \ref{fig:qqplotR1} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `fig:qqplotR1' on page undefined on input line 107. + +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 103--108 +[] + [] + + +Overfull \hbox (5.03125pt too wide) in paragraph at lines 103--108 +\OML/cmm/m/it/10 p$ + [] + + +Overfull \hbox (6.43404pt too wide) in paragraph at lines 103--108 +\OML/cmm/m/it/10 $ + [] + + +Overfull \hbox (12.07903pt too wide) in paragraph at lines 103--108 +\OML/cmm/m/it/10 R[]$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 103--108 +[] + [] + + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.109 \begin{figure} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.109 \begin{figure}[ + !ht] +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.111 \includegraphics + [height=4in, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no Y in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no f in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.113 \caption + {Graphique Quantile-Quantile} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no G in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no Q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no Q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.115 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 109--116 +[] + [] + +! Undefined control sequence. +l.117 \section + {Estimation} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.117 \section{E + stimation} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 117--119 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.120 D + ans cette section, on estime les paramètres de la distribution de +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 120--123 +[] + [] + + +Overfull \hbox (12.26738pt too wide) in paragraph at lines 120--123 +\OML/cmm/m/it/10 R[]$ + [] + +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ...ence `sec:GMMtwostep' on page \thepage + \space undefined\on@line . +l.124 ... itérative (Section \ref{sec:GMMtwostep} + ) +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:GMMtwostep' on page undefined on input line 124. + + +Missing character: There is no ) in font nullfont! + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 124--125 +[] + [] + +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...ce `tab:methodesquad' on page \thepage + \space undefined\on@line . +l.126 \ref{tab:methodesquad} + ) +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:methodesquad' on page undefined on input line 12 +6. + +Missing character: There is no ) in font nullfont! + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 125--127 +[] + [] + +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ...sec:equationsquadopt' on page \thepage + \space undefined\on@line . +l.128 \ref{sec:equationsquadopt} + ) et +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:equationsquadopt' on page undefined on input lin +e 128. + +Missing character: There is no ) in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 127--129 +[] + [] + +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ...ence `sec:eqoptmodif' on page \thepage + \space undefined\on@line . +l.130 \ref{sec:eqoptmodif} + ). +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:eqoptmodif' on page undefined on input line 130. + + +Missing character: There is no ) in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 129--131 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.134 l'aide des équations \eqref + {eq:ptdepartGAL}. On effectue ensuite une +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...`tab:premiereoptimR1' on page \thepage + \space undefined\on@line . +l.137 table \ref{tab:premiereoptimR1} + présente les paramètres obtenus +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:premiereoptimR1' on page undefined on input line + 137. + +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 133--139 +[] + [] + + +Overfull \hbox (9.18057pt too wide) in paragraph at lines 133--139 +\OML/cmm/m/it/10 ^^R[]$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 133--139 +[] + [] + + +Overfull \hbox (9.18057pt too wide) in paragraph at lines 133--139 +\OML/cmm/m/it/10 ^^R[]$ + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.140 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 9 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.154 \caption + {Paramètres $\theta_1$ de la première optimisation} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.156 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 140--157 +[] + [] + + +Overfull \hbox (62.08086pt too wide) in paragraph at lines 140--157 +[] + [] + + +Overfull \hbox (9.18057pt too wide) in paragraph at lines 140--157 +\OML/cmm/m/it/10 ^^R[]$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.160 \eqref + {eq:VarAsymptEstEE} en évaluant l'inverse de la +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +! Undefined control sequence. +l.162 \eqref + {eq:Crowder86-th3.3-def-2} et le gradient +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.163 \eqref + {eq:Crowder86-th3.3-def-1}: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 158--164 +[] + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.166 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathbf allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.168 \mathbf{S} + _T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no S in font nullfont! +! Missing $ inserted. + + $ +l.168 \mathbf{S}_ + T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.168 ...}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.169 1.13756e-03 & + -0.00207546& 0.000917316& 7.46064e-06\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.169 1.13756e-03 &-0.00207546& + 0.000917316& 7.46064e-06\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.169 1.13756e-03 &-0.00207546& 0.000917316& + 7.46064e-06\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.170 -2.07546e-03 & + 0.00984423& 0.002340623& 1.24328e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.170 -2.07546e-03 & 0.00984423& + 0.002340623& 1.24328e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.170 ... -2.07546e-03 & 0.00984423& 0.002340623& + 1.24328e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.171 9.17316e-04 & + 0.00234062& 0.003399880& 8.38932e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.171 9.17316e-04 & 0.00234062& + 0.003399880& 8.38932e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.171 9.17316e-04 & 0.00234062& 0.003399880& + 8.38932e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.172 7.46064e-06 & + 0.00124328& 0.000838932& 2.60840e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.172 7.46064e-06 & 0.00124328& + 0.000838932& 2.60840e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.172 7.46064e-06 & 0.00124328& 0.000838932& + 2.60840e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{itemize} on input line 164 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.173 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \begin{itemize} on input line 164 ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.174 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.174 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (50.52533pt too wide) in paragraph at lines 165--175 +[]\OT1/cmr/m/n/10 ([]; \OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 13756\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (17.77783pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 03 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (145.7678pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 002075460\OML/cmm/m/it/10 :\ +OT1/cmr/m/n/10 0009173167\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 46064\OML/cmm/m/it/1 +0 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 06[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 165--175 +\OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 07546\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 030\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 009844230\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0023406231\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 24328\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 03[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 9\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 17316\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 040\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 002340620\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0033998808\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 38932\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 04[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 7\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 46064\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 060\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001243280\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0008389322\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 60840\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 165--175 +\OT1/cmr/m/n/10 04$ + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.176 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathbf allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.178 \mathbf{S} + _T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no S in font nullfont! +! Missing $ inserted. + + $ +l.178 \mathbf{S}_ + T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.178 ...}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.179 1.47029e-03 & + -0.00212833& 0.00133597& 2.84688e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.179 1.47029e-03 &-0.00212833& + 0.00133597& 2.84688e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.179 1.47029e-03 &-0.00212833& 0.00133597& + 2.84688e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.180 -2.12833e-03 & + 0.01115599& 0.00277669& 1.95192e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.180 -2.12833e-03 & 0.01115599& + 0.00277669& 1.95192e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.180 -2.12833e-03 & 0.01115599& 0.00277669& + 1.95192e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.181 1.33597e-03 & + 0.00277669& 0.00396182& 1.18855e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.181 1.33597e-03 & 0.00277669& + 0.00396182& 1.18855e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.181 1.33597e-03 & 0.00277669& 0.00396182& + 1.18855e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.182 2.84688e-05 & + 0.00195192& 0.00118855& 4.92502e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.182 2.84688e-05 & 0.00195192& + 0.00118855& 4.92502e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.182 2.84688e-05 & 0.00195192& 0.00118855& + 4.92502e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{itemize} on input line 164 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.183 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \begin{itemize} on input line 164 ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.184 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.184 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (50.52533pt too wide) in paragraph at lines 175--185 +[]\OT1/cmr/m/n/10 ([]; \OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 47029\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (17.77783pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 03 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (140.76779pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 002128330\OML/cmm/m/it/10 :\ +OT1/cmr/m/n/10 001335972\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 84688\OML/cmm/m/it/10 + e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 05[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 175--185 +\OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 12833\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (150.76782pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 030\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 011155990\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 002776691\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 95192\OML/cmm/m/it/ +10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 03[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 33597\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (150.76782pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 030\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 002776690\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 003961821\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 18855\OML/cmm/m/it/ +10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 03[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 84688\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (150.76782pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 050\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001951920\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 001188554\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 92502\OML/cmm/m/it/ +10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 175--185 +\OT1/cmr/m/n/10 04$ + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.186 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathbf allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.188 \mathbf{S} + _T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no S in font nullfont! +! Missing $ inserted. + + $ +l.188 \mathbf{S}_ + T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.188 ...}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.189 1.03165e-03& + -0.00146603& 0.001145269& 6.63861e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.189 1.03165e-03& -0.00146603& + 0.001145269& 6.63861e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.189 1.03165e-03& -0.00146603& 0.001145269& + 6.63861e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.190 -1.46603e-03& + 0.00812868& 0.001989247& 1.17588e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.190 -1.46603e-03& 0.00812868& + 0.001989247& 1.17588e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.190 ... -1.46603e-03& 0.00812868& 0.001989247& + 1.17588e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.191 1.14527e-03& + 0.00198925& 0.003435163& 8.33622e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.191 1.14527e-03& 0.00198925& + 0.003435163& 8.33622e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.191 1.14527e-03& 0.00198925& 0.003435163& + 8.33622e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.192 6.63861e-05& + 0.00117588& 0.000833622& 2.71162e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.192 6.63861e-05& 0.00117588& + 0.000833622& 2.71162e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.192 6.63861e-05& 0.00117588& 0.000833622& + 2.71162e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{itemize} on input line 164 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.193 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \begin{itemize} on input line 164 ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.194 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.194 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (50.52533pt too wide) in paragraph at lines 185--195 +[]\OT1/cmr/m/n/10 ([]; \OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 03165\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (17.77783pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 03 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (145.7678pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001466030\OML/cmm/m/it/10 :\ +OT1/cmr/m/n/10 0011452696\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 63861\OML/cmm/m/it/1 +0 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 05[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 185--195 +\OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 46603\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 030\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 008128680\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0019892471\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 17588\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 03[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 14527\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 030\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001989250\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0034351638\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 33622\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 04[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 6\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 63861\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 050\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001175880\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0008336222\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 71162\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 185--195 +\OT1/cmr/m/n/10 04$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 197--199 +[] + [] + + +Overfull \hbox (16.98955pt too wide) in paragraph at lines 197--199 +\OML/cmm/m/it/10 \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (13.33337pt too wide) in paragraph at lines 197--199 +\OT1/cmr/m/n/10 5%$ + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! + +Overfull \hbox (233.07465pt too wide) in paragraph at lines 200--211 +[] + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! + +Overfull \hbox (233.07465pt too wide) in paragraph at lines 212--223 +[] + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 9 in font nullfont! + +Overfull \hbox (233.07465pt too wide) in paragraph at lines 224--235 +[] + [] + +Missing character: There is no à in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ´ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ... `tab:secondeoptimR1' on page \thepage + \space undefined\on@line . +l.243 \ref{tab:secondeoptimR1} + sont donc ceux obtenus avec la convergence de +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:secondeoptimR1' on page undefined on input line +243. + +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ...ence `sec:GMMtwostep' on page \thepage + \space undefined\on@line . +l.244 ...©ratif de la section \ref{sec:GMMtwostep} + avec un +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:GMMtwostep' on page undefined on input line 244. + + +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.245 critère d'arrêt $\epsilon=10^{-7}$ \eqref + {eq:criterearret}. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 238--246 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 238--246 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 238--246 +[] + [] + + +Overfull \hbox (14.61455pt too wide) in paragraph at lines 238--246 +\OML/cmm/m/it/10 ^^O \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (20.73618pt too wide) in paragraph at lines 238--246 +\OT1/cmr/m/n/10 10[]$ + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.247 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.259 \caption + {Paramètres $\theta_1$ de la première optimisation} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.261 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 247--262 +[] + [] + + +Overfull \hbox (62.08086pt too wide) in paragraph at lines 247--262 +[] + [] + + +Overfull \hbox (9.18057pt too wide) in paragraph at lines 247--262 +\OML/cmm/m/it/10 ^^R[]$ + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.266 ... à la convergence de l'algorithme \eqref + {matricevcovparamGMMnc} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.268 moments \eqref + {eq:matponderationproduith} et le gradient +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.269 \eqref + {eq:gradientGMM}. On obtient donc: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.270 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathbf allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.272 \mathbf{S} + _T^{*}(\hat\theta_{OPT};R_1^{*}) = +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no S in font nullfont! +! Missing $ inserted. + + $ +l.272 \mathbf{S}_ + T^{*}(\hat\theta_{OPT};R_1^{*}) = +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.273 \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.274 0.00203708& + 0.00438553& 0.00174636& 0.00154237\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.274 0.00203708& 0.00438553& + 0.00174636& 0.00154237\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.274 0.00203708& 0.00438553& 0.00174636& + 0.00154237\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.275 0.00438553& + 0.01207044& 0.00239640& 0.00384905\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.275 0.00438553& 0.01207044& + 0.00239640& 0.00384905\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.275 0.00438553& 0.01207044& 0.00239640& + 0.00384905\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.276 0.00174636& + 0.00239640& 0.00220402& 0.00104816\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.276 0.00174636& 0.00239640& + 0.00220402& 0.00104816\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.276 0.00174636& 0.00239640& 0.00220402& + 0.00104816\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.277 0.00154237& + 0.00384905& 0.00104816& 0.00127406 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.277 0.00154237& 0.00384905& + 0.00104816& 0.00127406 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.277 0.00154237& 0.00384905& 0.00104816& + 0.00127406 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.278 \end{bmatrix} + . +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.279 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.279 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 263--280 +[] + [] + + +Overfull \hbox (64.80432pt too wide) in paragraph at lines 263--280 +[]\OT1/cmr/m/n/10 ([]; \OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (191.1117pt too wide) in paragraph at lines 263--280 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 002037080\OML/cmm/m/it/10 :\ +OT1/cmr/m/n/10 004385530\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001746360\OML/cmm/m/i +t/10 :\OT1/cmr/m/n/10 00154237[] + [] + + +Overfull \hbox (191.1117pt too wide) in paragraph at lines 263--280 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 004385530\OML/cmm/m/it/10 :\ +OT1/cmr/m/n/10 012070440\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 002396400\OML/cmm/m/i +t/10 :\OT1/cmr/m/n/10 00384905[] + [] + + +Overfull \hbox (191.1117pt too wide) in paragraph at lines 263--280 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001746360\OML/cmm/m/it/10 :\ +OT1/cmr/m/n/10 002396400\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 002204020\OML/cmm/m/i +t/10 :\OT1/cmr/m/n/10 00104816[] + [] + + +Overfull \hbox (193.88948pt too wide) in paragraph at lines 263--280 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001542370\OML/cmm/m/it/10 :\ +OT1/cmr/m/n/10 003849050\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001048160\OML/cmm/m/i +t/10 :\OT1/cmr/m/n/10 00127406\OML/cmm/m/it/10 :$ + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 281--283 +[] + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.285 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathbf allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.287 \mathbf{S} + _T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no S in font nullfont! +! Missing $ inserted. + + $ +l.287 \mathbf{S}_ + T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.287 ...}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.288 1.13314e-03& + -0.00206602& 0.000919367& 7.53773e-06\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.288 1.13314e-03& -0.00206602& + 0.000919367& 7.53773e-06\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.288 1.13314e-03& -0.00206602& 0.000919367& + 7.53773e-06\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.289 -2.06602e-03 & + 0.00983078& 0.002332240 &1.24238e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.289 -2.06602e-03 & 0.00983078& + 0.002332240 &1.24238e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.289 ... -2.06602e-03 & 0.00983078& 0.002332240 & + 1.24238e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.290 9.19367e-04 & + 0.00233224& 0.003395736& 8.36472e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.290 9.19367e-04 & 0.00233224& + 0.003395736& 8.36472e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.290 9.19367e-04 & 0.00233224& 0.003395736& + 8.36472e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.291 7.53773e-06 & + 0.00124238& 0.000836472& 2.60254e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.291 7.53773e-06 & 0.00124238& + 0.000836472& 2.60254e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.291 7.53773e-06 & 0.00124238& 0.000836472& + 2.60254e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{itemize} on input line 283 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.292 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \begin{itemize} on input line 283 ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.293 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.293 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (50.52533pt too wide) in paragraph at lines 284--294 +[]\OT1/cmr/m/n/10 ([]; \OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 13314\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (17.77783pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 03 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (145.7678pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 002066020\OML/cmm/m/it/10 :\ +OT1/cmr/m/n/10 0009193677\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 53773\OML/cmm/m/it/1 +0 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 06[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 284--294 +\OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 06602\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 030\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 009830780\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0023322401\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 24238\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 03[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 9\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 19367\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 040\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 002332240\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0033957368\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 36472\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 04[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 7\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 53773\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 060\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001242380\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0008364722\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 60254\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 284--294 +\OT1/cmr/m/n/10 04$ + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.295 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathbf allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.297 \mathbf{S} + _T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no S in font nullfont! +! Missing $ inserted. + + $ +l.297 \mathbf{S}_ + T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.297 ...}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.298 1.47327e-03& + -0.00212557& 0.00134221& 2.89119e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.298 1.47327e-03& -0.00212557& + 0.00134221& 2.89119e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.298 1.47327e-03& -0.00212557& 0.00134221& + 2.89119e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.299 -2.12557e-03 & + 0.01116770 &0.00277803& 1.96072e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.299 -2.12557e-03 & 0.01116770 & + 0.00277803& 1.96072e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.299 -2.12557e-03 & 0.01116770 &0.00277803& + 1.96072e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.300 1.34221e-03 & + 0.00277803& 0.00396652& 1.19169e-03 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.300 1.34221e-03 & 0.00277803& + 0.00396652& 1.19169e-03 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.300 1.34221e-03 & 0.00277803& 0.00396652& + 1.19169e-03 \\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.301 2.89119e-05 & + 0.00196072& 0.00119169& 4.95537e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.301 2.89119e-05 & 0.00196072& + 0.00119169& 4.95537e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.301 2.89119e-05 & 0.00196072& 0.00119169& + 4.95537e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{itemize} on input line 283 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.302 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \begin{itemize} on input line 283 ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.303 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.303 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (50.52533pt too wide) in paragraph at lines 294--304 +[]\OT1/cmr/m/n/10 ([]; \OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 47327\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (17.77783pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 03 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (140.76779pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 002125570\OML/cmm/m/it/10 :\ +OT1/cmr/m/n/10 001342212\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 89119\OML/cmm/m/it/10 + e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 05[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 294--304 +\OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 12557\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (150.76782pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 030\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 011167700\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 002778031\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 96072\OML/cmm/m/it/ +10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 03[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 34221\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (150.76782pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 030\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 002778030\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 003966521\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 19169\OML/cmm/m/it/ +10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 03[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 89119\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (150.76782pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 050\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001960720\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 001191694\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 95537\OML/cmm/m/it/ +10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 294--304 +\OT1/cmr/m/n/10 04$ + [] + + +! LaTeX Error: The font size command \normalsize is not defined: + there is probably something wrong with the class file. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +[1 +! Undefined control sequence. + \newlabel{chap:application}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{chap:application}{{}{}} +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} +! Undefined control sequence. + \newlabel{sec:analysepA}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:analysepA}{{}{}} +! Undefined control sequence. + \newlabel{fig:seriechronoR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{fig:seriechronoR1}{{}{}} +! Undefined control sequence. + \newlabel{tab:statordreR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{tab:statordreR1}{{}{}} +! Undefined control sequence. + \newlabel{tab:statmomentsR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{tab:statmomentsR1}{{}{}} +! Undefined control sequence. + \newlabel{fig:distributionR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{fig:distributionR1}{{}{}} +! Undefined control sequence. + \newlabel{tab:eppspulleyR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{tab:eppspulleyR1}{{}{}} +! Undefined control sequence. + \newlabel{fig:qqplotR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{fig:qqplotR1}{{}{}} +! Undefined control sequence. + \newlabel{sec:estimation}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:estimation}{{}{}} +! Undefined control sequence. + ...label{tab:premiereoptimR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{tab:premiereoptimR1}{{}{}} +! Undefined control sequence. + \newlabel{eq:vcov1gaussR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:vcov1gaussR1}{{}{}} +! Undefined control sequence. + \newlabel{eq:vcov1eeR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:vcov1eeR1}{{}{}} +! Undefined control sequence. + \newlabel{eq:vcov1eemodR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:vcov1eemodR1}{{}{}} +! Undefined control sequence. + \newlabel{tab:secondeoptimR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{tab:secondeoptimR1}{{}{}} +! Undefined control sequence. + \newlabel{eq:vcov1gmmR1}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:vcov1gmmR1}{{}{}} +! Undefined control sequence. + ...label{eq:matvcov2R1-gauss}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:matvcov2R1-gauss}{{}{}} +! Undefined control sequence. + \newlabel{eq:matvcov2R1-ee}{{}{\thepage + }} +l.304 \item P + our la méthode de l'équation d'estimation optimale modifiée: +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:matvcov2R1-ee}{{}{}} +] +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.305 \begin{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \mathbf allowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.307 \mathbf{S} + _T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no S in font nullfont! +! Missing $ inserted. + + $ +l.307 \mathbf{S}_ + T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: Environment bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.307 ...}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. +l.308 1.03121e-03& + -0.00146899& 0.001142696& 6.60715e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.308 1.03121e-03& -0.00146899& + 0.001142696& 6.60715e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.308 1.03121e-03& -0.00146899& 0.001142696& + 6.60715e-05\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.309 -1.46899e-03 & + 0.00812703 &0.001987649& 1.17298e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.309 -1.46899e-03 & 0.00812703 & + 0.001987649& 1.17298e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.309 ... -1.46899e-03 & 0.00812703 &0.001987649& + 1.17298e-03\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.310 1.14270e-03 & + 0.00198765& 0.003432414& 8.32388e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.310 1.14270e-03 & 0.00198765& + 0.003432414& 8.32388e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.310 1.14270e-03 & 0.00198765& 0.003432414& + 8.32388e-04\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.311 6.60715e-05 & + 0.00117298& 0.000832388& 2.70299e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.311 6.60715e-05 & 0.00117298& + 0.000832388& 2.70299e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.311 6.60715e-05 & 0.00117298& 0.000832388& + 2.70299e-04 +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{itemize} on input line 283 ended by \end{bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.312 \end{bmatrix} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \begin{itemize} on input line 283 ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.313 \end{align} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.313 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +Overfull \hbox (50.52533pt too wide) in paragraph at lines 304--314 +[]\OT1/cmr/m/n/10 ([]; \OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 03121\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (17.77783pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 03 \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (145.7678pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001468990\OML/cmm/m/it/10 :\ +OT1/cmr/m/n/10 0011426966\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 60715\OML/cmm/m/it/1 +0 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 05[] + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 304--314 +\OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 46899\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 030\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 008127030\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0019876491\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 17298\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 03[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 14270\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 030\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001987650\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0034324148\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 32388\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 04[] + [] + + +Overfull \hbox (45.21194pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 6\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 60715\OML/cmm/m/it/10 e \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (155.76784pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 050\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 001172980\OML/cmm/m/it/10 +:\OT1/cmr/m/n/10 0008323882\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 70299\OML/cmm/m/it +/10 e \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 304--314 +\OT1/cmr/m/n/10 04$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 316--317 +[] + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! + +Overfull \hbox (233.07465pt too wide) in paragraph at lines 318--329 +[] + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 7 in font nullfont! + +Overfull \hbox (233.07465pt too wide) in paragraph at lines 329--340 +[] + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! + +Overfull \hbox (233.07465pt too wide) in paragraph at lines 340--351 +[] + [] + +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 2 in font nullfont! + +Overfull \hbox (233.07465pt too wide) in paragraph at lines 351--362 +[] + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +! Undefined control sequence. +l.364 En utilisant la propriété \eqref + {eq:transparamGALNS}, on retrouve les +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...tresdonneesorigineR1' on page \thepage + \space undefined\on@line . +l.367 \ref{tab:parametresdonneesorigineR1} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:parametresdonneesorigineR1' on page undefined on + input line 367. + +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 364--368 +[] + [] + + +Overfull \hbox (29.61458pt too wide) in paragraph at lines 364--368 +\OML/cmm/m/it/10 R\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (77.19952pt too wide) in paragraph at lines 364--368 +[]\OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) + + [] + + +Overfull \hbox (44.20395pt too wide) in paragraph at lines 364--368 +\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 ]\OML/cm +m/m/it/10 ; t \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (37.2221pt too wide) in paragraph at lines 364--368 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; [] ; \OT1/cmr/m/n/10 50$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 364--368 +[] + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.369 \begin{table} + [!ht] \centering +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.380 \caption + {Paramètres des données $R_1$} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.382 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 369--383 +[] + [] + + +Overfull \hbox (62.08086pt too wide) in paragraph at lines 369--383 +[] + [] + + +Overfull \hbox (12.07903pt too wide) in paragraph at lines 369--383 +\OML/cmm/m/it/10 R[]$ + [] + +! Undefined control sequence. +l.384 \section + {Approximation} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no A in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 384--386 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 387--392 +[] + [] + + +Overfull \hbox (14.23376pt too wide) in paragraph at lines 387--392 +\OML/cmm/m/it/10 r \OMS/cmsy/m/n/10 2 + [] + + +Overfull \hbox (171.66692pt too wide) in paragraph at lines 387--392 +[]$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.394 \eqref + {eq:saddlepoint}. On illustre graphiquement cette équation pour +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...quationptselle0.01R1' on page \thepage + \space undefined\on@line . +l.395 ...la figure \ref{fig:equationptselle0.01R1} + : +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `fig:equationptselle0.01R1' on page undefined on inpu +t line 395. + +Missing character: There is no : in font nullfont! + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.396 \begin{figure} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.398 \includegraphics + [scale=0.5]{../contenus/maxima/pointdeselleGMM.png} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.399 \caption + {Équation du point de selle pour $r=0.01$} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.401 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 393--402 +[] + [] + + +Overfull \hbox (15.34488pt too wide) in paragraph at lines 393--402 +\OML/cmm/m/it/10 r \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (17.77783pt too wide) in paragraph at lines 393--402 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 01$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 393--402 +[] + [] + + +Overfull \hbox (15.34488pt too wide) in paragraph at lines 393--402 +\OML/cmm/m/it/10 r \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (17.77783pt too wide) in paragraph at lines 393--402 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 01$ + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.406 \eqref + {eq:approximationsaddlepointordre1} % et celle +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +! Undefined control sequence. +l.410 de densité ${f}_{R}(r)$ \eqref + {eq:densitekotz2001}, on obtient les +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...proximationdensiteR1' on page \thepage + \space undefined\on@line . +l.411 ...la table \ref{tab:approximationdensiteR1} + . Dans +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:approximationdensiteR1' on page undefined on inp +ut line 411. + +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.413 l'intégrale \eqref + {eq:normalisationsaddle1}. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 403--414 +[] + [] + + +Overfull \hbox (15.24301pt too wide) in paragraph at lines 403--414 +[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (42.77791pt too wide) in paragraph at lines 403--414 +\OT1/cmr/m/n/10 56\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 050951$ + [] + + +Overfull \hbox (30.38425pt too wide) in paragraph at lines 403--414 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (24.02312pt too wide) in paragraph at lines 403--414 +\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 403--414 +[] + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.417 \begin{table} + [ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 9 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.433 \caption + {Approximation de la densité de $R_1$} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.435 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 417--436 +[] + [] + + +Overfull \hbox (162.62717pt too wide) in paragraph at lines 417--436 +[] + [] + + +Overfull \hbox (12.07903pt too wide) in paragraph at lines 417--436 +\OML/cmm/m/it/10 R[]$ + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.439 \eqref + {eq:approximationsaddlepointREPordre1} % et de second ordre +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no R in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...pproximationrepartR1' on page \thepage + \space undefined\on@line . +l.443 ... la table \ref{tab:approximationrepartR1} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:approximationrepartR1' on page undefined on inpu +t line 443. + +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 437--444 +[] + [] + + +Overfull \hbox (31.91895pt too wide) in paragraph at lines 437--444 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 437--444 +[] + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.447 \begin{table} + [ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 4 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.463 \caption + {Approximation de la fonction de répartition de $R_1$} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.465 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 447--466 +[] + [] + + +Overfull \hbox (224.40665pt too wide) in paragraph at lines 447--466 +[] + [] + + +Overfull \hbox (12.07903pt too wide) in paragraph at lines 447--466 +\OML/cmm/m/it/10 R[]$ + [] + +! Undefined control sequence. +l.467 \section + {Graphiques} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no G in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 467--469 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. + ...ence `fig:densite1R1' on page \thepage + \space undefined\on@line . +l.470 ...quement, aux figures \ref{fig:densite1R1} + et +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `fig:densite1R1' on page undefined on input line 470. + + +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. + ...ence `fig:densite3R1' on page \thepage + \space undefined\on@line . +l.471 \ref{fig:densite3R1} + , la fonction de densité de la distribution de +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `fig:densite3R1' on page undefined on input line 471. + + +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ... `tab:courbesdensite' on page \thepage + \space undefined\on@line . +l.474 ...timale. La table \ref{tab:courbesdensite} + décrit +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:courbesdensite' on page undefined on input line +474. + +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.476 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no E in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.489 \caption + {Courbes de densité} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.491 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 470--492 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 470--492 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 470--492 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 470--492 +[] + [] + + +Overfull \hbox (402.3008pt too wide) in paragraph at lines 470--492 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.495 \eqref + {eq:normalisationsaddle1} pour les quatre méthodes se trouve à +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...ntapproxpointselleR1' on page \thepage + \space undefined\on@line . +l.496 la table \ref{tab:intapproxpointselleR1} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:intapproxpointselleR1' on page undefined on inpu +t line 496. + +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.497 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.509 \caption + {Valeur de l'intégrale de l'approximation de la densité pa... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no V in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.511 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 493--512 +[] + [] + + +Overfull \hbox (4.32756pt too wide) in paragraph at lines 493--512 +\OML/cmm/m/it/10 c$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 493--512 +[] + [] + + +Overfull \hbox (144.57686pt too wide) in paragraph at lines 493--512 +[] + [] + + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.513 \begin{figure} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.515 \includegraphics + [height=6in, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no Y in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no f in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.517 \caption + {Densité de $R_1^{*}$ selon la méthode des moments +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.520 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 513--521 +[] + [] + + +Overfull \hbox (12.26738pt too wide) in paragraph at lines 513--521 +\OML/cmm/m/it/10 R[]$ + [] + + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.522 \begin{figure} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.524 \includegraphics + [height=6in, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no Y in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no f in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.526 \caption + {Densité de $R_1^{*}$ selon la méthode de l'équation +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.529 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 522--530 +[] + [] + + +Overfull \hbox (12.26738pt too wide) in paragraph at lines 522--530 +\OML/cmm/m/it/10 R[]$ + [] + + +! LaTeX Error: The font size command \normalsize is not defined: + there is probably something wrong with the class file. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.531 \clearpage + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +[2 +! Undefined control sequence. + ...label{eq:matvcov2R1-eemod}{{}{\thepage + }} +l.531 \clearpage + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{eq:matvcov2R1-eemod}{{}{}} +! Undefined control sequence. + ...arametresdonneesorigineR1}{{}{\thepage + }} +l.531 \clearpage + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{tab:parametresdonneesorigineR1}{{}{}} +! Undefined control sequence. + ...label{sec:approximationR1}{{}{\thepage + }} +l.531 \clearpage + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:approximationR1}{{}{}} +! Undefined control sequence. + ...fig:equationptselle0.01R1}{{}{\thepage + }} +l.531 \clearpage + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{fig:equationptselle0.01R1}{{}{}} +! Undefined control sequence. + ...ab:approximationdensiteR1}{{}{\thepage + }} +l.531 \clearpage + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{tab:approximationdensiteR1}{{}{}} +! Undefined control sequence. + ...tab:approximationrepartR1}{{}{\thepage + }} +l.531 \clearpage + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{tab:approximationrepartR1}{{}{}} +! Undefined control sequence. + \newlabel{sec:graphiques}{{}{\thepage + }} +l.531 \clearpage + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{sec:graphiques}{{}{}} +! Undefined control sequence. + \newlabel{tab:courbesdensite}{{}{\thepage + }} +l.531 \clearpage + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{tab:courbesdensite}{{}{}} +! Undefined control sequence. + ...tab:intapproxpointselleR1}{{}{\thepage + }} +l.531 \clearpage + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{tab:intapproxpointselleR1}{{}{}} +! Undefined control sequence. + \newlabel{fig:densite1R1}{{}{\thepage + }} +l.531 \clearpage + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{fig:densite1R1}{{}{}} +! Undefined control sequence. + \newlabel{fig:densite3R1}{{}{\thepage + }} +l.531 \clearpage + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{fig:densite3R1}{{}{}} +] +! Undefined control sequence. +l.532 \section + {Tests statistiques} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 532--534 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no R in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.538 formule d'inversion \eqref + {eq:approxinvfncaract}. On se rappelle que +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ´ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...ence `tab:testchi2R1' on page \thepage + \space undefined\on@line . +l.544 \ref{tab:testchi2R1} + à partir de la définition \eqref{eq:statchi2}. On +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:testchi2R1' on page undefined on input line 544. + + +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.544 ...hi2R1} à partir de la définition \eqref + {eq:statchi2}. On +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.546 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 9 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.558 \caption + {Test du $\chi^2$} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.560 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 535--561 +[] + [] + + +Overfull \hbox (10.74304pt too wide) in paragraph at lines 535--561 +\OML/cmm/m/it/10 ^^_[]$ + [] + + +Overfull \hbox (12.39168pt too wide) in paragraph at lines 535--561 +\OML/cmm/m/it/10 Q[]$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 535--561 +[] + [] + + +Overfull \hbox (16.98955pt too wide) in paragraph at lines 535--561 +\OML/cmm/m/it/10 \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (13.33337pt too wide) in paragraph at lines 535--561 +\OT1/cmr/m/n/10 5%$ + [] + + +Overfull \hbox (172.78941pt too wide) in paragraph at lines 535--561 +[] + [] + + +Overfull \hbox (10.74304pt too wide) in paragraph at lines 535--561 +\OML/cmm/m/it/10 ^^_[]$ + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 562--565 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...erence `tab:testKSR1' on page \thepage + \space undefined\on@line . +l.571 présentée à la table \ref{tab:testKSR1} + à l'aide de la définition +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:testKSR1' on page undefined on input line 571. + +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.572 \eqref + {eq:statks}. En utilisant un seuil de tolérance de $\alpha=5\%$, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 566--574 +[] + [] + + +Overfull \hbox (12.26738pt too wide) in paragraph at lines 566--574 +\OML/cmm/m/it/10 R[]$ + [] + + +Overfull \hbox (16.75142pt too wide) in paragraph at lines 566--574 +\OML/cmm/m/it/10 D[]$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 566--574 +[] + [] + + +Overfull \hbox (16.98955pt too wide) in paragraph at lines 566--574 +\OML/cmm/m/it/10 \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (13.33337pt too wide) in paragraph at lines 566--574 +\OT1/cmr/m/n/10 5%$ + [] + + +Overfull \hbox (37.7779pt too wide) in paragraph at lines 566--574 +\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 194286$ + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.575 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 8 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.587 \caption + {Test de Kolmogorov-Smirnov} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.589 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 575--590 +[] + [] + + +Overfull \hbox (177.14915pt too wide) in paragraph at lines 575--590 +[] + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 591--594 +[] + [] + +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. + ...sec:test-de-distance' on page \thepage + \space undefined\on@line . +l.599 ... à la section \ref{sec:test-de-distance} + . Avec un seuil +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `sec:test-de-distance' on page undefined on input lin +e 599. + +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...erence `tab:testDMR1' on page \thepage + \space undefined\on@line . +l.602 \ref{tab:testDMR1} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:testDMR1' on page undefined on input line 602. + +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 595--603 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 595--603 +[] + [] + + +Overfull \hbox (16.98955pt too wide) in paragraph at lines 595--603 +\OML/cmm/m/it/10 \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (13.33337pt too wide) in paragraph at lines 595--603 +\OT1/cmr/m/n/10 5%$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 595--603 +[] + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.604 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.616 \caption + {Test de distance minimale basé sur la fonction génératri... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.618 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 604--619 +[] + [] + + +Overfull \hbox (210.91512pt too wide) in paragraph at lines 604--619 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 620--623 +[] + [] + +! Undefined control sequence. +l.624 \section + {Évaluation d'options} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 624--626 +[] + [] + +Missing character: There is no à in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...acteristiqueoptionR1' on page \thepage + \space undefined\on@line . +l.629 \ref{tab:caracteristiqueoptionR1} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:caracteristiqueoptionR1' on page undefined on in +put line 629. + +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 627--630 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 627--630 +[] + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.631 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no à in font cmbx10! +Missing character: There is no © in font cmbx10! +Missing character: There is no T in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no % in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no % in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no % in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.644 \caption + {Caractéristiques de l'option} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no C in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.646 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 631--647 +[] + [] + + +Overfull \hbox (104.58417pt too wide) in paragraph at lines 631--647 +[] + [] + +Missing character: There is no à in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...tresdonneesorigineR1' on page \thepage + \space undefined\on@line . +l.649 \ref{tab:parametresdonneesorigineR1} + , en utilisant l'équation +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:parametresdonneesorigineR1' on page undefined on + input line 649. + +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.651 généralisée \eqref + {eq:martingaleGAL}, on obtient l'ensemble +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. + ...:paramrisqueneutreR1' on page \thepage + \space undefined\on@line . +l.654 ...s. La table \ref{tab:paramrisqueneutreR1} + présente +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `tab:paramrisqueneutreR1' on page undefined on input +line 654. + +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.656 \begin{table} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 4 in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.668 \caption + {Paramètres neutres au risque} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.670 \end{table} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 648--671 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 648--671 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 648--671 +[] + [] + + +Overfull \hbox (22.57411pt too wide) in paragraph at lines 648--671 +[] + [] + +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. + ...rence `fig:prix1R1-1' on page \thepage + \space undefined\on@line . +l.674 optimale aux figures \ref{fig:prix1R1-1} + et \ref{fig:prix1R1-3}. On +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `fig:prix1R1-1' on page undefined on input line 674. + +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. + ...rence `fig:prix1R1-3' on page \thepage + \space undefined\on@line . +l.674 ...ref{fig:prix1R1-1} et \ref{fig:prix1R1-3} + . On +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `fig:prix1R1-3' on page undefined on input line 674. + +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no H in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.681 \begin{figure} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.683 \includegraphics + [height=6in, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no Y in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no f in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.685 \caption + {Prix de l'option selon les paramètres estimés avec la +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.688 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 672--689 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 672--689 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 672--689 +[] + [] + + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.690 \begin{figure} + [!ht] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no ! in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.692 \includegraphics + [height=6in, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no Y in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no f in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.694 \caption + {Prix de l'option selon les paramètres estimés avec la +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.697 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 690--698 +[] + [] + +) +! Emergency stop. +<*> chapitre9.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 53 strings out of 493307 + 904 string characters out of 6139904 + 67685 words of memory out of 5000000 + 3606 multiletter control sequences out of 15000+600000 + 3948 words of font info for 15 fonts, out of 8000000 for 9000 + 957 hyphenation exceptions out of 8191 + 18i,9n,21p,141b,186s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/memoire/chapitre9.tex b/memoire/chapitre9.tex new file mode 100644 index 0000000..1efa428 --- /dev/null +++ b/memoire/chapitre9.tex @@ -0,0 +1,702 @@ +\chapter{Exemple d'application} +\label{chap:application}% + +On considère un échantillon $S_1$ formé de l'ensemble des prix +$S_1(t)$ à la fermeture du titre Abbey National entre le 31 juillet et +le 8 octobre 1991. La table \ref{prixabbeyn} présente l'ensemble des 50 +observations. Cet échantillon a notamment été étudié précédemment par +\cite{buckle1995bayesian}. + +\section{Description des données} +\label{sec:analysepA} + +On évalue tout d'abord les rendements quotidiens $R_1$ à l'aide de +l'équation \eqref{eq:rendementlogprix}. On obtient alors 49 +observations du processus des rendements $R(t)$, que l'on présente à +la figure \ref{fig:seriechronoR1} sous forme de série chronologique. + +\begin{figure}[!ht] + \centering + \includegraphics[height=4in, + width=4in]{../contenus/r/code/ABBEYN-chronologie.pdf} + \caption{Représentation en série chronologique de l'échantillon + $R_1$} + \label{fig:seriechronoR1} +\end{figure} + +On énumère d'abord quelques propriétés de cet échantillon qui pourront +compléter l'analyse. À la table \ref{tab:statordreR1}, on présente +quelques statistiques d'ordre. De plus, à la table +\ref{tab:statmomentsR1}, on retrouve quelques valeurs relatives aux +premiers moments. + +\begin{table}[!ht] + \centering + \begin{tabular}{cccccc} + \hline + \textbf{Statistique d'ordre} & \textbf{Valeur} \\ + \hline + Minimum & -0.027500\\ + 1er quartile & -0.009790\\ + Médiane & -0.003260\\ + 3e Quartile & 0.006620\\ + Maximum & 0.043400\\ + \hline + + \end{tabular} + \caption{Statistiques d'ordre de l'échantillon $R_1$} + \label{tab:statordreR1} +\end{table} + +\begin{table}[!ht] + \centering + \begin{tabular}{cccc} + \hline + \textbf{Statistique} & \textbf{Valeur} \\ + \hline + Moyenne & 0.000206\\ + Variance & 0.000169\\ + Coefficient d'asymétrie & 0.977563\\ + Coefficient d'aplatissement & 4.597114 \\ + \hline + \end{tabular} + \caption{Valeurs relatives aux premiers moments de l'échantillon $R_1$} + \label{tab:statmomentsR1} +\end{table} + +On présente maintenant la distribution des rendements sous la forme +d'une courbe de densité à la figure \ref{fig:distributionR1}. + +\begin{figure}[!ht] + \centering + \includegraphics[height=4in,width=4in]{../contenus/r/code/ABBEYN-histogramme.pdf} + \caption{Distribution de la variable aléatoire $R_1$} + \label{fig:distributionR1} +\end{figure} + +À l'aide du test de normalité d'Epps-Pulley (Section +\ref{sec:test-de-epps}), on vérifie si la distribution des rendements +$R_1$ est significativement différente de la normale. On fixe le seuil +de tolérance $\alpha = 5\% $. + +On évalue d'abord la statistique $EP_T$ +\eqref{eq:statistiqueEppsPulley}, puis la statistique modifiée +$EP_T^{*}$ \eqref{eq:EppsPulleyMod} étant donné que $T>10$. La table +\ref{tab:eppspulleyR1} présente les résultats. + +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Statistique} & \textbf{Valeur} \\ + \hline + $EP_T$ & 0.626033 \\ + $EP_T^{*}$ & 0.635568 \\ + $Z_T$ & 2.44824 \\ + $p$ & 0.007178 \\ + \hline + \end{tabular} + \caption{Test de normalité d'Epps-Pulley pour $R_1$} + \label{tab:eppspulleyR1} +\end{table} + +Étant donné que la valeur $p$ est inférieure au seuil $\alpha$, on +rejette l'hypothèse de normalité de l'échantillon $R_1$. On peut +vérifier cette affirmation à l'aide du graphique de comparaison des +quantiles empiriques avec ceux de la loi normale présenté à la figure +\ref{fig:qqplotR1}. + +\begin{figure}[!ht] + \centering + \includegraphics[height=4in, + width=4in]{../contenus/r/code/ABBEYN-qq.pdf} + \caption{Graphique Quantile-Quantile} + \label{fig:qqplotR1} +\end{figure} + +\section{Estimation} +\label{sec:estimation} + +Dans cette section, on estime les paramètres de la distribution de +Laplace asymétrique généralisée pour l'échantillon de données centrées +et réduites $R_1^{*}$, à l'aide des méthodes: +\begin{itemize} +\item des moments généralisée itérative (Section \ref{sec:GMMtwostep}) +\item d'estimation gaussienne de Whittle (table + \ref{tab:methodesquad}) +\item de l'équation d'estimation optimale de Crowder (Section + \ref{sec:equationsquadopt}) et +\item de l'équation d'estimation optimale modifiée (Section + \ref{sec:eqoptmodif}). +\end{itemize} + +On évalue d'abord le vecteur de paramètres initiaux $\theta_0$ à +l'aide des équations \eqref{eq:ptdepartGAL}. On effectue ensuite une +première optimisation à l'aide de chacune des trois méthodes afin de +pouvoir évaluer la matrice de pondération qui servira à la seconde. La +table \ref{tab:premiereoptimR1} présente les paramètres obtenus +$\theta_1$. + +\begin{table}[!ht] + \centering + \begin{tabular}{lcccc} + \hline + \textbf{Méthode} & $\theta$ & $\sigma$ & $\mu$ & $\tau$ \\ + \hline + Paramètres initiaux & -0.612081 & 0.515932 & 0.325854 & 1.878388 \\ + \hline + Moments généralisée & -0.641646 & 0.625908 & 0.326366 & 1.965995 \\ + Estimation gaussienne & -0.776204 & 0.581154 & 0.385120 & 2.015437 \\ + Équation d'estimation optimale & -0.660300 & 0.645678 & 0.376654 & 1.753069 \\ + Équation d'estimation optimale modifiée & -0.711439 & 0.606642 & 0.362932 & 1.960299 \\ + \hline + \end{tabular} + \caption{Paramètres $\theta_1$ de la première optimisation} + \label{tab:premiereoptimR1} +\end{table} + +On obtient la matrice de variance-covariance des paramètres pour les +méthodes basées sur une équation d'estimation +\eqref{eq:VarAsymptEstEE} en évaluant l'inverse de la +variance-covariance des éléments composant celle-ci +\eqref{eq:Crowder86-th3.3-def-2} et le gradient +\eqref{eq:Crowder86-th3.3-def-1}: +\begin{itemize} +\item Pour la méthode d'estimation gaussienne: + \begin{align} + \label{eq:vcov1gaussR1} + \mathbf{S}_T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} + 1.13756e-03 &-0.00207546& 0.000917316& 7.46064e-06\\ + -2.07546e-03 & 0.00984423& 0.002340623& 1.24328e-03\\ + 9.17316e-04 & 0.00234062& 0.003399880& 8.38932e-04\\ + 7.46064e-06 & 0.00124328& 0.000838932& 2.60840e-04 + \end{bmatrix} + \end{align} +\item Pour la méthode de l'équation d'estimation optimale: + \begin{align} + \label{eq:vcov1eeR1} + \mathbf{S}_T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} + 1.47029e-03 &-0.00212833& 0.00133597& 2.84688e-05\\ + -2.12833e-03 & 0.01115599& 0.00277669& 1.95192e-03\\ + 1.33597e-03 & 0.00277669& 0.00396182& 1.18855e-03\\ + 2.84688e-05 & 0.00195192& 0.00118855& 4.92502e-04 + \end{bmatrix} + \end{align} +\item Pour la méthode de l'équation d'estimation optimale modifiée: + \begin{align} + \label{eq:vcov1eemodR1} + \mathbf{S}_T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} + 1.03165e-03& -0.00146603& 0.001145269& 6.63861e-05\\ + -1.46603e-03& 0.00812868& 0.001989247& 1.17588e-03\\ + 1.14527e-03& 0.00198925& 0.003435163& 8.33622e-04\\ + 6.63861e-05& 0.00117588& 0.000833622& 2.71162e-04 + \end{bmatrix} + \end{align} +\end{itemize} + +On peut ainsi construire des intervalles de confiance pour les +paramètres estimés. On utilise un seuil de tolérance de $\alpha=5\%$: +\begin{itemize} +\item Pour la méthode d'estimation gaussienne:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.8328 & -0.7762 & -0.7197 \\ + $\sigma$ & 0.4148 & 0.5812 & 0.7475 \\ + $\mu$ & 0.2874 & 0.3851 & 0.4829 \\ + $\tau$ & 1.9884 & 2.0154 & 2.0425 \\ + \hline + \end{tabular}\\ + +\item Pour la méthode de l'équation d'estimation optimale:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.7246 & -0.6603 & -0.5960 \\ + $\sigma$ & 0.4686 & 0.6457 & 0.8228 \\ + $\mu$ & 0.2711 & 0.3767 & 0.4822 \\ + $\tau$ & 1.7159 & 1.7531 & 1.7903 \\ + \hline + \end{tabular}\\ + +\item Pour la méthode de l'équation d'estimation optimale modifiée:\\ \\ +\begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.7653 & -0.7114 & -0.6576 \\ + $\sigma$ & 0.4555 & 0.6066 & 0.7578 \\ + $\mu$ & 0.2647 & 0.3629 & 0.4612 \\ + $\tau$ & 1.9327 & 1.9603 & 1.9879 \\ + \hline +\end{tabular}\\ + +\end{itemize} + +À l'aide de l'inverse de la variance-covariance des conditions de +moments et des équations d'estimation, on peut effectuer une seconde +optimisation afin d'obtenir des estimateurs convergents. Pour la +méthode des moments généralisée, on utilisera plutôt une procédure +itérative. Les paramètres présentés à la table +\ref{tab:secondeoptimR1} sont donc ceux obtenus avec la convergence de +l'algorithme itératif de la section \ref{sec:GMMtwostep} avec un +critère d'arrêt $\epsilon=10^{-7}$ \eqref{eq:criterearret}. + +\begin{table}[!ht] + \centering + \begin{tabular}{lcccc} + \hline + \textbf{Méthode} & $\theta$ & $\sigma$ & $\mu$ & $\tau$ \\ + \hline + Moments généralisée & -0.640067 & 0.625431 & 0.324311 & 1.973623 \\ + Estimation gaussienne & -0.775071 & 0.581413 & 0.384342 & 2.016619 \\ + Équation d'estimation optimale & -0.658697 & 0.646516 & 0.376251 & 1.750685 \\ + Équation d'estimation optimale modifiée & -0.712450 & 0.606193 & 0.363196 & 1.961614 \\ + \hline + \end{tabular} + \caption{Paramètres $\theta_1$ de la première optimisation} + \label{tab:secondeoptimR1} +\end{table} + +Pour la méthode des moments généralisée, puisque l'on utilise une +procédure itérative, présenter la première matrice de +variance-covariance des paramètres n'est pas pertinent. On obtient +celle à la convergence de l'algorithme \eqref{matricevcovparamGMMnc} +en évaluant l'inverse de la variance-covariance des conditions de +moments \eqref{eq:matponderationproduith} et le gradient +\eqref{eq:gradientGMM}. On obtient donc: +\begin{align} + \label{eq:vcov1gmmR1} + \mathbf{S}_T^{*}(\hat\theta_{OPT};R_1^{*}) = + \begin{bmatrix} + 0.00203708& 0.00438553& 0.00174636& 0.00154237\\ + 0.00438553& 0.01207044& 0.00239640& 0.00384905\\ + 0.00174636& 0.00239640& 0.00220402& 0.00104816\\ + 0.00154237& 0.00384905& 0.00104816& 0.00127406 + \end{bmatrix}. +\end{align} + +Pour les méthodes basées sur une équation d'estimation, on obtient, +pour la seconde optimisation, les variances-covariances suivantes: +\begin{itemize} +\item Pour la méthode d'estimation gaussienne: + \begin{align} + \label{eq:matvcov2R1-gauss} + \mathbf{S}_T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} + 1.13314e-03& -0.00206602& 0.000919367& 7.53773e-06\\ + -2.06602e-03 & 0.00983078& 0.002332240 &1.24238e-03\\ + 9.19367e-04 & 0.00233224& 0.003395736& 8.36472e-04\\ + 7.53773e-06 & 0.00124238& 0.000836472& 2.60254e-04 + \end{bmatrix} + \end{align} +\item Pour la méthode de l'équation d'estimation optimale: + \begin{align} + \label{eq:matvcov2R1-ee} + \mathbf{S}_T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} + 1.47327e-03& -0.00212557& 0.00134221& 2.89119e-05\\ + -2.12557e-03 & 0.01116770 &0.00277803& 1.96072e-03\\ + 1.34221e-03 & 0.00277803& 0.00396652& 1.19169e-03 \\ + 2.89119e-05 & 0.00196072& 0.00119169& 4.95537e-04 + \end{bmatrix} + \end{align} +\item Pour la méthode de l'équation d'estimation optimale modifiée: + \begin{align} + \label{eq:matvcov2R1-eemod} + \mathbf{S}_T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} + 1.03121e-03& -0.00146899& 0.001142696& 6.60715e-05\\ + -1.46899e-03 & 0.00812703 &0.001987649& 1.17298e-03\\ + 1.14270e-03 & 0.00198765& 0.003432414& 8.32388e-04\\ + 6.60715e-05 & 0.00117298& 0.000832388& 2.70299e-04 + \end{bmatrix} + \end{align} +\end{itemize} + +On peut donc construire des intervalles de confiance: +\begin{itemize} +\item Pour la méthode des moments généralisée:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.715736& -0.640067& -0.564397\\ + $\sigma$ & 0.441236 & 0.625431 & 0.809627\\ + $\mu$ & 0.245602 & 0.324311 & 0.403020\\ + $\tau$ & 1.913780 & 1.973623 & 2.033466\\ + \hline + \end{tabular} \\ +\item Pour la méthode d'estimation gaussienne:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.8315 & -0.7751 & -0.7186 \\ + $\sigma$ & 0.4152 & 0.5814 & 0.7476 \\ + $\mu$ & 0.2866 & 0.3843 & 0.4820 \\ + $\tau$ & 1.9896 & 2.0166 & 2.0437 \\ + \hline + \end{tabular} \\ +\item Pour la méthode de l'équation d'estimation optimale:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.7230 & -0.6587 & -0.5943 \\ + $\sigma$ & 0.4693 & 0.6465 & 0.8237 \\ + $\mu$ & 0.2707 & 0.3763 & 0.4818 \\ + $\tau$ & 1.7134 & 1.7507 & 1.7880 \\ + \hline + \end{tabular} \\ +\item Pour la méthode de l'équation d'estimation optimale modifiée:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.7663 & -0.7124 & -0.6586 \\ + $\sigma$ & 0.4551 & 0.6062 & 0.7573 \\ + $\mu$ & 0.2650 & 0.3632 & 0.4614 \\ + $\tau$ & 1.9341 & 1.9616 & 1.9892 \\ + \hline + \end{tabular} \\ +\end{itemize} + +En utilisant la propriété \eqref{eq:transparamGALNS}, on retrouve les +paramètres, correspondants aux données $R(t)= \sqrt{Var[R_1]} R^{*}(t) ++ E[R_1], t=1,\ldots,50$, présentés à la table +\ref{tab:parametresdonneesorigineR1}. + +\begin{table}[!ht] \centering + \begin{tabular}{lcccc} + \hline + \textbf{Méthode} & $\theta$ & $\sigma$ & $\mu$ & $\tau$ \\ + \hline + Moments généralisée& -0.008119 & 0.008134 & 0.002983 & 1.973623 \\ + Estimation gaussienne& -0.009875 & 0.007562 & 0.003535 & 2.016619 \\ + Équation d'estimation optimale& -0.008361 & 0.008409 & 0.003460 & 1.750685 \\ + Équation d'estimation optimale modifiée& -0.009060 & 0.007884 & 0.003340 & 1.961614 \\ + \hline + \end{tabular} + \caption{Paramètres des données $R_1$} + \label{tab:parametresdonneesorigineR1} +\end{table} + +\section{Approximation} +\label{sec:approximationR1} + +On effectue l'approximation des fonctions de densité et de répartition +pour un ensemble de points $r \in +\left\{-0.03,-0.02,-0.01,0,0.01,0.02,0.03 \right\}$ à l'aide de la +méthode du point de selle. On utilise les paramètres obtenus par la +méthode des moments généralisée. + +On résout d'abord l'équation du point de selle +\eqref{eq:saddlepoint}. On illustre graphiquement cette équation pour +$r=0.01$ à la figure \ref{fig:equationptselle0.01R1} : +\begin{figure}[!ht] + \centering + \includegraphics[scale=0.5]{../contenus/maxima/pointdeselleGMM.png} + \caption{Équation du point de selle pour $r=0.01$} + \label{fig:equationptselle0.01R1} +\end{figure} + +Le point de selle prend alors une valeur de $\hat{s}=56.050951$ dans +cette situation. En utilisant l'approximation de premier ordre +$\hat{f}_{R,1}(r)$ +\eqref{eq:approximationsaddlepointordre1} % et celle +% de second ordre $\hat{f}_{R,2}(r)$ +% \eqref{eq:approximationsaddlepointordre2} +de la fonction de densité, et en comparant le résutat avec la fonction +de densité ${f}_{R}(r)$ \eqref{eq:densitekotz2001}, on obtient les +résultats présentés à la table \ref{tab:approximationdensiteR1}. Dans +les deux cas, la fonction de densité a été normalisée à l'aide de +l'intégrale \eqref{eq:normalisationsaddle1}. + +% latex table generated in R 2.15.2 by xtable 1.7-1 package Mon Aug 19 +% 16:36:59 2013 +\begin{table}[ht] + \centering + \begin{tabular}{cccc} + & \multicolumn{2}{c}{\textbf{Densité}} & \multicolumn{1}{c}{\textbf{Erreur relative}} \\ + \hline + $r$ & $\hat{f}_{R,1}(r)$ & ${f}_{R}(r)$ & $(\hat{f}_{R,1}(r)-{f}_{R}(r))/{{f}_{R}(r)}$ \\ + \hline + -0.03 & 1.561365 & 2.065547 & -0.244091 \\ + -0.02 & 9.038616 & 10.702896 & -0.155498 \\ + -0.01 & 31.216317 & 37.581108 & -0.169361 \\ + 0.00 & 33.015869 & 31.073988 & 0.062492 \\ + 0.01 & 15.941633 & 12.450742 & 0.280376 \\ + 0.02 & 5.988373 & 4.120017 & 0.453483 \\ + 0.03 & 2.026838 & 1.246205 & 0.626409 \\ + \hline + \end{tabular} + \caption{Approximation de la densité de $R_1$} + \label{tab:approximationdensiteR1} +\end{table} + +On évalue ensuite la valeur de la fonction de répartition à l'aide de +l'approximation de premier ordre $\hat{F}_{R,1}(r)$ +\eqref{eq:approximationsaddlepointREPordre1} % et de second ordre +% $\hat{F}_{R,2}(r)$ \eqref{eq:approximationsaddlepointREPordre2} +. On compare celle-ci à la valeur de la fonction de répartition +obtenue en intégrant numériquement la fonction de densité. Les +résultats sont présentés à la table \ref{tab:approximationrepartR1}. + +% latex table generated in R 2.15.2 by xtable 1.7-1 package Mon Aug 19 +% 16:49:26 2013 +\begin{table}[ht] + \centering + \begin{tabular}{cccc} + & \multicolumn{2}{c}{\textbf{Fonction de répartition}} & \multicolumn{1}{c}{\textbf{Erreur relative}} \\ + \hline + $r$ & $\hat{F}_{R,1}(r)$ & ${F}_{R}(r)$ & $(\hat{F}_{R,1}(r)-{F}_{R}(r))/{{F}_{R}(r)}$ \\ + \hline + -0.03 & 0.007402 & 0.011577 & -0.360606 \\ + -0.02 & 0.048942 & 0.064944 & -0.246402 \\ + -0.01 & 0.249911 & 0.292034 & -0.144241 \\ + 0.00 & 0.625225 & 0.681167 & -0.082126 \\ + 0.01 & 0.857830 & 0.889832 & -0.035965 \\ + 0.02 & 0.951508 & 0.965916 & -0.014916 \\ + 0.03 & 0.984393 & 0.990080 & -0.005744 \\ + \hline + \end{tabular} + \caption{Approximation de la fonction de répartition de $R_1$} + \label{tab:approximationrepartR1} +\end{table} + +\section{Graphiques} +\label{sec:graphiques} + +On illustre graphiquement, aux figures \ref{fig:densite1R1} et +\ref{fig:densite3R1}, la fonction de densité de la distribution de +Laplace asymétrique généralisée avec les paramètres estimés par la +méthode des moments généralisée et la méthode de l'équation +d'estimation optimale. La table \ref{tab:courbesdensite} décrit +chacune des courbes. +\begin{table}[!ht] + \centering + \begin{tabular}{lp{12cm}} + \hline + \textbf{Abbréviation} & \textbf{Description} \\ + \hline + Emp. & Données empiriques. \\ + Norm. & Densité de la distribution normale ayant les mêmes moyenne et variance que la variable aléatoire $R$. \\ + Estim. & Densité de la distribution de Laplace asymétrique généralisée avec les paramètres estimés.\\ + Pt selle o.1 & Approximation de premier ordre avec la méthode du point de selle. \\ + FFT & Transformée de Fourier rapide. \\ + \hline + \end{tabular} + \caption{Courbes de densité} + \label{tab:courbesdensite} +\end{table} + +On remarquera au passage que la méthode du point de selle nécessite +une normalisation. La valeur de l'intégrale $c$ +\eqref{eq:normalisationsaddle1} pour les quatre méthodes se trouve à +la table \ref{tab:intapproxpointselleR1}. +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Méthode} & \textbf{Valeur de l'intégrale $c$}\\ + \hline + Moments généralisée & 0.920148 \\ + Estimation gaussienne & 0.932246 \\ + Équation d'estimation optimale & 0.917326 \\ + Équation d'estimation optimale modifiée & 0.925781 \\ + \hline + \end{tabular} + \caption{Valeur de l'intégrale de l'approximation de la densité par la méthode du point de selle } + \label{tab:intapproxpointselleR1} +\end{table} + +\begin{figure}[!ht] + \centering + \includegraphics[height=6in, + width=6in]{../contenus/r/code/ABBEYN-densiteGALmu-7.pdf} + \caption{Densité de $R_1^{*}$ selon la méthode des moments + généralisée} + \label{fig:densite1R1} +\end{figure} + +\begin{figure}[!ht] + \centering + \includegraphics[height=6in, + width=6in]{../contenus/r/code/ABBEYN-densiteGALmu-5.pdf} + \caption{Densité de $R_1^{*}$ selon la méthode de l'équation + d'estimation optimale} + \label{fig:densite3R1} +\end{figure} + +\clearpage +\section{Tests statistiques} +\label{sec:tests-statistiques} + +On effectue le test du $\chi^2$ en utilisant sept classes optimales +déterminées par l'algorithme du logiciel GNU R. On obtient la fonction +de répartition à partir de la fonction caractéristique en utilisant la +formule d'inversion \eqref{eq:approxinvfncaract}. On se rappelle que +cet algorithme produit surtout des erreurs aux extrémités de la +distribution. Par contre, ce test donne davantage d'importance aux +classes ayant un plus grand nombre de données par construction. On +utilisera donc l'inversion plutôt que la méthode du point de selle. On +obtient la statistique $Q_{6}$ présentée à la table +\ref{tab:testchi2R1} à partir de la définition \eqref{eq:statchi2}. On +utilise un seuil de tolérance $\alpha = 5\%$. +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Méthode} & \textbf{Valeur de la statistique $Q_{6}$} \\ + \hline + Moments généralisée & 0.484919 \\ + Estimation gaussienne & 0.473527 \\ + Équation d'estimation optimale & 0.531888 \\ + Équation d'estimation optimale modifiée & 0.494769 \\ + \hline + \end{tabular} + \caption{Test du $\chi^2$} + \label{tab:testchi2R1} +\end{table} + +Ce test ne permet pas de rejeter l'hypothèse de la distribution de +Laplace asymétrique généralisée avec les paramètres obtenus pour +aucune des méthodes d'estimation. + +On effectue maintenant le test de Kolmogorov-Smirnov. Comme celui-ci +est basé sur la fonction de répartition, on utilisera la même méthode +que précédemment, impliquant l'inversion de la fonction +caractéristique. On évalue la fonction de répartition en chaque point +de l'échantillon $R_1^{*}$ et l'on obtient la statistique $D_{49}$ +présentée à la table \ref{tab:testKSR1} à l'aide de la définition +\eqref{eq:statks}. En utilisant un seuil de tolérance de $\alpha=5\%$, +on obtient une valeur critique de $0.194286$. + +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Méthode} & \textbf{Valeur de la statistique $D_{49}$} \\ + \hline + Moments généralisée & 0.0684329 \\ + Estimation gaussienne & 0.0784436 \\ + Équation d'estimation optimale & 0.0758317 \\ + Équation d'estimation optimale modifiée & 0.074668 \\ + \hline + \end{tabular} + \caption{Test de Kolmogorov-Smirnov} + \label{tab:testKSR1} +\end{table} + +Encore une fois, on ne peut pas rejeter l'hypothèse de la distribution +de Laplace asymétrique généralisée avec les paramètres estimés pour +chacune des méthodes. + +Ces deux tests sont approximatifs, puisque les paramètres n'ont pas +été estimés en minimisant la statistique utilisée. Cependant, on +obtient un test asymptotiquement exact en utilisant une statistique de +distance minimale basée sur la fonction génératrice des moments, tel +que développé à la section \ref{sec:test-de-distance}. Avec un seuil +de tolérance de $\alpha=5\%$, on obtient une valeur critique de +67.5048. Les statistiques obtenues sont présentées à la table +\ref{tab:testDMR1}. + +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Méthode} & \textbf{Valeur de la statistique $Td(F_{49},F_{\theta})$} \\ + \hline + Moments généralisée & 0.226891 \\ + Estimation gaussienne & 0.110573 \\ + Équation d'estimation optimale & 0.074020 \\ + Équation d'estimation optimale modifiée & 0.108625 \\ + \hline + \end{tabular} + \caption{Test de distance minimale basé sur la fonction génératrice des moments} + \label{tab:testDMR1} +\end{table} + +On ne peut pas rejeter l'hypothèse de la distribution de Laplace +asymétrique généralisée avec les paramètres estimés pour chacune des +méthodes avec ce test. + +\section{Évaluation d'options} +\label{sec:evaluation-doptions} + +À titre d'exemple, on évaluera une option européenne dont les +différentes caractéristiques figurent à la table +\ref{tab:caracteristiqueoptionR1}. + +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Caractéristique} & \textbf{Valeur} \\ + \hline + Type & Option de vente \\ + Échéance ($T$) & 30 jours \\ + Valeur actuelle du titre ($S(0)$) & 299 \\ + Prix d'exercice ($K$) & entre 95\% et 105\% de la valeur actuelle \\ + Taux sans risque annuel ($r_f$) & 5\% \\ + \hline + \end{tabular} + \caption{Caractéristiques de l'option} + \label{tab:caracteristiqueoptionR1} +\end{table} + +À partir des paramètres de la table +\ref{tab:parametresdonneesorigineR1}, en utilisant l'équation +martingale appliquée à la distribution de Laplace asymétrique +généralisée \eqref{eq:martingaleGAL}, on obtient l'ensemble +correspondant pour la mesure neutre au risque. On rappelle que +celle-ci n'est pas unique étant donné que le processus de Laplace est +un processus de sauts. La table \ref{tab:paramrisqueneutreR1} présente +ces paramètres. +\begin{table}[!ht] + \centering + \begin{tabular}{lcccc} + \hline + & $\theta$ & $\sigma$ & $\mu$ & $\tau$ \\ + \hline + Moments généralisée & -0.005824 & 0.008134 & 0.002983 & 1.973623 \\ + Estimation gaussienne & -0.007062 & 0.007562 & 0.003535 & 2.016619 \\ + Équation d'estimation optimale & -0.005993 & 0.008409 & 0.003460 & 1.750685 \\ + Équation d'estimation optimale modifiée & -0.006487 & 0.007884 & 0.003340 & 1.961614 \\ + \hline + \end{tabular} + \caption{Paramètres neutres au risque} + \label{tab:paramrisqueneutreR1} +\end{table} + +On présente les graphiques de la valeur du prix de l'option de vente +pour les méthodes des moments généralisée et de l'équation d'estimation +optimale aux figures \ref{fig:prix1R1-1} et \ref{fig:prix1R1-3}. On +peut facilement remarquer le manque de précision de l'approche de +Carr-Madan, qui s'approche de la courbe de Black-Scholes lorsque le +titre est dans la monnaie et qui se met à osciller dès que le titre +est hors de la monnaie. Les méthodes de Epps et de Heston donnent des +résultats très similaires, et l'approximation du point de selle +d'ordre 1 est très précise dans ce contexte. +\begin{figure}[!ht] + \centering + \includegraphics[height=6in, + width=6in]{../contenus/r/code/ABBEYN-callGAL-7.pdf} + \caption{Prix de l'option selon les paramètres estimés avec la + méthode des moments généralisée} + \label{fig:prix1R1-1} +\end{figure} + +\begin{figure}[!ht] + \centering + \includegraphics[height=6in, + width=6in]{../contenus/r/code/ABBEYN-callGAL-5.pdf} + \caption{Prix de l'option selon les paramètres estimés avec la + méthode de l'équation d'estimation optimale} + \label{fig:prix1R1-3} +\end{figure} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/conclusion.aux b/memoire/conclusion.aux new file mode 100644 index 0000000..3d38646 --- /dev/null +++ b/memoire/conclusion.aux @@ -0,0 +1,80 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{toc}{\contentsline {chapter}{Conclusion}{113}{section*.52}} +\@setckpt{conclusion}{ +\setcounter{page}{114} +\setcounter{equation}{0} +\setcounter{enumi}{4} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{9} +\setcounter{section}{6} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{132} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{8} +\setcounter{lofdepth}{1} +\setcounter{table}{15} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{30} +\setcounter{Hfootnote}{3} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{104} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/conclusion.tex b/memoire/conclusion.tex new file mode 100644 index 0000000..942a0ea --- /dev/null +++ b/memoire/conclusion.tex @@ -0,0 +1,47 @@ +\chapter*{Conclusion} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Conclusion} % dans TdM + +%% Revenir sur le papier de Derman, résumé des avantages et +%% inconvénients du modèle, des méthodes d'estimation et d'évaluation +%% d'options + +En guise de conclusion, j'aimerais tout d'abord effectuer un retour +sur les différents éléments introduits au début du premier chapitre, +concernant le risque de modélisation. Le modèle présenté n'en est pas +exempt, bien au contraire. Cependant, il +nécessite moins d'hypothèses restrictives que les autres modèles +présentés pour être valide, bien qu'il exige toujours l'indépendance +des observations. Il tient compte de la possibilité de sauts tout en +conservant une composante de mouvement aléatoire, ce qui décrit +adéquatement les observations empiriques à ce jour. Comme tout modèle +paramétrique, il reste dépendant du nombre et de la qualité des +données disponibles. Étant donné que l'utilisation d'algorithmes +d'optimisation numérique est inévitable, il subsiste un risque +important autour de l'estimation des paramètres et de l'approximation de la distribution. De plus, étant donné qu'il n'existe pas +de mesure neutre au risque unique, l'arbitrage de modèle reste +possible et doit être considéré. L'utilisation d'un échantillon de +données instables à travers le temps peut produire des résultats +inattendus, surtout au niveau de la distribution de la volatilité +historique, un aspect qui pourra être approfondi ultérieurement. +Enfin, subsiste toujours le risque d'erreurs de nature informatique +qui pourraient produire de faux résultats. + +Ce retour permet de constater qu'il y a toujours place à +l'amélioration des outils développés. Entre autres, il pourrait être +pertinent d'étudier les différences entre le comportement à court et à long terme du modèle. En se basant sur la théorie de +l'utilité, on pourrait développer une meilleure approche pour +déterminer les paramètres de la distribution neutre au risque. Il +pourrait aussi être intéressant de développer des mesures de risque +cohérentes pour les processus de Lévy, notamment +avec les avancées de celles basées sur l'entropie. L'extension +multivariée de ce modèle n'a toujours pas été développée dans la +littérature, alors il pourrait être pertinent de s'y attarder, entre +autres pour étudier les titres indiciels et optimiser la composition +de portefeuilles. Enfin, il pourrait être intéressant d'aborder le +problème inverse de l'estimation des paramètres à partir des prix des +produits dérivés observés sur les marchés financiers. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/deed.aux b/memoire/deed.aux new file mode 100644 index 0000000..1e6f2af --- /dev/null +++ b/memoire/deed.aux @@ -0,0 +1,80 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{toc}{\contentsline {chapter}{Contrat de partage}{137}{section*.64}} +\@setckpt{deed}{ +\setcounter{page}{138} +\setcounter{equation}{0} +\setcounter{enumi}{2} +\setcounter{enumii}{2} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{-1} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{3} +\setcounter{section}{0} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{156} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{1} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{63} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{3} +\setcounter{Item}{35} +\setcounter{Hfootnote}{3} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{135} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/deed.log b/memoire/deed.log new file mode 100644 index 0000000..aee4344 --- /dev/null +++ b/memoire/deed.log @@ -0,0 +1,1022 @@ +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.7.6) 19 SEP 2013 20:25 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**\input deed.tex +(./deed.tex + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1 \begin{center} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 1--1 +[] + [] + +! Undefined control sequence. +l.2 \includegraphics + [height=7mm,keepaspectratio=true]{by-sa}\\% +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.6 \href + {http://creativecommons.org/licenses/by-sa/2.5/ca/deed.fr}{% +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no / in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 5--12 +[] + [] + +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ; in font nullfont! + +Overfull \hbox (42.53448pt too wide) in paragraph at lines 13--15 +\OT1/cmr/bx/n/10 partager + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 13--15 +\OT1/cmr/m/n/10 ^^[ + [] + +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ; in font nullfont! + +Overfull \hbox (38.86095pt too wide) in paragraph at lines 15--16 +\OT1/cmr/bx/n/10 remixer + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 15--16 +\OT1/cmr/m/n/10 ^^[ + [] + +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 16--17 +\OT1/cmr/m/n/10 ^^[ + [] + +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 19--20 +[] + [] + + +! LaTeX Error: Environment tabularx undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.21 \begin{tabularx} + {\linewidth}{@{}lX@{}} +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing number, treated as zero. + + } +l.21 \begin{tabularx}{\linewidth} + {@{}lX@{}} +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + } +l.21 \begin{tabularx}{\linewidth} + {@{}lX@{}} +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +Missing character: There is no @ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no @ in font nullfont! +! Undefined control sequence. + \includegraphics + [height=11mm,keepaspectratio=true]{by} +l.23 ...ics[height=11mm,keepaspectratio=true]{by}} + & +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no y in font nullfont! +! Misplaced alignment tab character &. +l.23 ...s[height=11mm,keepaspectratio=true]{by}} & + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no . in font nullfont! +! Undefined control sequence. + \includegraphics + [height=11mm,keepaspectratio=true]{sa} +l.29 ...ics[height=11mm,keepaspectratio=true]{sa}} + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +! Misplaced alignment tab character &. +l.30 & + \textbf{Partage à l'identique} --- Si vous modifiez, transformez +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +Missing character: There is no à in font cmbx10! +Missing character: There is no   in font cmbx10! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{tabularx}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.34 \end{tabularx} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +) +! Emergency stop. +<*> \input deed.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 14 strings out of 493485 + 199 string characters out of 3143527 + 48818 words of memory out of 3000000 + 3420 multiletter control sequences out of 15000+200000 + 3948 words of font info for 15 fonts, out of 3000000 for 9000 + 957 hyphenation exceptions out of 8191 + 14i,1n,13p,105b,110s stack positions out of 5000i,500n,10000p,200000b,50000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/memoire/deed.tex b/memoire/deed.tex new file mode 100644 index 0000000..093a15e --- /dev/null +++ b/memoire/deed.tex @@ -0,0 +1,46 @@ +\chapter*{Contrat de partage} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Contrat de + partage} % dans TdM + +Cette création est mise à disposition selon le contrat +\emph{Attribution - Partage dans les Mêmes Conditions 4.0 International} +de Creative Commons disponible à l'adresse +\url{http://creativecommons.org/licenses/by-sa/4.0/deed.fr} + +En vertu de ce contrat, vous êtes autorisé à: +\begin{itemize} +\item \textbf{Partager} --- copier, distribuer et communiquer le + matériel par tous moyens et sous tous formats +\item \textbf{Adapter} --- remixer, transformer et créer à partir du + matériel pour toute utilisation, y compris commerciale. +\end{itemize} + +L'Offrant ne peux retirer les autorisations concédées par la +licence tant que vous appliquez les termes de cette licence. \\ + +Selon les conditions suivantes:\\ + + \begin{tabularx}{\linewidth}{@{}lX@{}} + \raisebox{-9mm}[0mm][13mm]{% + \includegraphics[height=11mm,keepaspectratio=true]{by}} & + \textbf{Attribution} --- Vous devez créditer l'Oeuvre, intégrer un lien vers la licence et indiquer si des modifications ont été effectuées à l'Oeuvre. Vous devez indiquer ces informations par tous les moyens possibles mais vous ne pouvez pas suggérer que l'Offrant vous soutient ou soutient la façon dont vous avez utilisé son Oeuvre. \\ \\ + \raisebox{-9mm}{\includegraphics[height=11mm,keepaspectratio=true]{sa}} + & \textbf{Partage à l'identique} --- Dans le cas où vous effectuez + un remix, que vous transformez, ou créez à partir du matériel + composant l'Oeuvre originale, vous devez diffuser l'Oeuvre + modifiée dans les même conditions, c'est-à-dire avec la même + licence avec laquelle l'Oeuvre originale a été diffusée. + \end{tabularx} \\ \\ + \textbf{Aucune autre restriction} --- Vous n'êtes pas autorisé à + appliquer des conditions légales ou des mesures techniques qui + restreindraient légalement autrui à utiliser l'Oeuvre dans les + conditions décrites par la licence. + + \begin{center} + \includegraphics[height=7mm,keepaspectratio=true]{by-sa}\\% + \end{center} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/gabarit-maitrise.aux b/memoire/gabarit-maitrise.aux new file mode 100644 index 0000000..918ed20 --- /dev/null +++ b/memoire/gabarit-maitrise.aux @@ -0,0 +1,122 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand*{\memsetcounter}[2]{} +\catcode `:\active +\catcode `;\active +\catcode `!\active +\catcode `?\active +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} +\bibstyle{plainnatmod} +\select@language{french} +\@writefile{toc}{\select@language{french}} +\@writefile{lof}{\select@language{french}} +\@writefile{lot}{\select@language{french}} +\select@language{french} +\@writefile{toc}{\select@language{french}} +\@writefile{lof}{\select@language{french}} +\@writefile{lot}{\select@language{french}} +\@input{resume.aux} +\@input{abstract.aux} +\@writefile{toc}{\contentsline {chapter}{Table des mati{\`e}res}{vii}{section*.5}} +\citation{madan1990variance} +\@writefile{toc}{\contentsline {chapter}{Liste des tableaux}{xi}{section*.6}} +\citation{mitchell1916critique} +\@writefile{toc}{\contentsline {chapter}{Liste des figures}{xiii}{section*.7}} +\@input{remerciements.aux} +\@input{introduction.aux} +\@input{chapitre1.aux} +\@input{chapitre2.aux} +\@input{chapitre3.aux} +\@input{chapitre4.aux} +\@input{chapitre5.aux} +\@input{chapitre6.aux} +\@input{chapitre7.aux} +\@input{chapitre8.aux} +\@input{chapitre9.aux} +\@input{conclusion.aux} +\@input{annexe1.aux} +\@input{annexe2.aux} +\@input{annexe3.aux} +\bibdata{memoire} +\bibcite{abramowitz1965handbook}{{1}{1965}{{Abramowitz et Stegun}}{{}}} +\bibcite{applebaum2004levy}{{2}{2004}{{Applebaum}}{{}}} +\bibcite{bachelier1900theorie}{{3}{1900}{{Bachelier}}{{}}} +\bibcite{barndorff2001levy}{{4}{2001}{{Barndorff-Nielsen et~al.}}{{Barndorff-Nielsen, Mikosch, et Resnick}}} +\bibcite{berkson1980minimum}{{5}{1980}{{Berkson}}{{}}} +\bibcite{bingham2004risk}{{6}{2004}{{Bingham et Kiesel}}{{}}} +\bibcite{black1976pricing}{{7}{1976}{{Black}}{{}}} +\bibcite{black1973pricing}{{8}{1973}{{Black et Scholes}}{{}}} +\bibcite{buckle1995bayesian}{{9}{1995}{{Buckle}}{{}}} +\bibcite{butler2007saddlepoint}{{10}{2007}{{Butler}}{{}}} +\bibcite{carr1999option}{{11}{1999}{{Carr et Madan}}{{}}} +\bibcite{crowder1986consistency}{{12}{1986}{{Crowder}}{{}}} +\bibcite{crowder1987linear}{{13}{1987}{{Crowder}}{{}}} +\@writefile{toc}{\contentsline {chapter}{Bibliographie}{131}{section*.62}} +\bibcite{daniels1954saddlepoint}{{14}{1954}{{Daniels}}{{}}} +\bibcite{derman1996modelrisk}{{15}{1996}{{Derman}}{{}}} +\bibcite{dodge2004statistique}{{16}{2004}{{Dodge}}{{}}} +\bibcite{epps2007pricing}{{17}{2007}{{Epps}}{{}}} +\bibcite{epps1983test}{{18}{1983}{{Epps et Pulley}}{{}}} +\bibcite{everitt2006cambridge}{{19}{2006}{{Everitt et Skrondal}}{{}}} +\bibcite{fama1993common}{{20}{1993}{{Fama et French}}{{}}} +\bibcite{feuerverger1981efficiency}{{21}{1981}{{Feuerverger et McDunnough}}{{}}} +\bibcite{fox1986large}{{22}{1986}{{Fox et Taqqu}}{{}}} +\bibcite{gil1951note}{{23}{1951}{{Gil-Pelaez}}{{}}} +\bibcite{gourieroux1989statistique}{{24}{1989}{{Gourieroux et Monfort}}{{}}} +\bibcite{hall2005generalized}{{25}{2005}{{Hall}}{{}}} +\bibcite{hamilton1994time}{{26}{1994}{{Hamilton}}{{}}} +\bibcite{hansen1982large}{{27}{1982}{{Hansen}}{{}}} +\bibcite{henze1990approximation}{{28}{1990}{{Henze}}{{}}} +\bibcite{heston1993closed}{{29}{1993}{{Heston}}{{}}} +\bibcite{hinkley1977estimation}{{30}{1977}{{Hinkley et Revankar}}{{}}} +\bibcite{hogg1978introduction}{{31}{1978}{{Hogg et Craig}}{{}}} +\bibcite{itkin2005pricing}{{32}{2005}{{Itkin}}{{}}} +\bibcite{kotz2001laplace}{{33}{2001}{{Kotz et~al.}}{{Kotz, Kozubowski, et Podg{\'o}rski}}} +\bibcite{KOUTROUVELIS01011980}{{34}{1980}{{Koutrouvelis}}{{}}} +\bibcite{kozubowski1999class}{{35}{1999}{{Kozubowski et Podg{\'o}rski}}{{}}} +\bibcite{kozubowski2001asymmetric}{{36}{2001}{{Kozubowski et Podg{\'o}rski}}{{}}} +\bibcite{kyprianou2007introductory}{{37}{2007}{{Kyprianou}}{{}}} +\bibcite{lugannani1980saddle}{{38}{1980}{{Lugannani et Rice}}{{}}} +\bibcite{lukacs1960characteristic}{{39}{1960}{{Lukacs}}{{}}} +\bibcite{luong1987minimum}{{40}{1987}{{Luong et Thompson}}{{}}} +\bibcite{madan1990variance}{{41}{1990}{{Madan et Seneta}}{{}}} +\bibcite{madan1998variance}{{42}{1998}{{Madan et~al.}}{{Madan, Carr, et Chang}}} +\bibcite{mandelbrot1963variation}{{43}{1963}{{Mandelbrot}}{{}}} +\bibcite{merton1976option}{{44}{1976}{{Merton}}{{}}} +\bibcite{mitchell1916critique}{{45}{1916}{{Mitchell}}{{}}} +\bibcite{newey1994large}{{46}{1994}{{Newey et McFadden}}{{}}} +\bibcite{newey1987hypothesis}{{47}{1987}{{Newey et West}}{{}}} +\bibcite{praetz1972distribution}{{48}{1972}{{Praetz}}{{}}} +\bibcite{press1967compound}{{49}{1967}{{Press}}{{}}} +\bibcite{sato1999levy}{{50}{1999}{{Sato}}{{}}} +\bibcite{schoutens2003levy}{{51}{2003}{{Schoutens}}{{}}} +\bibcite{RpackageVarianceGamma}{{52}{2012}{{Scott et Dong}}{{}}} +\bibcite{seneta2004fitting}{{53}{2004}{{Seneta}}{{}}} +\bibcite{shapiro1965analysis}{{54}{1965}{{Shapiro et Wilk}}{{}}} +\bibcite{shephard1991characteristic}{{55}{1991}{{Shephard}}{{}}} +\bibcite{spiegel1999schaum}{{56}{1999}{{Spiegel et Liu}}{{}}} +\bibcite{stuart1987kendall}{{57}{1987}{{Stuart et Ord}}{{}}} +\bibcite{teschl2004topics}{{58}{2004}{{Teschl}}{{}}} +\bibcite{walterlevy}{{59}{}{{Walter}}{{}}} +\bibcite{wang2003evaluating}{{60}{2003}{{Wang et~al.}}{{Wang, Tsang, et Marsaglia}}} +\bibcite{wendel1961non}{{61}{1961}{{Wendel}}{{}}} +\bibcite{wolfowitz1957minimum}{{62}{1957}{{Wolfowitz}}{{}}} +\bibcite{wooldridge2001econometric}{{63}{2001}{{Wooldridge}}{{}}} +\@input{deed.aux} +\memsetcounter{lastsheet}{155} +\memsetcounter{lastpage}{137} diff --git a/memoire/gabarit-maitrise.bbl b/memoire/gabarit-maitrise.bbl new file mode 100644 index 0000000..438bce6 --- /dev/null +++ b/memoire/gabarit-maitrise.bbl @@ -0,0 +1,367 @@ +\begin{thebibliography}{63} +\providecommand{\natexlab}[1]{#1} +\providecommand{\url}[1]{\texttt{#1}} +\expandafter\ifx\csname urlstyle\endcsname\relax + \providecommand{\doi}[1]{doi: #1}\else + \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi + +\bibitem[Abramowitz et Stegun(1965)]{abramowitz1965handbook} +Milton Abramowitz et Irene~A Stegun. +\newblock \emph{Handbook of mathematical functions: with formulas, graphs, and + mathematical tables}, volume~55. +\newblock Dover publications, 1965. + +\bibitem[Applebaum(2004)]{applebaum2004levy} +David Applebaum. +\newblock L{\'e}vy processes: From probability to finance and quantum groups. +\newblock \emph{Notices of the AMS}, 51\penalty0 (11):\penalty0 1336--1347, + 2004. + +\bibitem[Bachelier(1900)]{bachelier1900theorie} +Louis Bachelier. +\newblock \emph{Th{\'e}orie de la sp{\'e}culation}. +\newblock Gauthier-Villars, 1900. + +\bibitem[Barndorff-Nielsen et~al.(2001)Barndorff-Nielsen, Mikosch, et + Resnick]{barndorff2001levy} +O.E.E. Barndorff-Nielsen, T.E. Mikosch, et S.I.E. Resnick. +\newblock \emph{L{\'e}vy Processes: Theory and Applications}. +\newblock Birkhäuser, 2001. +\newblock ISBN 9780817641672. +\newblock URL \url{http://books.google.ca/books?id=ExpTdTauXMwC}. + +\bibitem[Berkson(1980)]{berkson1980minimum} +Joseph Berkson. +\newblock Minimum chi-square, not maximum likelihood! +\newblock \emph{The Annals of Statistics}, pages 457--487, 1980. + +\bibitem[Bingham et Kiesel(2004)]{bingham2004risk} +Nicholas~H Bingham et R{\"u}diger Kiesel. +\newblock \emph{Risk-neutral valuation: Pricing and hedging of financial + derivatives}. +\newblock Springer, 2004. + +\bibitem[Black(1976)]{black1976pricing} +Fischer Black. +\newblock The pricing of commodity contracts. +\newblock \emph{Journal of financial economics}, 3\penalty0 (1):\penalty0 + 167--179, 1976. + +\bibitem[Black et Scholes(1973)]{black1973pricing} +Fischer Black et Myron Scholes. +\newblock The pricing of options and corporate liabilities. +\newblock \emph{The journal of political economy}, pages 637--654, 1973. + +\bibitem[Buckle(1995)]{buckle1995bayesian} +DJ~Buckle. +\newblock Bayesian inference for stable distributions. +\newblock \emph{Journal of the American Statistical Association}, 90\penalty0 + (430):\penalty0 605--613, 1995. + +\bibitem[Butler(2007)]{butler2007saddlepoint} +Ronald~W Butler. +\newblock \emph{Saddlepoint approximations with applications}, volume~22. +\newblock Cambridge University Press, 2007. + +\bibitem[Carr et Madan(1999)]{carr1999option} +Peter Carr et Dilip Madan. +\newblock Option valuation using the fast fourier transform. +\newblock \emph{Journal of Computational Finance}, 2\penalty0 (4):\penalty0 + 61--73, 1999. + +\bibitem[Crowder(1986)]{crowder1986consistency} +Martin Crowder. +\newblock On consistency and inconsistency of estimating equations. +\newblock \emph{Econometric Theory}, pages 305--330, 1986. + +\bibitem[Crowder(1987)]{crowder1987linear} +Martin Crowder. +\newblock On linear and quadratic estimating functions. +\newblock \emph{Biometrika}, 74\penalty0 (3):\penalty0 591--597, 1987. + +\bibitem[Daniels(1954)]{daniels1954saddlepoint} +Henry~E Daniels. +\newblock Saddlepoint approximations in statistics. +\newblock \emph{The Annals of Mathematical Statistics}, pages 631--650, 1954. + +\bibitem[Derman(1996)]{derman1996modelrisk} +Emanuel Derman. +\newblock Model risk. +\newblock Technical report, Goldman Sachs, 1996. + +\bibitem[Dodge(2004)]{dodge2004statistique} +Yadolah Dodge. +\newblock \emph{Statistique: dictionnaire encyclop{\'e}dique}. +\newblock Springer Verlag France, 2004. + +\bibitem[Epps(2007)]{epps2007pricing} +Thomas~W Epps. +\newblock \emph{Pricing derivative securities}. +\newblock World Scientific Publishing Company Incorporated, 2007. + +\bibitem[Epps et Pulley(1983)]{epps1983test} +Thomas~W Epps et Lawrence~B Pulley. +\newblock A test for normality based on the empirical characteristic function. +\newblock \emph{Biometrika}, 70\penalty0 (3):\penalty0 723--726, 1983. + +\bibitem[Everitt et Skrondal(2006)]{everitt2006cambridge} +Brian Everitt et Anders Skrondal. +\newblock \emph{The Cambridge dictionary of statistics}, volume~4. +\newblock Cambridge University Press Cambridge, 2006. + +\bibitem[Fama et French(1993)]{fama1993common} +Eugene~F Fama et Kenneth~R French. +\newblock Common risk factors in the returns on stocks and bonds. +\newblock \emph{Journal of financial economics}, 33\penalty0 (1):\penalty0 + 3--56, 1993. + +\bibitem[Feuerverger et McDunnough(1981)]{feuerverger1981efficiency} +Andrey Feuerverger et Philip McDunnough. +\newblock On the efficiency of empirical characteristic function procedures. +\newblock \emph{Journal of the Royal Statistical Society. Series B + (Methodological)}, pages 20--27, 1981. + +\bibitem[Fox et Taqqu(1986)]{fox1986large} +Robert Fox et Murad~S Taqqu. +\newblock Large-sample properties of parameter estimates for strongly dependent + stationary gaussian time series. +\newblock \emph{The Annals of Statistics}, 14\penalty0 (2):\penalty0 517--532, + 1986. + +\bibitem[Gil-Pelaez(1951)]{gil1951note} +J~Gil-Pelaez. +\newblock Note on the inversion theorem. +\newblock \emph{Biometrika}, 38\penalty0 (3-4):\penalty0 481--482, 1951. + +\bibitem[Gourieroux et Monfort(1989)]{gourieroux1989statistique} +Christian Gourieroux et Alain Monfort. +\newblock \emph{Statistique et mod{\`e}les {\'e}conom{\'e}triques: Notions + g{\'e}n{\'e}rales, estimation, pr{\'e}vision, algorithmes}, volume~1. +\newblock Economica, 1989. + +\bibitem[Hall(2005)]{hall2005generalized} +Alastair~R Hall. +\newblock \emph{Generalized method of moments}. +\newblock Oxford University Press Oxford, 2005. + +\bibitem[Hamilton(1994)]{hamilton1994time} +James~Douglas Hamilton. +\newblock \emph{Time series analysis}, volume~2. +\newblock Cambridge Univ Press, 1994. + +\bibitem[Hansen(1982)]{hansen1982large} +Lars~Peter Hansen. +\newblock Large sample properties of generalized method of moments estimators. +\newblock \emph{Econometrica: Journal of the Econometric Society}, pages + 1029--1054, 1982. + +\bibitem[Henze(1990)]{henze1990approximation} +N~Henze. +\newblock An approximation to the limit distribution of the epps-pulley test + statistic for normality. +\newblock \emph{Metrika}, 37\penalty0 (1):\penalty0 7--18, 1990. + +\bibitem[Heston(1993)]{heston1993closed} +Steven~L Heston. +\newblock A closed-form solution for options with stochastic volatility with + applications to bond and currency options. +\newblock \emph{Review of financial studies}, 6\penalty0 (2):\penalty0 + 327--343, 1993. + +\bibitem[Hinkley et Revankar(1977)]{hinkley1977estimation} +David~V Hinkley et Nagesh~S Revankar. +\newblock Estimation of the pareto law from underreported data: A further + analysis. +\newblock \emph{Journal of Econometrics}, 5\penalty0 (1):\penalty0 1--11, 1977. + +\bibitem[Hogg et Craig(1978)]{hogg1978introduction} +R.V. Hogg et A.T. Craig. +\newblock \emph{Introduction to mathematical statistics}. +\newblock Macmillan, 1978. +\newblock ISBN 9780029789902. +\newblock URL \url{http://books.google.ca/books?id=OZYQAQAAIAAJ}. + +\bibitem[Itkin(2005)]{itkin2005pricing} +Andrey Itkin. +\newblock Pricing options with vg model using fft. +\newblock \emph{arXiv preprint physics/0503137}, 2005. + +\bibitem[Kotz et~al.(2001)Kotz, Kozubowski, et Podg{\'o}rski]{kotz2001laplace} +S.~Kotz, T.J. Kozubowski, et K.~Podg{\'o}rski. +\newblock \emph{The Laplace Distribution and Generalizations: A Revisit With + Applications to Communications, Exonomics, Engineering, and Finance}. +\newblock Progress in Mathematics Series. Birkh{\"a}user, 2001. +\newblock ISBN 9780817641665. +\newblock URL \url{http://books.google.ca/books?id=cb8B07hwULUC}. + +\bibitem[Koutrouvelis(1980)]{KOUTROUVELIS01011980} +I.~A. Koutrouvelis. +\newblock A goodness-of-fit test of simple hypotheses based on the empirical + characteristic function. +\newblock \emph{Biometrika}, 67\penalty0 (1):\penalty0 238--240, 1980. +\newblock \doi{10.1093/biomet/67.1.238}. +\newblock URL \url{http://biomet.oxfordjournals.org/content/67/1/238.abstract}. + +\bibitem[Kozubowski et Podg{\'o}rski(1999)]{kozubowski1999class} +Tomasz~J Kozubowski et Krzysztof Podg{\'o}rski. +\newblock A class of asymmetric distributions. +\newblock \emph{Actuarial Research Clearing House}, 1:\penalty0 113--134, 1999. + +\bibitem[Kozubowski et Podg{\'o}rski(2001)]{kozubowski2001asymmetric} +Tomasz~J Kozubowski et Krzysztof Podg{\'o}rski. +\newblock Asymmetric laplace laws and modeling financial data. +\newblock \emph{Mathematical and computer modelling}, 34\penalty0 (9):\penalty0 + 1003--1021, 2001. + +\bibitem[Kyprianou(2007)]{kyprianou2007introductory} +Andreas~E Kyprianou. +\newblock \emph{Introductory lectures on fluctuations of L{\'e}vy processes + with applications}. +\newblock Springer, 2007. + +\bibitem[Lugannani et Rice(1980)]{lugannani1980saddle} +Robert Lugannani et Stephen Rice. +\newblock Saddle point approximation for the distribution of the sum of + independent random variables. +\newblock \emph{Advances in applied probability}, pages 475--490, 1980. + +\bibitem[Lukacs(1960)]{lukacs1960characteristic} +Eugene Lukacs. +\newblock \emph{Characteristic functions}, volume~4. +\newblock Griffin London, 1960. + +\bibitem[Luong et Thompson(1987)]{luong1987minimum} +A~Luong et ME~Thompson. +\newblock Minimum-distance methods based on quadratic distances for transforms. +\newblock \emph{Canadian Journal of Statistics}, 15\penalty0 (3):\penalty0 + 239--251, 1987. + +\bibitem[Madan et Seneta(1990)]{madan1990variance} +Dilip~B Madan et Eugene Seneta. +\newblock The variance gamma (vg) model for share market returns. +\newblock \emph{Journal of business}, pages 511--524, 1990. + +\bibitem[Madan et~al.(1998)Madan, Carr, et Chang]{madan1998variance} +Dilip~B Madan, Peter~P Carr, et Eric~C Chang. +\newblock The variance gamma process and option pricing. +\newblock \emph{European Finance Review}, 2\penalty0 (1):\penalty0 79--105, + 1998. + +\bibitem[Mandelbrot(1963)]{mandelbrot1963variation} +B.~Mandelbrot. +\newblock The variation of certain speculative prices. +\newblock \emph{Journal of business}, pages 394--419, 1963. + +\bibitem[Merton(1976)]{merton1976option} +Robert~C Merton. +\newblock Option pricing when underlying stock returns are discontinuous. +\newblock \emph{Journal of financial economics}, 3\penalty0 (1):\penalty0 + 125--144, 1976. + +\bibitem[Mitchell(1916)]{mitchell1916critique} +Wesley~C Mitchell. +\newblock A critique of index numbers of the prices of stocks. +\newblock \emph{The Journal of Political Economy}, pages 625--693, 1916. + +\bibitem[Newey et McFadden(1994)]{newey1994large} +Whitney~K Newey et Daniel McFadden. +\newblock Large sample estimation and hypothesis testing. +\newblock \emph{Handbook of econometrics}, 4:\penalty0 2111--2245, 1994. + +\bibitem[Newey et West(1987)]{newey1987hypothesis} +Whitney~K Newey et Kenneth~D West. +\newblock Hypothesis testing with efficient method of moments estimation. +\newblock \emph{International Economic Review}, 28\penalty0 (3):\penalty0 + 777--787, 1987. + +\bibitem[Praetz(1972)]{praetz1972distribution} +Peter~D Praetz. +\newblock The distribution of share price changes. +\newblock \emph{Journal of business}, pages 49--55, 1972. + +\bibitem[Press(1967)]{press1967compound} +S~James Press. +\newblock A compound events model for security prices. +\newblock \emph{Journal of Business}, pages 317--335, 1967. + +\bibitem[Sato(1999)]{sato1999levy} +K.~Sato. +\newblock \emph{L{\'e}vy Processes and Infinitely Divisible Distributions}. +\newblock Cambridge Studies in Advanced Mathematics. Cambridge University + Press, 1999. +\newblock ISBN 9780521553025. +\newblock URL \url{http://books.google.ca/books?id=CwT5BNG0-owC}. + +\bibitem[Schoutens(2003)]{schoutens2003levy} +Wim Schoutens. +\newblock \emph{L{\'e}vy processes in Finance}. +\newblock Wiley, 2003. + +\bibitem[Scott et Dong(2012)]{RpackageVarianceGamma} +David Scott et Christine~Yang Dong. +\newblock \emph{VarianceGamma: The Variance Gamma Distribution}, 2012. +\newblock URL \url{http://CRAN.R-project.org/package=VarianceGamma}. +\newblock R package version 0.3-1. + +\bibitem[Seneta(2004)]{seneta2004fitting} +Eugene Seneta. +\newblock Fitting the variance-gamma model to financial data. +\newblock \emph{Journal of Applied Probability}, pages 177--187, 2004. + +\bibitem[Shapiro et Wilk(1965)]{shapiro1965analysis} +Samuel~Sanford Shapiro et Martin~B Wilk. +\newblock An analysis of variance test for normality (complete samples). +\newblock \emph{Biometrika}, 52\penalty0 (3/4):\penalty0 591--611, 1965. + +\bibitem[Shephard(1991)]{shephard1991characteristic} +Neil~G Shephard. +\newblock From characteristic function to distribution function: a simple + framework for the theory. +\newblock \emph{Econometric Theory}, 7\penalty0 (04):\penalty0 519--529, 1991. + +\bibitem[Spiegel et Liu(1999)]{spiegel1999schaum} +Murray~R Spiegel et John Liu. +\newblock \emph{Schaum's mathematical handbook of formulas and tables}, volume + 1000. +\newblock McGraw-Hill, 1999. + +\bibitem[Stuart et Ord(1987)]{stuart1987kendall} +Alan Stuart et J~Keith Ord. +\newblock \emph{Kendall’s advanced theory of statistics, Vol. 1}. +\newblock Oxford University Press, New York, 1987. + +\bibitem[Teschl(2004)]{teschl2004topics} +Gerald Teschl. +\newblock Topics in real and functional analysis. +\newblock \emph{unpublished, available online at + \url{http://www.mat.univie.ac.at/\~gerald}}, 2004. + +\bibitem[Walter()]{walterlevy} +Christian Walter. +\newblock Levy-stability-under-addition and fractal structure of markets: + imlications for the actuaries and emphasized axamination of matif national + contract. + +\bibitem[Wang et~al.(2003)Wang, Tsang, et Marsaglia]{wang2003evaluating} +Jingbo Wang, Wai~Wan Tsang, et George Marsaglia. +\newblock Evaluating kolmogorov's distribution. +\newblock \emph{Journal of Statistical Software}, 8\penalty0 (18):\penalty0 + 1--4, 2003. + +\bibitem[Wendel(1961)]{wendel1961non} +JG~Wendel. +\newblock The non-absolute convergence of gil-pelaez'inversion integral. +\newblock \emph{The Annals of Mathematical Statistics}, 32\penalty0 + (1):\penalty0 338--339, 1961. + +\bibitem[Wolfowitz(1957)]{wolfowitz1957minimum} +Jacob Wolfowitz. +\newblock The minimum distance method. +\newblock \emph{The Annals of Mathematical Statistics}, pages 75--88, 1957. + +\bibitem[Wooldridge(2001)]{wooldridge2001econometric} +Jeffrey~M Wooldridge. +\newblock \emph{Econometric analysis of cross section and panel data}. +\newblock MIT press, 2001. + +\end{thebibliography} diff --git a/memoire/gabarit-maitrise.blg b/memoire/gabarit-maitrise.blg new file mode 100644 index 0000000..aae7281 --- /dev/null +++ b/memoire/gabarit-maitrise.blg @@ -0,0 +1,68 @@ +This is BibTeX, Version 0.99d (TeX Live 2012/Debian) +Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 +The top-level auxiliary file: gabarit-maitrise.aux +The style file: plainnatmod.bst +A level-1 auxiliary file: resume.aux +A level-1 auxiliary file: abstract.aux +A level-1 auxiliary file: remerciements.aux +A level-1 auxiliary file: introduction.aux +A level-1 auxiliary file: chapitre1.aux +A level-1 auxiliary file: chapitre2.aux +A level-1 auxiliary file: chapitre3.aux +A level-1 auxiliary file: chapitre4.aux +A level-1 auxiliary file: chapitre5.aux +A level-1 auxiliary file: chapitre6.aux +A level-1 auxiliary file: chapitre7.aux +A level-1 auxiliary file: chapitre8.aux +A level-1 auxiliary file: chapitre9.aux +A level-1 auxiliary file: conclusion.aux +A level-1 auxiliary file: annexe1.aux +A level-1 auxiliary file: annexe2.aux +A level-1 auxiliary file: annexe3.aux +A level-1 auxiliary file: deed.aux +Database file #1: memoire.bib +Warning--empty booktitle in walterlevy +Warning--empty year in walterlevy +Warning--empty year in walterlevy +You've used 63 entries, + 2773 wiz_defined-function locations, + 974 strings with 13683 characters, +and the built_in function-call counts, 25492 in all, are: += -- 2374 +> -- 963 +< -- 80 ++ -- 345 +- -- 282 +* -- 1720 +:= -- 3843 +add.period$ -- 201 +call.type$ -- 63 +change.case$ -- 263 +chr.to.int$ -- 63 +cite$ -- 129 +duplicate$ -- 1378 +empty$ -- 2363 +format.name$ -- 391 +if$ -- 5467 +int.to.chr$ -- 1 +int.to.str$ -- 1 +missing$ -- 81 +newline$ -- 333 +num.names$ -- 252 +pop$ -- 596 +preamble$ -- 1 +purify$ -- 220 +quote$ -- 0 +skip$ -- 1041 +stack$ -- 0 +substring$ -- 1124 +swap$ -- 192 +text.length$ -- 21 +text.prefix$ -- 0 +top$ -- 0 +type$ -- 593 +warning$ -- 3 +while$ -- 278 +width$ -- 0 +write$ -- 830 +(There were 3 warnings) diff --git a/memoire/gabarit-maitrise.lof b/memoire/gabarit-maitrise.lof new file mode 100644 index 0000000..b0e0abb --- /dev/null +++ b/memoire/gabarit-maitrise.lof @@ -0,0 +1,30 @@ +\select@language {french} +\select@language {french} +\addvspace {10pt} +\contentsline {figure}{\numberline {1.1}{\ignorespaces Mod\IeC {\`e}le de Bachelier: probabilit\IeC {\'e} compos\IeC {\'e}e\relax }}{8}{figure.caption.12} +\contentsline {figure}{\numberline {1.2}{\ignorespaces Distribution des rendements annuels de 40 titres boursiers, de 1890 \IeC {\`a} 1915, Table XVIII de \cite {mitchell1916critique}\relax }}{10}{figure.caption.13} +\addvspace {10pt} +\contentsline {figure}{\numberline {2.1}{\ignorespaces Premier incr\IeC {\'e}ment d'un processus subordonn\IeC {\'e}\relax }}{23}{figure.caption.16} +\contentsline {figure}{\numberline {2.2}{\ignorespaces Simulation d'un processus de Wiener subordonn\IeC {\'e} par un processus gamma\relax }}{24}{figure.caption.17} +\contentsline {figure}{\numberline {2.3}{\ignorespaces Fonction de densit\IeC {\'e} de la distribution Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e avec diff\IeC {\'e}rents param\IeC {\`e}tres: $GAL(y;\theta ,\sigma ,\kappa ,\tau )$\relax }}{31}{figure.caption.19} +\contentsline {figure}{\numberline {2.4}{\ignorespaces Histogramme et estimateur de densit\IeC {\'e} par noyau de 2500 r\IeC {\'e}alisations de la variable al\IeC {\'e}atoire $Y\sim GAL(\theta =0,\sigma =1,\kappa =2,\tau =1)$\relax }}{34}{figure.caption.20} +\addvspace {10pt} +\contentsline {figure}{\numberline {3.1}{\ignorespaces Probabilit\IeC {\'e} \IeC {\`a} l'extr\IeC {\'e}mit\IeC {\'e} du support\relax }}{40}{figure.caption.23} +\contentsline {figure}{\numberline {3.2}{\ignorespaces Distribution normale tronqu\IeC {\'e}e \IeC {\`a} $w_0$\relax }}{41}{figure.caption.24} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\contentsline {figure}{\numberline {9.1}{\ignorespaces Repr\IeC {\'e}sentation en s\IeC {\'e}rie chronologique de l'\IeC {\'e}chantillon $R_1$\relax }}{98}{figure.caption.28} +\contentsline {figure}{\numberline {9.2}{\ignorespaces Distribution de la variable al\IeC {\'e}atoire $R_1$\relax }}{99}{figure.caption.31} +\contentsline {figure}{\numberline {9.3}{\ignorespaces Graphique Quantile-Quantile\relax }}{100}{figure.caption.33} +\contentsline {figure}{\numberline {9.4}{\ignorespaces \IeC {\'E}quation du point de selle pour $r=0.01$\relax }}{104}{figure.caption.37} +\contentsline {figure}{\numberline {9.5}{\ignorespaces Densit\IeC {\'e} de $R_1^{*}$ selon la m\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{106}{figure.caption.42} +\contentsline {figure}{\numberline {9.6}{\ignorespaces Densit\IeC {\'e} de $R_1^{*}$ selon la m\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale\relax }}{107}{figure.caption.43} +\contentsline {figure}{\numberline {9.7}{\ignorespaces Prix de l'option selon les param\IeC {\`e}tres estim\IeC {\'e}s avec la m\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{110}{figure.caption.49} +\contentsline {figure}{\numberline {9.8}{\ignorespaces Prix de l'option selon les param\IeC {\`e}tres estim\IeC {\'e}s avec la m\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale\relax }}{111}{figure.caption.50} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} diff --git a/memoire/gabarit-maitrise.log b/memoire/gabarit-maitrise.log new file mode 100644 index 0000000..8bd72ed --- /dev/null +++ b/memoire/gabarit-maitrise.log @@ -0,0 +1,2272 @@ +This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2013.11.30) 2 DEC 2013 21:09 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**gabarit-maitrise.tex +(./gabarit-maitrise.tex +LaTeX2e <2011/06/27> +Babel <3.9f> and hyphenation patterns for 78 languages loaded. +(./ulthese.cls +Document Class: ulthese 2012/09/30 v1.0 Classe pour les theses et memoires de l +'Universite Laval +(/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty +Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC) +) +(/usr/share/texlive/texmf-dist/tex/latex/memoir/memoir.cls +Document Class: memoir 2013/05/30 v3.7b configurable book, report, article docu +ment class +\onelineskip=\skip41 +\lxvchars=\skip42 +\xlvchars=\skip43 +\@memcnta=\count79 +\c@@memmarkcntra=\count80 +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifpdf.sty +Package: ifpdf 2011/01/30 v2.3 Provides the ifpdf switch (HO) +Package ifpdf Info: pdfTeX in PDF mode is detected. +) +(/usr/share/texlive/texmf-dist/tex/latex/ifetex/ifetex.sty +Package: ifetex 2011/12/15 v1.2 ifetex + +(/usr/share/texlive/texmf-dist/tex/plain/ifetex/ifetex.tex)) +(/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional +) +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifluatex.sty +Package: ifluatex 2010/03/01 v1.3 Provides the ifluatex switch (HO) +Package ifluatex Info: LuaTeX not detected. +) +\stockheight=\skip44 +\stockwidth=\skip45 +\trimtop=\skip46 +\trimedge=\skip47 + +(/usr/share/texlive/texmf-dist/tex/latex/etex-pkg/etex.sty +Package: etex 1998/03/26 v2.0 eTeX basic definition package (PEB) +\et@xins=\count81 +) +(/usr/share/texlive/texmf-dist/tex/latex/memoir/mem11.clo +File: mem11.clo 2008/01/30 v0.3 memoir class 11pt size option +) +\binding=\skip48 +\spinemargin=\skip49 +\foremargin=\skip50 +\uppermargin=\skip51 +\lowermargin=\skip52 +\headdrop=\skip53 +\normalrulethickness=\skip54 +\mem@maxheadheight=\skip55 +\mem@maxfootheight=\skip56 +\headwidth=\skip57 +\c@storedpagenumber=\count82 +\memPD=\dimen102 +\m@mabparskip=\skip58 +\thanksmarkwidth=\skip59 +\thanksmarksep=\skip60 +\droptitle=\skip61 +\c@book=\count83 +\c@part=\count84 +\c@chapter=\count85 +\c@section=\count86 +\c@subsection=\count87 +\c@subsubsection=\count88 +\c@paragraph=\count89 +\c@subparagraph=\count90 +\beforechapskip=\skip62 +\midchapskip=\skip63 +\afterchapskip=\skip64 +\chapindent=\skip65 +\bottomsectionskip=\skip66 +\secindent=\skip67 +\beforesecskip=\skip68 +\aftersecskip=\skip69 +\subsecindent=\skip70 +\beforesubsecskip=\skip71 +\aftersubsecskip=\skip72 +\subsubsecindent=\skip73 +\beforesubsubsecskip=\skip74 +\aftersubsubsecskip=\skip75 +\paraindent=\skip76 +\beforeparaskip=\skip77 +\afterparaskip=\skip78 +\subparaindent=\skip79 +\beforesubparaskip=\skip80 +\aftersubparaskip=\skip81 +\pfbreakskip=\skip82 +\c@@ppsavesec=\count91 +\c@@ppsaveapp=\count92 +\ragrparindent=\dimen103 +\everylistparindent=\dimen104 +\parsepi=\skip83 +\topsepi=\skip84 +\itemsepi=\skip85 +\parsepii=\skip86 +\topsepii=\skip87 +\topsepiii=\skip88 +\itemsepii=\skip89 +\itemsepiii=\skip90 +\partopsepii=\skip91 +\partopsepiii=\skip92 +\m@msavetopsep=\skip93 +\m@msavepartopsep=\skip94 +\@enLab=\toks14 +\abstitleskip=\skip95 +\absleftindent=\skip96 +\abs@leftindent=\dimen105 +\absrightindent=\skip97 +\absparindent=\skip98 +\absparsep=\skip99 +\c@vslineno=\count93 +\c@poemline=\count94 +\c@modulo@vs=\count95 +\c@memfvsline=\count96 +\vleftskip=\skip100 +\vrightskip=\skip101 +\stanzaskip=\skip102 +\versewidth=\skip103 +\vgap=\skip104 +\vindent=\skip105 +\vleftmargin=\dimen106 +\c@verse=\count97 +\c@chrsinstr=\count98 +\beforepoemtitleskip=\skip106 +\afterpoemtitleskip=\skip107 +\c@poem=\count99 +\beforePoemTitleskip=\skip108 +\midPoemTitleskip=\skip109 +\afterPoemTitleskip=\skip110 +\col@sep=\dimen107 +\extrarowheight=\dimen108 +\NC@list=\toks15 +\extratabsurround=\skip111 +\backup@length=\skip112 +\TX@col@width=\dimen109 +\TX@old@table=\dimen110 +\TX@old@col=\dimen111 +\TX@target=\dimen112 +\TX@delta=\dimen113 +\TX@cols=\count100 +\TX@ftn=\toks16 +\heavyrulewidth=\dimen114 +\lightrulewidth=\dimen115 +\cmidrulewidth=\dimen116 +\belowrulesep=\dimen117 +\belowbottomsep=\dimen118 +\aboverulesep=\dimen119 +\abovetopsep=\dimen120 +\cmidrulesep=\dimen121 +\cmidrulekern=\dimen122 +\defaultaddspace=\dimen123 +\@cmidla=\count101 +\@cmidlb=\count102 +\@aboverulesep=\dimen124 +\@belowrulesep=\dimen125 +\@thisruleclass=\count103 +\@lastruleclass=\count104 +\@thisrulewidth=\dimen126 +\ctableftskip=\skip113 +\ctabrightskip=\skip114 +\abovecolumnspenalty=\count105 +\@linestogo=\count106 +\@cellstogo=\count107 +\@cellsincolumn=\count108 +\crtok=\toks17 +\@mincolumnwidth=\dimen127 +\c@newflo@tctr=\count109 +\mem@margin@floatbox=\box26 +\@contcwidth=\skip115 +\@contindw=\skip116 +\abovecaptionskip=\skip117 +\belowcaptionskip=\skip118 +\subfloattopskip=\skip119 +\subfloatcapskip=\skip120 +\subfloatcaptopadj=\skip121 +\subfloatbottomskip=\skip122 +\subfloatlabelskip=\skip123 +\subfloatcapmargin=\dimen128 +\c@@contsubnum=\count110 +\m@mscap@capbox=\box27 +\m@mscap@fbox=\box28 +\sidecapsep=\dimen129 +\sidecapwidth=\dimen130 +\m@m@tempdima=\dimen131 +\m@mscapraise=\dimen132 +\sidecapraise=\dimen133 +\m@mscapmainwidth=\dimen134 +\m@mscaplkern=\dimen135 +\beforeepigraphskip=\skip124 +\afterepigraphskip=\skip125 +\epigraphwidth=\skip126 +\epigraphrule=\skip127 +LaTeX Info: Redefining \em on input line 7492. +LaTeX Info: Redefining \emph on input line 7500. +\tocentryskip=\skip128 +\tocbaseline=\skip129 +\cftparskip=\skip130 +\cftbeforebookskip=\skip131 +\cftbookindent=\dimen136 +\cftbooknumwidth=\dimen137 +\cftbeforepartskip=\skip132 +\cftpartindent=\skip133 +\cftpartnumwidth=\skip134 +\cftbeforechapterskip=\skip135 +\cftchapterindent=\skip136 +\cftchapternumwidth=\skip137 +\cftbeforesectionskip=\skip138 +\cftsectionindent=\skip139 +\cftsectionnumwidth=\skip140 +\cftbeforesubsectionskip=\skip141 +\cftsubsectionindent=\skip142 +\cftsubsectionnumwidth=\skip143 +\cftbeforesubsubsectionskip=\skip144 +\cftsubsubsectionindent=\skip145 +\cftsubsubsectionnumwidth=\skip146 +\cftbeforeparagraphskip=\skip147 +\cftparagraphindent=\skip148 +\cftparagraphnumwidth=\skip149 +\cftbeforesubparagraphskip=\skip150 +\cftsubparagraphindent=\skip151 +\cftsubparagraphnumwidth=\skip152 +\prechapterprecisshift=\dimen138 +\c@maxsecnumdepth=\count111 +\bibindent=\dimen139 +\bibitemsep=\skip153 +\indexcolsep=\skip154 +\indexrule=\skip155 +\indexmarkstyle=\toks18 +\@indexbox=\insert233 +\glossarycolsep=\dimen140 +\glossaryrule=\dimen141 +\sideparvshift=\skip156 +\sideins=\insert232 +\sidebartopsep=\skip157 +\sidebarhsep=\skip158 +\sidebarvsep=\skip159 +\sidebarwidth=\skip160 +\footmarkwidth=\skip161 +\footmarksep=\skip162 +\footparindent=\skip163 +\footinsdim=\skip164 +\footinsv@r=\insert231 +\@mpfootinsv@r=\insert230 +\m@m@k=\count112 +\m@m@h=\dimen142 +\m@mipn@skip=\skip165 +\sidefootins=\insert229 +\sidefootadjust=\skip166 +\sidefootheight=\skip167 +\sidefoothsep=\skip168 +\sidefootvsep=\skip169 +\sidefootwidth=\skip170 +\m@mdownsf=\skip171 +\c@sidefootnote=\count113 +\sidefootmarkwidth=\skip172 +\sidefootmarksep=\skip173 +\sidefootparindent=\skip174 +\c@pagenote=\count114 +\c@pagenoteshadow=\count115 +\mem@pn@lastkern=\skip175 +\every@verbatim=\toks19 +\afterevery@verbatim=\toks20 +\verbatim@line=\toks21 +\tab@position=\count116 +\verbatim@in@stream=\read1 +\verbatimindent=\skip176 +\verbatim@out=\write3 +\bvboxsep=\skip177 +\c@memfbvline=\count117 +\c@bvlinectr=\count118 +\bvnumlength=\skip178 +\fb@frw=\dimen143 +\fb@frh=\dimen144 +\FrameRule=\dimen145 +\FrameSep=\dimen146 +\c@cp@cntr=\count119 +LaTeX Info: Redefining \: on input line 11958. +LaTeX Info: Redefining \! on input line 11960. +\c@ism@mctr=\count120 +\c@xsm@mctr=\count121 +\c@csm@mctr=\count122 +\c@ksm@mctr=\count123 +\c@xksm@mctr=\count124 +\c@cksm@mctr=\count125 +\c@msm@mctr=\count126 +\c@xmsm@mctr=\count127 +\c@cmsm@mctr=\count128 +\c@bsm@mctr=\count129 +\c@workm@mctr=\count130 +\c@sheetsequence=\count131 +\c@lastsheet=\count132 +\c@lastpage=\count133 +\c@figure=\count134 +\c@lofdepth=\count135 +\cftbeforefigureskip=\skip179 +\cftfigureindent=\skip180 +\cftfigurenumwidth=\skip181 +\c@table=\count136 +\c@lotdepth=\count137 +\cftbeforetableskip=\skip182 +\cfttableindent=\skip183 +\cfttablenumwidth=\skip184 +Package abstract [2008/07/23] emulated by memoir. +Package appendix [2008/07/23] emulated by memoir. +Package array [2008/09/09] emulated by memoir. +Package booktabs [2008/07/23] emulated by memoir. +Package ccaption [2008/07/23] emulated by memoir. +Package changepage [2008/07/23] emulated by memoir. +Package chngcntr [2008/07/23] emulated by memoir. +Package chngpage [2008/07/23] emulated by memoir. +Package crop emulated by memoir. +Package dcolumn [2008/07/23] emulated by memoir. +Package delarray [2008/07/23] emulated by memoir. +Package enumerate [2008/07/23] emulated by memoir. +Package epigraph [2008/07/23] emulated by memoir. +Package ifmtarg [2008/07/23] emulated by memoir. +Package index [2008/07/23] emulated by memoir. +Package makeidx [2008/07/23] emulated by memoir. +Package moreverb [2008/07/23] emulated by memoir. +Package mparhack [2008/07/23] emulated by memoir. +Package needspace [2008/07/23] emulated by memoir. +Package newfile [2008/07/23] emulated by memoir. +Package nextpage [2008/07/23] emulated by memoir. +Package pagenote [2008/07/23] emulated by memoir. +Package parskip [2008/07/23] emulated by memoir. +Package patchcmd [2008/07/23] emulated by memoir. +Package setspace [2008/07/23] emulated by memoir. +Package shortvrb [2008/07/23] emulated by memoir. +Package showidx [2008/07/23] emulated by memoir. +Package tabularx [2008/07/23] emulated by memoir. +Package titleref [2008/07/23] emulated by memoir. +Package titling [2008/07/23] emulated by memoir. +Package tocbibind [2008/07/23] emulated by memoir. +Package tocloft [2008/07/23] emulated by memoir. +Package tocvsec2 [2008/07/23] emulated by memoir. +Package verbatim [2008/07/23] emulated by memoir. +Package verse [2008/07/23] emulated by memoir. + +(/usr/share/texlive/texmf-dist/tex/latex/memoir/mempatch.sty +Package: mempatch 2009/07/24 v6.0f Patches for memoir class v1.6180339 +)) +(/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +Package: fontenc 2005/09/27 v1.99g Standard LaTeX package + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.def +File: t1enc.def 2005/09/27 v1.99g Standard LaTeX file +LaTeX Font Info: Redeclaring font encoding T1 on input line 43. +)) +(/usr/share/texlive/texmf-dist/tex/latex/natbib/natbib.sty +Package: natbib 2010/09/13 8.31b (PWD, AO) +\bibhang=\skip185 +\bibsep=\skip186 +LaTeX Info: Redefining \cite on input line 694. +\c@NAT@ctr=\count138 +) +(/usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +Package: babel 2013/05/16 v3.9f The Babel package + +(/usr/share/texlive/texmf-dist/tex/generic/babel-english/english.ldf +Language: english 2012/08/20 v3.3p English support from the babel system + +(/usr/share/texlive/texmf-dist/tex/generic/babel/babel.def +File: babel.def 2013/05/16 v3.9f Babel common definitions +\babel@savecnt=\count139 +\U@D=\dimen147 +) +\l@canadian = a dialect from \language\l@american +\l@australian = a dialect from \language\l@british +\l@newzealand = a dialect from \language\l@british +) +(/usr/share/texlive/texmf-dist/tex/generic/babel-french/frenchb.ldf +Language: frenchb 2013/07/06 v2.6e French support from the babel system +\FB@interchartokenstateORI=\count140 +Package babel Info: Making : an active character on input line 240. +Package babel Info: Making ; an active character on input line 241. +Package babel Info: Making ! an active character on input line 242. +Package babel Info: Making ? an active character on input line 243. +\FB@Mht=\dimen148 +\std@mcc=\count141 +\dec@mcc=\count142 +\listindentFB=\skip187 +\labelwidthFB=\skip188 +\leftmarginFB=\skip189 +\parindentFFN=\dimen149 +)) +(/usr/share/texlive/texmf-dist/tex/latex/carlisle/scalefnt.sty) +(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +Package: keyval 1999/03/16 v1.13 key=value parser (DPC) +\KV@toks@=\toks22 +) +(/usr/share/texlive/texmf-dist/tex/latex/numprint/numprint.sty +Package: numprint 2012/08/20 v1.39 Print numbers (HH) +\c@nprt@mantissa@digitsbefore=\count143 +\c@nprt@mantissa@digitsafter=\count144 +\c@nprt@exponent@digitsbefore=\count145 +\c@nprt@exponent@digitsafter=\count146 +\nprt@digitwidth=\skip190 +\nprt@sepwidth=\skip191 +\nprt@decimalwidth=\skip192 +\nprt@blockwidth=\skip193 +\nprt@digittoks=\toks23 +\nprt@pretoks=\toks24 +\nprt@posttoks=\toks25 +\nprt@thisdigit=\count147 +\nprt@curpos=\count148 +\nprt@rndpos=\count149 +\c@nprt@digitsfirstblock=\count150 +\c@nprt@blockcnt=\count151 +\c@nprt@cntprint=\count152 + +No configuration file `numprint.cfg' found.) +(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty +Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR) + +(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty +Package: graphics 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR) + +(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty +Package: trig 1999/03/16 v1.09 sin cos tan (DPC) +) +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/graphics.cfg +File: graphics.cfg 2010/04/23 v1.9 graphics configuration of TeX Live +) +Package graphics Info: Driver file: pdftex.def on input line 91. + +(/usr/share/texlive/texmf-dist/tex/latex/pdftex-def/pdftex.def +File: pdftex.def 2011/05/27 v0.06d Graphics/color for pdfTeX + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty +Package: infwarerr 2010/04/08 v1.3 Providing info/warning/error messages (HO) +) +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty +Package: ltxcmds 2011/11/09 v1.22 LaTeX kernel commands for general use (HO) +) +\Gread@gobject=\count153 +)) +\Gin@req@height=\dimen150 +\Gin@req@width=\dimen151 +) +(/usr/share/texmf/tex/latex/xcolor/xcolor.sty +Package: xcolor 2007/01/21 v2.11 LaTeX color extensions (UK) + +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg +File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive +) +Package xcolor Info: Driver file: pdftex.def on input line 225. +Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1337. +Package xcolor Info: Model `hsb' substituted by `rgb' on input line 1341. +Package xcolor Info: Model `RGB' extended on input line 1353. +Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1355. +Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1356. +Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1357. +Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1358. +Package xcolor Info: Model `Gray' substituted by `gray' on input line 1359. +Package xcolor Info: Model `wave' substituted by `hsb' on input line 1360. +) +(/usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty +Package: textcomp 2005/09/27 v1.99g Standard LaTeX package +Package textcomp Info: Sub-encoding information: +(textcomp) 5 = only ISO-Adobe without \textcurrency +(textcomp) 4 = 5 + \texteuro +(textcomp) 3 = 4 + \textohm +(textcomp) 2 = 3 + \textestimated + \textcurrency +(textcomp) 1 = TS1 - \textcircled - \t +(textcomp) 0 = TS1 (full) +(textcomp) Font families with sub-encoding setting implement +(textcomp) only a restricted character set as indicated. +(textcomp) Family '?' is the default used for unknown fonts. +(textcomp) See the documentation for details. +Package textcomp Info: Setting ? sub-encoding to TS1/1 on input line 71. + +(/usr/share/texlive/texmf-dist/tex/latex/base/ts1enc.def +File: ts1enc.def 2001/06/05 v3.0e (jk/car/fm) Standard LaTeX file +) +LaTeX Info: Redefining \oldstylenums on input line 266. +Package textcomp Info: Setting cmr sub-encoding to TS1/0 on input line 281. +Package textcomp Info: Setting cmss sub-encoding to TS1/0 on input line 282. +Package textcomp Info: Setting cmtt sub-encoding to TS1/0 on input line 283. +Package textcomp Info: Setting cmvtt sub-encoding to TS1/0 on input line 284. +Package textcomp Info: Setting cmbr sub-encoding to TS1/0 on input line 285. +Package textcomp Info: Setting cmtl sub-encoding to TS1/0 on input line 286. +Package textcomp Info: Setting ccr sub-encoding to TS1/0 on input line 287. +Package textcomp Info: Setting ptm sub-encoding to TS1/4 on input line 288. +Package textcomp Info: Setting pcr sub-encoding to TS1/4 on input line 289. +Package textcomp Info: Setting phv sub-encoding to TS1/4 on input line 290. +Package textcomp Info: Setting ppl sub-encoding to TS1/3 on input line 291. +Package textcomp Info: Setting pag sub-encoding to TS1/4 on input line 292. +Package textcomp Info: Setting pbk sub-encoding to TS1/4 on input line 293. +Package textcomp Info: Setting pnc sub-encoding to TS1/4 on input line 294. +Package textcomp Info: Setting pzc sub-encoding to TS1/4 on input line 295. +Package textcomp Info: Setting bch sub-encoding to TS1/4 on input line 296. +Package textcomp Info: Setting put sub-encoding to TS1/5 on input line 297. +Package textcomp Info: Setting uag sub-encoding to TS1/5 on input line 298. +Package textcomp Info: Setting ugq sub-encoding to TS1/5 on input line 299. +Package textcomp Info: Setting ul8 sub-encoding to TS1/4 on input line 300. +Package textcomp Info: Setting ul9 sub-encoding to TS1/4 on input line 301. +Package textcomp Info: Setting augie sub-encoding to TS1/5 on input line 302. +Package textcomp Info: Setting dayrom sub-encoding to TS1/3 on input line 303. +Package textcomp Info: Setting dayroms sub-encoding to TS1/3 on input line 304. + +Package textcomp Info: Setting pxr sub-encoding to TS1/0 on input line 305. +Package textcomp Info: Setting pxss sub-encoding to TS1/0 on input line 306. +Package textcomp Info: Setting pxtt sub-encoding to TS1/0 on input line 307. +Package textcomp Info: Setting txr sub-encoding to TS1/0 on input line 308. +Package textcomp Info: Setting txss sub-encoding to TS1/0 on input line 309. +Package textcomp Info: Setting txtt sub-encoding to TS1/0 on input line 310. +Package textcomp Info: Setting lmr sub-encoding to TS1/0 on input line 311. +Package textcomp Info: Setting lmdh sub-encoding to TS1/0 on input line 312. +Package textcomp Info: Setting lmss sub-encoding to TS1/0 on input line 313. +Package textcomp Info: Setting lmssq sub-encoding to TS1/0 on input line 314. +Package textcomp Info: Setting lmvtt sub-encoding to TS1/0 on input line 315. +Package textcomp Info: Setting qhv sub-encoding to TS1/0 on input line 316. +Package textcomp Info: Setting qag sub-encoding to TS1/0 on input line 317. +Package textcomp Info: Setting qbk sub-encoding to TS1/0 on input line 318. +Package textcomp Info: Setting qcr sub-encoding to TS1/0 on input line 319. +Package textcomp Info: Setting qcs sub-encoding to TS1/0 on input line 320. +Package textcomp Info: Setting qpl sub-encoding to TS1/0 on input line 321. +Package textcomp Info: Setting qtm sub-encoding to TS1/0 on input line 322. +Package textcomp Info: Setting qzc sub-encoding to TS1/0 on input line 323. +Package textcomp Info: Setting qhvc sub-encoding to TS1/0 on input line 324. +Package textcomp Info: Setting futs sub-encoding to TS1/4 on input line 325. +Package textcomp Info: Setting futx sub-encoding to TS1/4 on input line 326. +Package textcomp Info: Setting futj sub-encoding to TS1/4 on input line 327. +Package textcomp Info: Setting hlh sub-encoding to TS1/3 on input line 328. +Package textcomp Info: Setting hls sub-encoding to TS1/3 on input line 329. +Package textcomp Info: Setting hlst sub-encoding to TS1/3 on input line 330. +Package textcomp Info: Setting hlct sub-encoding to TS1/5 on input line 331. +Package textcomp Info: Setting hlx sub-encoding to TS1/5 on input line 332. +Package textcomp Info: Setting hlce sub-encoding to TS1/5 on input line 333. +Package textcomp Info: Setting hlcn sub-encoding to TS1/5 on input line 334. +Package textcomp Info: Setting hlcw sub-encoding to TS1/5 on input line 335. +Package textcomp Info: Setting hlcf sub-encoding to TS1/5 on input line 336. +Package textcomp Info: Setting pplx sub-encoding to TS1/3 on input line 337. +Package textcomp Info: Setting pplj sub-encoding to TS1/3 on input line 338. +Package textcomp Info: Setting ptmx sub-encoding to TS1/4 on input line 339. +Package textcomp Info: Setting ptmj sub-encoding to TS1/4 on input line 340. +) +(/usr/share/texlive/texmf-dist/tex/latex/psnfss/helvet.sty +Package: helvet 2005/04/12 PSNFSS-v9.2a (WaS) +) + +****************************************************** +Stock height and width: 794.96999pt by 614.295pt +Top and edge trims: 0pt and 0pt +Page height and width: 794.96999pt by 614.295pt +Text height and width: 636.60028pt by 443pt +Spine and edge margins: 99.58464pt and 71.13188pt +Upper and lower margins: 71.13188pt and 87.23782pt +Headheight and headsep: 13.6pt and 19.8738pt +Footskip: 56.9055pt +Columnsep and columnseprule: 10pt and 0pt +Marginparsep and marginparwidth: 7pt and 57.13188pt +Sidecapsep and sidecapwidth: 7pt and 111pt +Sidebarhsep and sidebarwidth: 7pt and 111pt +Sidebarvsep and sidebartopsep: 13.6pt and 0pt +Sidebarheight: 539.9202pt +Sidefoothsep and sidefootwidth: 7pt and 111pt +Sidefootvsep and sidefootheight: 13.6pt and 539.9202pt +****************************************************** + +) (/usr/share/texlive/texmf-dist/tex/latex/koma-script/scrtime.sty +Package: scrtime 2012/07/29 v3.11b KOMA-Script package (time of LaTeX run) + +(/usr/share/texlive/texmf-dist/tex/latex/koma-script/scrkbase.sty +Package: scrkbase 2012/07/29 v3.11b KOMA-Script package (KOMA-Script-dependent +basics and keyval usage) + +(/usr/share/texlive/texmf-dist/tex/latex/koma-script/scrbase.sty +Package: scrbase 2012/07/29 v3.11b KOMA-Script package (KOMA-Script-independent + basics and keyval usage) + +(/usr/share/texlive/texmf-dist/tex/latex/koma-script/scrlfile.sty +Package: scrlfile 2012/06/15 v3.12 KOMA-Script package (loading files) + +Package scrlfile, 2012/06/15 v3.12 KOMA-Script package (loading files) + Copyright (C) Markus Kohm + + +LaTeX Warning: Command \InputIfFileExists has changed. + Check if current package is valid. + +)))) (/usr/share/texmf/tex/latex/pgf/frontendlayer/tikz.sty +(/usr/share/texmf/tex/latex/pgf/basiclayer/pgf.sty +(/usr/share/texmf/tex/latex/pgf/utilities/pgfrcs.sty +(/usr/share/texmf/tex/generic/pgf/utilities/pgfutil-common.tex +\pgfutil@everybye=\toks26 +) +(/usr/share/texmf/tex/generic/pgf/utilities/pgfutil-latex.def +\pgfutil@abb=\box29 + +(/usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty +Package: everyshi 2001/05/15 v3.00 EveryShipout Package (MS) +)) +(/usr/share/texmf/tex/generic/pgf/utilities/pgfrcs.code.tex +Package: pgfrcs 2010/10/25 v2.10 (rcs-revision 1.24) +)) +Package: pgf 2008/01/15 v2.10 (rcs-revision 1.12) + +(/usr/share/texmf/tex/latex/pgf/basiclayer/pgfcore.sty +(/usr/share/texmf/tex/latex/pgf/systemlayer/pgfsys.sty +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys.code.tex +Package: pgfsys 2010/06/30 v2.10 (rcs-revision 1.37) + +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeys.code.tex +\pgfkeys@pathtoks=\toks27 +\pgfkeys@temptoks=\toks28 + +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex +\pgfkeys@tmptoks=\toks29 +)) +\pgf@x=\dimen152 +\pgf@y=\dimen153 +\pgf@xa=\dimen154 +\pgf@ya=\dimen155 +\pgf@xb=\dimen156 +\pgf@yb=\dimen157 +\pgf@xc=\dimen158 +\pgf@yc=\dimen159 +\w@pgf@writea=\write4 +\r@pgf@reada=\read2 +\c@pgf@counta=\count154 +\c@pgf@countb=\count155 +\c@pgf@countc=\count156 +\c@pgf@countd=\count157 + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgf.cfg +File: pgf.cfg 2008/05/14 (rcs-revision 1.7) +) +Package pgfsys Info: Driver file for pgf: pgfsys-pdftex.def on input line 900. + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys-pdftex.def +File: pgfsys-pdftex.def 2009/05/22 (rcs-revision 1.26) + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def +File: pgfsys-common-pdf.def 2008/05/19 (rcs-revision 1.10) +))) +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +File: pgfsyssoftpath.code.tex 2008/07/18 (rcs-revision 1.7) +\pgfsyssoftpath@smallbuffer@items=\count158 +\pgfsyssoftpath@bigbuffer@items=\count159 +) +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +File: pgfsysprotocol.code.tex 2006/10/16 (rcs-revision 1.4) +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcore.code.tex +Package: pgfcore 2010/04/11 v2.10 (rcs-revision 1.7) + +(/usr/share/texmf/tex/generic/pgf/math/pgfmath.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathcalc.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathutil.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathparser.code.tex +\pgfmath@dimen=\dimen160 +\pgfmath@count=\count160 +\pgfmath@box=\box30 +\pgfmath@toks=\toks30 +\pgfmath@stack@operand=\toks31 +\pgfmath@stack@operation=\toks32 +) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.random.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.base.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.round.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex))) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfloat.code.tex +\c@pgfmathroundto@lastzeros=\count161 +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex +File: pgfcorepoints.code.tex 2010/04/09 (rcs-revision 1.20) +\pgf@picminx=\dimen161 +\pgf@picmaxx=\dimen162 +\pgf@picminy=\dimen163 +\pgf@picmaxy=\dimen164 +\pgf@pathminx=\dimen165 +\pgf@pathmaxx=\dimen166 +\pgf@pathminy=\dimen167 +\pgf@pathmaxy=\dimen168 +\pgf@xx=\dimen169 +\pgf@xy=\dimen170 +\pgf@yx=\dimen171 +\pgf@yy=\dimen172 +\pgf@zx=\dimen173 +\pgf@zy=\dimen174 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex +File: pgfcorepathconstruct.code.tex 2010/08/03 (rcs-revision 1.24) +\pgf@path@lastx=\dimen175 +\pgf@path@lasty=\dimen176 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex +File: pgfcorepathusage.code.tex 2008/04/22 (rcs-revision 1.12) +\pgf@shorten@end@additional=\dimen177 +\pgf@shorten@start@additional=\dimen178 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex +File: pgfcorescopes.code.tex 2010/09/08 (rcs-revision 1.34) +\pgfpic=\box31 +\pgf@hbox=\box32 +\pgf@layerbox@main=\box33 +\pgf@picture@serial@count=\count162 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex +File: pgfcoregraphicstate.code.tex 2008/04/22 (rcs-revision 1.9) +\pgflinewidth=\dimen179 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex +File: pgfcoretransformations.code.tex 2009/06/10 (rcs-revision 1.11) +\pgf@pt@x=\dimen180 +\pgf@pt@y=\dimen181 +\pgf@pt@temp=\dimen182 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorequick.code.tex +File: pgfcorequick.code.tex 2008/10/09 (rcs-revision 1.3) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex +File: pgfcoreobjects.code.tex 2006/10/11 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex +File: pgfcorepathprocessing.code.tex 2008/10/09 (rcs-revision 1.8) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex +File: pgfcorearrows.code.tex 2008/04/23 (rcs-revision 1.11) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex +File: pgfcoreshade.code.tex 2008/11/23 (rcs-revision 1.13) +\pgf@max=\dimen183 +\pgf@sys@shading@range@num=\count163 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex +File: pgfcoreimage.code.tex 2010/03/25 (rcs-revision 1.16) + +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex +File: pgfcoreexternal.code.tex 2010/09/01 (rcs-revision 1.17) +\pgfexternal@startupbox=\box34 +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex +File: pgfcorelayers.code.tex 2010/08/27 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex +File: pgfcoretransparency.code.tex 2008/01/17 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex +File: pgfcorepatterns.code.tex 2009/07/02 (rcs-revision 1.3) +))) +(/usr/share/texmf/tex/generic/pgf/modules/pgfmoduleshapes.code.tex +File: pgfmoduleshapes.code.tex 2010/09/09 (rcs-revision 1.13) +\pgfnodeparttextbox=\box35 +) +(/usr/share/texmf/tex/generic/pgf/modules/pgfmoduleplot.code.tex +File: pgfmoduleplot.code.tex 2010/10/22 (rcs-revision 1.8) +) +(/usr/share/texmf/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +Package: pgfcomp-version-0-65 2007/07/03 v2.10 (rcs-revision 1.7) +\pgf@nodesepstart=\dimen184 +\pgf@nodesepend=\dimen185 +) +(/usr/share/texmf/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +Package: pgfcomp-version-1-18 2007/07/23 v2.10 (rcs-revision 1.1) +)) +(/usr/share/texmf/tex/latex/pgf/utilities/pgffor.sty +(/usr/share/texmf/tex/latex/pgf/utilities/pgfkeys.sty +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeys.code.tex)) +(/usr/share/texmf/tex/generic/pgf/utilities/pgffor.code.tex +Package: pgffor 2010/03/23 v2.10 (rcs-revision 1.18) +\pgffor@iter=\dimen186 +\pgffor@skip=\dimen187 +\pgffor@stack=\toks33 +\pgffor@toks=\toks34 +)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +Package: tikz 2010/10/13 v2.10 (rcs-revision 1.76) + +(/usr/share/texmf/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex +File: pgflibraryplothandlers.code.tex 2010/05/31 v2.10 (rcs-revision 1.15) +\pgf@plot@mark@count=\count164 +\pgfplotmarksize=\dimen188 +) +\tikz@lastx=\dimen189 +\tikz@lasty=\dimen190 +\tikz@lastxsaved=\dimen191 +\tikz@lastysaved=\dimen192 +\tikzleveldistance=\dimen193 +\tikzsiblingdistance=\dimen194 +\tikz@figbox=\box36 +\tikz@tempbox=\box37 +\tikztreelevel=\count165 +\tikznumberofchildren=\count166 +\tikznumberofcurrentchild=\count167 +\tikz@fig@count=\count168 + +(/usr/share/texmf/tex/generic/pgf/modules/pgfmodulematrix.code.tex +File: pgfmodulematrix.code.tex 2010/08/24 (rcs-revision 1.4) +\pgfmatrixcurrentrow=\count169 +\pgfmatrixcurrentcolumn=\count170 +\pgf@matrix@numberofcolumns=\count171 +) +\tikz@expandcount=\count172 + +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopat +hs.code.tex +File: tikzlibrarytopaths.code.tex 2008/06/17 v2.10 (rcs-revision 1.2) +))) (/usr/share/texlive/texmf-dist/tex/latex/multirow/multirow.sty +\bigstrutjot=\dimen195 +) +(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +Package: inputenc 2008/03/30 v1.1d Input encoding file +\inpenc@prehook=\toks35 +\inpenc@posthook=\toks36 + +(/usr/share/texlive/texmf-dist/tex/latex/base/utf8.def +File: utf8.def 2008/04/05 v1.1m UTF-8 support for inputenc +Now handling font encoding OML ... +... no UTF-8 mapping file for font encoding OML +Now handling font encoding T1 ... +... processing UTF-8 mapping file for font encoding T1 + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu +File: t1enc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A1 (decimal 161) + defining Unicode char U+00A3 (decimal 163) + defining Unicode char U+00AB (decimal 171) + defining Unicode char U+00BB (decimal 187) + defining Unicode char U+00BF (decimal 191) + defining Unicode char U+00C0 (decimal 192) + defining Unicode char U+00C1 (decimal 193) + defining Unicode char U+00C2 (decimal 194) + defining Unicode char U+00C3 (decimal 195) + defining Unicode char U+00C4 (decimal 196) + defining Unicode char U+00C5 (decimal 197) + defining Unicode char U+00C6 (decimal 198) + defining Unicode char U+00C7 (decimal 199) + defining Unicode char U+00C8 (decimal 200) + defining Unicode char U+00C9 (decimal 201) + defining Unicode char U+00CA (decimal 202) + defining Unicode char U+00CB (decimal 203) + defining Unicode char U+00CC (decimal 204) + defining Unicode char U+00CD (decimal 205) + defining Unicode char U+00CE (decimal 206) + defining Unicode char U+00CF (decimal 207) + defining Unicode char U+00D0 (decimal 208) + defining Unicode char U+00D1 (decimal 209) + defining Unicode char U+00D2 (decimal 210) + defining Unicode char U+00D3 (decimal 211) + defining Unicode char U+00D4 (decimal 212) + defining Unicode char U+00D5 (decimal 213) + defining Unicode char U+00D6 (decimal 214) + defining Unicode char U+00D8 (decimal 216) + defining Unicode char U+00D9 (decimal 217) + defining Unicode char U+00DA (decimal 218) + defining Unicode char U+00DB (decimal 219) + defining Unicode char U+00DC (decimal 220) + defining Unicode char U+00DD (decimal 221) + defining Unicode char U+00DE (decimal 222) + defining Unicode char U+00DF (decimal 223) + defining Unicode char U+00E0 (decimal 224) + defining Unicode char U+00E1 (decimal 225) + defining Unicode char U+00E2 (decimal 226) + defining Unicode char U+00E3 (decimal 227) + defining Unicode char U+00E4 (decimal 228) + defining Unicode char U+00E5 (decimal 229) + defining Unicode char U+00E6 (decimal 230) + defining Unicode char U+00E7 (decimal 231) + defining Unicode char U+00E8 (decimal 232) + defining Unicode char U+00E9 (decimal 233) + defining Unicode char U+00EA (decimal 234) + defining Unicode char U+00EB (decimal 235) + defining Unicode char U+00EC (decimal 236) + defining Unicode char U+00ED (decimal 237) + defining Unicode char U+00EE (decimal 238) + defining Unicode char U+00EF (decimal 239) + defining Unicode char U+00F0 (decimal 240) + defining Unicode char U+00F1 (decimal 241) + defining Unicode char U+00F2 (decimal 242) + defining Unicode char U+00F3 (decimal 243) + defining Unicode char U+00F4 (decimal 244) + defining Unicode char U+00F5 (decimal 245) + defining Unicode char U+00F6 (decimal 246) + defining Unicode char U+00F8 (decimal 248) + defining Unicode char U+00F9 (decimal 249) + defining Unicode char U+00FA (decimal 250) + defining Unicode char U+00FB (decimal 251) + defining Unicode char U+00FC (decimal 252) + defining Unicode char U+00FD (decimal 253) + defining Unicode char U+00FE (decimal 254) + defining Unicode char U+00FF (decimal 255) + defining Unicode char U+0102 (decimal 258) + defining Unicode char U+0103 (decimal 259) + defining Unicode char U+0104 (decimal 260) + defining Unicode char U+0105 (decimal 261) + defining Unicode char U+0106 (decimal 262) + defining Unicode char U+0107 (decimal 263) + defining Unicode char U+010C (decimal 268) + defining Unicode char U+010D (decimal 269) + defining Unicode char U+010E (decimal 270) + defining Unicode char U+010F (decimal 271) + defining Unicode char U+0110 (decimal 272) + defining Unicode char U+0111 (decimal 273) + defining Unicode char U+0118 (decimal 280) + defining Unicode char U+0119 (decimal 281) + defining Unicode char U+011A (decimal 282) + defining Unicode char U+011B (decimal 283) + defining Unicode char U+011E (decimal 286) + defining Unicode char U+011F (decimal 287) + defining Unicode char U+0130 (decimal 304) + defining Unicode char U+0131 (decimal 305) + defining Unicode char U+0132 (decimal 306) + defining Unicode char U+0133 (decimal 307) + defining Unicode char U+0139 (decimal 313) + defining Unicode char U+013A (decimal 314) + defining Unicode char U+013D (decimal 317) + defining Unicode char U+013E (decimal 318) + defining Unicode char U+0141 (decimal 321) + defining Unicode char U+0142 (decimal 322) + defining Unicode char U+0143 (decimal 323) + defining Unicode char U+0144 (decimal 324) + defining Unicode char U+0147 (decimal 327) + defining Unicode char U+0148 (decimal 328) + defining Unicode char U+014A (decimal 330) + defining Unicode char U+014B (decimal 331) + defining Unicode char U+0150 (decimal 336) + defining Unicode char U+0151 (decimal 337) + defining Unicode char U+0152 (decimal 338) + defining Unicode char U+0153 (decimal 339) + defining Unicode char U+0154 (decimal 340) + defining Unicode char U+0155 (decimal 341) + defining Unicode char U+0158 (decimal 344) + defining Unicode char U+0159 (decimal 345) + defining Unicode char U+015A (decimal 346) + defining Unicode char U+015B (decimal 347) + defining Unicode char U+015E (decimal 350) + defining Unicode char U+015F (decimal 351) + defining Unicode char U+0160 (decimal 352) + defining Unicode char U+0161 (decimal 353) + defining Unicode char U+0162 (decimal 354) + defining Unicode char U+0163 (decimal 355) + defining Unicode char U+0164 (decimal 356) + defining Unicode char U+0165 (decimal 357) + defining Unicode char U+016E (decimal 366) + defining Unicode char U+016F (decimal 367) + defining Unicode char U+0170 (decimal 368) + defining Unicode char U+0171 (decimal 369) + defining Unicode char U+0178 (decimal 376) + defining Unicode char U+0179 (decimal 377) + defining Unicode char U+017A (decimal 378) + defining Unicode char U+017B (decimal 379) + defining Unicode char U+017C (decimal 380) + defining Unicode char U+017D (decimal 381) + defining Unicode char U+017E (decimal 382) + defining Unicode char U+200C (decimal 8204) + defining Unicode char U+2013 (decimal 8211) + defining Unicode char U+2014 (decimal 8212) + defining Unicode char U+2018 (decimal 8216) + defining Unicode char U+2019 (decimal 8217) + defining Unicode char U+201A (decimal 8218) + defining Unicode char U+201C (decimal 8220) + defining Unicode char U+201D (decimal 8221) + defining Unicode char U+201E (decimal 8222) + defining Unicode char U+2030 (decimal 8240) + defining Unicode char U+2031 (decimal 8241) + defining Unicode char U+2039 (decimal 8249) + defining Unicode char U+203A (decimal 8250) + defining Unicode char U+2423 (decimal 9251) +) +Now handling font encoding OT1 ... +... processing UTF-8 mapping file for font encoding OT1 + +(/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu +File: ot1enc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A1 (decimal 161) + defining Unicode char U+00A3 (decimal 163) + defining Unicode char U+00B8 (decimal 184) + defining Unicode char U+00BF (decimal 191) + defining Unicode char U+00C5 (decimal 197) + defining Unicode char U+00C6 (decimal 198) + defining Unicode char U+00D8 (decimal 216) + defining Unicode char U+00DF (decimal 223) + defining Unicode char U+00E6 (decimal 230) + defining Unicode char U+00EC (decimal 236) + defining Unicode char U+00ED (decimal 237) + defining Unicode char U+00EE (decimal 238) + defining Unicode char U+00EF (decimal 239) + defining Unicode char U+00F8 (decimal 248) + defining Unicode char U+0131 (decimal 305) + defining Unicode char U+0141 (decimal 321) + defining Unicode char U+0142 (decimal 322) + defining Unicode char U+0152 (decimal 338) + defining Unicode char U+0153 (decimal 339) + defining Unicode char U+2013 (decimal 8211) + defining Unicode char U+2014 (decimal 8212) + defining Unicode char U+2018 (decimal 8216) + defining Unicode char U+2019 (decimal 8217) + defining Unicode char U+201C (decimal 8220) + defining Unicode char U+201D (decimal 8221) +) +Now handling font encoding OMS ... +... processing UTF-8 mapping file for font encoding OMS + +(/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu +File: omsenc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A7 (decimal 167) + defining Unicode char U+00B6 (decimal 182) + defining Unicode char U+00B7 (decimal 183) + defining Unicode char U+2020 (decimal 8224) + defining Unicode char U+2021 (decimal 8225) + defining Unicode char U+2022 (decimal 8226) +) +Now handling font encoding OMX ... +... no UTF-8 mapping file for font encoding OMX +Now handling font encoding U ... +... no UTF-8 mapping file for font encoding U +Now handling font encoding TS1 ... +... processing UTF-8 mapping file for font encoding TS1 + +(/usr/share/texlive/texmf-dist/tex/latex/base/ts1enc.dfu +File: ts1enc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A2 (decimal 162) + defining Unicode char U+00A3 (decimal 163) + defining Unicode char U+00A4 (decimal 164) + defining Unicode char U+00A5 (decimal 165) + defining Unicode char U+00A6 (decimal 166) + defining Unicode char U+00A7 (decimal 167) + defining Unicode char U+00A8 (decimal 168) + defining Unicode char U+00A9 (decimal 169) + defining Unicode char U+00AA (decimal 170) + defining Unicode char U+00AC (decimal 172) + defining Unicode char U+00AE (decimal 174) + defining Unicode char U+00AF (decimal 175) + defining Unicode char U+00B0 (decimal 176) + defining Unicode char U+00B1 (decimal 177) + defining Unicode char U+00B2 (decimal 178) + defining Unicode char U+00B3 (decimal 179) + defining Unicode char U+00B4 (decimal 180) + defining Unicode char U+00B5 (decimal 181) + defining Unicode char U+00B6 (decimal 182) + defining Unicode char U+00B7 (decimal 183) + defining Unicode char U+00B9 (decimal 185) + defining Unicode char U+00BA (decimal 186) + defining Unicode char U+00BC (decimal 188) + defining Unicode char U+00BD (decimal 189) + defining Unicode char U+00BE (decimal 190) + defining Unicode char U+00D7 (decimal 215) + defining Unicode char U+00F7 (decimal 247) + defining Unicode char U+0192 (decimal 402) + defining Unicode char U+02C7 (decimal 711) + defining Unicode char U+02D8 (decimal 728) + defining Unicode char U+02DD (decimal 733) + defining Unicode char U+0E3F (decimal 3647) + defining Unicode char U+2016 (decimal 8214) + defining Unicode char U+2020 (decimal 8224) + defining Unicode char U+2021 (decimal 8225) + defining Unicode char U+2022 (decimal 8226) + defining Unicode char U+2030 (decimal 8240) + defining Unicode char U+2031 (decimal 8241) + defining Unicode char U+203B (decimal 8251) + defining Unicode char U+203D (decimal 8253) + defining Unicode char U+2044 (decimal 8260) + defining Unicode char U+204E (decimal 8270) + defining Unicode char U+2052 (decimal 8274) + defining Unicode char U+20A1 (decimal 8353) + defining Unicode char U+20A4 (decimal 8356) + defining Unicode char U+20A6 (decimal 8358) + defining Unicode char U+20A9 (decimal 8361) + defining Unicode char U+20AB (decimal 8363) + defining Unicode char U+20AC (decimal 8364) + defining Unicode char U+20B1 (decimal 8369) + defining Unicode char U+2103 (decimal 8451) + defining Unicode char U+2116 (decimal 8470) + defining Unicode char U+2117 (decimal 8471) + defining Unicode char U+211E (decimal 8478) + defining Unicode char U+2120 (decimal 8480) + defining Unicode char U+2122 (decimal 8482) + defining Unicode char U+2126 (decimal 8486) + defining Unicode char U+2127 (decimal 8487) + defining Unicode char U+212E (decimal 8494) + defining Unicode char U+2190 (decimal 8592) + defining Unicode char U+2191 (decimal 8593) + defining Unicode char U+2192 (decimal 8594) + defining Unicode char U+2193 (decimal 8595) + defining Unicode char U+2329 (decimal 9001) + defining Unicode char U+232A (decimal 9002) + defining Unicode char U+2422 (decimal 9250) + defining Unicode char U+25E6 (decimal 9702) + defining Unicode char U+25EF (decimal 9711) + defining Unicode char U+266A (decimal 9834) +) + defining Unicode char U+00A9 (decimal 169) + defining Unicode char U+00AA (decimal 170) + defining Unicode char U+00AE (decimal 174) + defining Unicode char U+00BA (decimal 186) + defining Unicode char U+02C6 (decimal 710) + defining Unicode char U+02DC (decimal 732) + defining Unicode char U+200C (decimal 8204) + defining Unicode char U+2026 (decimal 8230) + defining Unicode char U+2122 (decimal 8482) + defining Unicode char U+2423 (decimal 9251) +)) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2013/01/14 v2.14 AMS math features +\@mathmargin=\skip194 + +For additional information on amsmath, use the `?' option. +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2000/06/29 v2.01 + +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 +\@emptytoks=\toks37 +\ex@=\dimen196 +)) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d +\pmbraise@=\dimen197 +) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +Package: amsopn 1999/12/14 v2.01 operator names +) +\inf@bad=\count173 +LaTeX Info: Redefining \frac on input line 210. +\uproot@=\count174 +\leftroot@=\count175 +LaTeX Info: Redefining \overline on input line 306. +\classnum@=\count176 +\DOTSCASE@=\count177 +LaTeX Info: Redefining \ldots on input line 378. +LaTeX Info: Redefining \dots on input line 381. +LaTeX Info: Redefining \cdots on input line 466. +\Mathstrutbox@=\box38 +\strutbox@=\box39 +\big@size=\dimen198 +LaTeX Font Info: Redeclaring font encoding OML on input line 566. +LaTeX Font Info: Redeclaring font encoding OMS on input line 567. +\macc@depth=\count178 +\c@MaxMatrixCols=\count179 +\dotsspace@=\muskip10 +\c@parentequation=\count180 +\dspbrk@lvl=\count181 +\tag@help=\toks38 +\row@=\count182 +\column@=\count183 +\maxfields@=\count184 +\andhelp@=\toks39 +\eqnshift@=\dimen199 +\alignsep@=\dimen200 +\tagshift@=\dimen201 +\tagwidth@=\dimen202 +\totwidth@=\dimen203 +\lineht@=\dimen204 +\@envbody=\toks40 +\multlinegap=\skip195 +\multlinetaggap=\skip196 +\mathdisplay@stack=\toks41 +LaTeX Info: Redefining \[ on input line 2665. +LaTeX Info: Redefining \] on input line 2666. +) +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 106. +) +(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty +Package: amsthm 2004/08/06 v2.20 +\thm@style=\toks42 +\thm@bodyfont=\toks43 +\thm@headfont=\toks44 +\thm@notefont=\toks45 +\thm@headpunct=\toks46 +\thm@preskip=\skip197 +\thm@postskip=\skip198 +\thm@headsep=\skip199 +\dth@everypar=\toks47 +) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thmtools.sty +Package: thmtools 2012/05/04 v63 +\thmt@toks=\toks48 +\c@thmt@dummyctr=\count185 + +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-patch.sty +Package: thm-patch 2012/05/04 v63 + +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/parseargs.sty +Package: parseargs 2012/05/04 v63 +\@parsespec=\toks49 +)) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-kv.sty +Package: thm-kv 2012/05/04 v63 +Package thm-kv Info: Theorem names will be uppercased on input line 40. + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvsetkeys.sty +Package: kvsetkeys 2012/04/25 v1.16 Key value parser (HO) + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/etexcmds.sty +Package: etexcmds 2011/02/16 v1.5 Avoid name clashes with e-TeX commands (HO) +Package etexcmds Info: Could not find \expanded. +(etexcmds) That can mean that you are not using pdfTeX 1.50 or +(etexcmds) that some package has redefined \expanded. +(etexcmds) In the latter case, load this package earlier. +)) +Package thm-kv Info: kvsetkeys patch (v1.16 or later) on input line 156. +) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-autoref.sty +Package: thm-autoref 2012/05/04 v63 + +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/aliasctr.sty +Package: aliasctr 2012/05/04 v63 + +(/usr/share/texlive/texmf-dist/tex/latex/carlisle/remreset.sty))) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-listof.sty +Package: thm-listof 2012/05/04 v63 +) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-restate.sty +Package: thm-restate 2012/05/04 v63 +) +(/usr/share/texlive/texmf-dist/tex/latex/thmtools/thm-amsthm.sty +Package: thm-amsthm 2012/05/04 v63 +\thmt@style@headstyle=\toks50 +)) +(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty +Package: hyperref 2012/11/06 v6.83m Hypertext links for LaTeX + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty +Package: hobsub-hyperref 2012/05/28 v1.13 Bundle oberdiek, subset hyperref (HO) + + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty +Package: hobsub-generic 2012/05/28 v1.13 Bundle oberdiek, subset generic (HO) +Package: hobsub 2012/05/28 v1.13 Construct package bundles (HO) +Package hobsub Info: Skipping package `infwarerr' (already loaded). +Package hobsub Info: Skipping package `ltxcmds' (already loaded). +Package hobsub Info: Skipping package `ifluatex' (already loaded). +Package: ifvtex 2010/03/01 v1.5 Detect VTeX and its facilities (HO) +Package ifvtex Info: VTeX not detected. +Package: intcalc 2007/09/27 v1.1 Expandable calculations with integers (HO) +Package hobsub Info: Skipping package `ifpdf' (already loaded). +Package hobsub Info: Skipping package `etexcmds' (already loaded). +Package hobsub Info: Skipping package `kvsetkeys' (already loaded). +Package: kvdefinekeys 2011/04/07 v1.3 Define keys (HO) +Package: pdftexcmds 2011/11/29 v0.20 Utility functions of pdfTeX for LuaTeX (HO +) +Package pdftexcmds Info: LuaTeX not detected. +Package pdftexcmds Info: \pdf@primitive is available. +Package pdftexcmds Info: \pdf@ifprimitive is available. +Package pdftexcmds Info: \pdfdraftmode found. +Package: pdfescape 2011/11/25 v1.13 Implements pdfTeX's escape features (HO) +Package: bigintcalc 2012/04/08 v1.3 Expandable calculations on big integers (HO +) +Package: bitset 2011/01/30 v1.1 Handle bit-vector datatype (HO) +Package: uniquecounter 2011/01/30 v1.2 Provide unlimited unique counter (HO) +) +Package hobsub Info: Skipping package `hobsub' (already loaded). +Package: letltxmacro 2010/09/02 v1.4 Let assignment for LaTeX macros (HO) +Package: hopatch 2012/05/28 v1.2 Wrapper for package hooks (HO) +Package: xcolor-patch 2011/01/30 xcolor patch +Package: atveryend 2011/06/30 v1.8 Hooks at the very end of document (HO) +Package atveryend Info: \enddocument detected (standard20110627). +Package: atbegshi 2011/10/05 v1.16 At begin shipout hook (HO) +Package: refcount 2011/10/16 v3.4 Data extraction from label references (HO) +Package: hycolor 2011/01/30 v1.7 Color options for hyperref/bookmark (HO) +) +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty +Package: auxhook 2011/03/04 v1.3 Hooks for auxiliary files (HO) +) +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty +Package: kvoptions 2011/06/30 v3.11 Key value format for package options (HO) +) +\@linkdim=\dimen205 +\Hy@linkcounter=\count186 +\Hy@pagecounter=\count187 + +(/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def +File: pd1enc.def 2012/11/06 v6.83m Hyperref: PDFDocEncoding definition (HO) +Now handling font encoding PD1 ... +... no UTF-8 mapping file for font encoding PD1 +) +\Hy@SavedSpaceFactor=\count188 + +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg +File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive +) +Package hyperref Info: Hyper figures OFF on input line 4443. +Package hyperref Info: Link nesting OFF on input line 4448. +Package hyperref Info: Hyper index ON on input line 4451. +Package hyperref Info: Plain pages OFF on input line 4458. +Package hyperref Info: Backreferencing OFF on input line 4463. +Package hyperref Info: Implicit mode ON; LaTeX internals redefined. +Package hyperref Info: Bookmarks ON on input line 4688. +\c@Hy@tempcnt=\count189 + +(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty +\Urlmuskip=\muskip11 +Package: url 2006/04/12 ver 3.3 Verb mode for urls, etc. +) +LaTeX Info: Redefining \url on input line 5041. +\XeTeXLinkMargin=\dimen206 +\Fld@menulength=\count190 +\Field@Width=\dimen207 +\Fld@charsize=\dimen208 +Package hyperref Info: Hyper figures OFF on input line 6295. +Package hyperref Info: Link nesting OFF on input line 6300. +Package hyperref Info: Hyper index ON on input line 6303. +Package hyperref Info: backreferencing OFF on input line 6310. +Package hyperref Info: Link coloring OFF on input line 6315. +Package hyperref Info: Link coloring with OCG OFF on input line 6320. +Package hyperref Info: PDF/A mode OFF on input line 6325. +LaTeX Info: Redefining \ref on input line 6365. +LaTeX Info: Redefining \pageref on input line 6369. +\Hy@abspage=\count191 +\c@Item=\count192 +\c@Hfootnote=\count193 +) +(/usr/share/texlive/texmf-dist/tex/latex/memoir/memhfixc.sty +Package: memhfixc 2013/05/30 v1.17 nameref/hyperref package fixes for memoir cl +ass +\c@memhycontfloat=\count194 +\c@Hpagenote=\count195 +) + +Package hyperref Message: Driver (autodetected): hpdftex. + +(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def +File: hpdftex.def 2012/11/06 v6.83m Hyperref driver for pdfTeX +\Fld@listcount=\count196 +\c@bookmark@seq@number=\count197 + +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty +Package: rerunfilecheck 2011/04/15 v1.7 Rerun checks for auxiliary files (HO) +Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2 +82. +) +\Hy@SectionHShift=\skip200 +) +Package hyperref Info: Option `colorlinks' set `true' on input line 26. + +(/usr/share/texlive/texmf-dist/tex/latex/cleveref/cleveref.sty +Package: cleveref 2013/03/22 v0.18.9 Intelligent cross-referencing +LaTeX Info: Redefining \Cref on input line 895. +Package cleveref Info: `hyperref' support loaded on input line 2163. +LaTeX Info: Redefining \cref on input line 2163. +LaTeX Info: Redefining \Cref on input line 2163. +LaTeX Info: Redefining \crefrange on input line 2163. +LaTeX Info: Redefining \Crefrange on input line 2163. +LaTeX Info: Redefining \cpageref on input line 2163. +LaTeX Info: Redefining \Cpageref on input line 2163. +LaTeX Info: Redefining \cpagerefrange on input line 2163. +LaTeX Info: Redefining \Cpagerefrange on input line 2163. +LaTeX Info: Redefining \labelcref on input line 2163. +LaTeX Info: Redefining \labelcpageref on input line 2163. +Package cleveref Info: `amsthm' support loaded on input line 2583. +) +(/usr/share/texlive/texmf-dist/tex/latex/caption/caption.sty +Package: caption 2013/05/02 v3.3-89 Customizing captions (AR) + +(/usr/share/texlive/texmf-dist/tex/latex/caption/caption3.sty +Package: caption3 2013/05/02 v1.6-88 caption3 kernel (AR) +Package caption3 Info: TeX engine: e-TeX on input line 57. +\captionmargin=\dimen209 +\captionmargin@=\dimen210 +\captionwidth=\dimen211 +\caption@tempdima=\dimen212 +\caption@indent=\dimen213 +\caption@parindent=\dimen214 +\caption@hangindent=\dimen215 +Package caption Info: The definition of \@makecaption has been changed, +(caption) frenchb will NOT customize it. +Package caption Info: Unknown document class (or package), +(caption) standard defaults will be used. +) + +Package caption Warning: Unsupported document class (or package) detected, +(caption) usage of the caption package is not recommended. +See the caption package documentation for explanation. + +Package caption Info: \@makecaption = \long macro:#1#2->\let \@memtempa \relax +\ifdim \prevdepth >-99\p@ \vskip \abovecaptionskip \else \def \@memtempa {\vbox + to\topskip {}}\fi \@@makecaption {#1}{#2}\vskip \belowcaptionskip . +\c@ContinuedFloat=\count198 +Package caption Info: hyperref package is loaded. +) (/usr/share/texlive/texmf-dist/tex/latex/auto-pst-pdf/auto-pst-pdf.sty +Package: auto-pst-pdf 2009/04/26 v0.6 Wrapper for pst-pdf + +(/usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty +Package: xkeyval 2012/10/14 v2.6b package option processing (HA) + +(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex +\XKV@toks=\toks51 +\XKV@tempa@toks=\toks52 +\XKV@depth=\count199 +File: xkeyval.tex 2012/10/14 v2.6b key=value parser (HA) +)) +(/usr/share/texlive/texmf-dist/tex/latex/ifplatform/ifplatform.sty +Package: ifplatform 2010/10/22 v0.4 Testing for the operating system + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/catchfile.sty +Package: catchfile 2011/03/01 v1.6 Catch the contents of a file (HO) +) + +Package ifplatform Warning: + shell escape is disabled, so I can only detect \ifwindows. + +) +\c@app@runs=\count200 + (/usr/share/texlive/texmf-dist/tex/latex/pst-pdf/pst-pdf.sty +Package: pst-pdf 2008/10/09 v1.1v PS graphics for pdfLaTeX (RN,HjG) +\c@pspicture=\count201 +Package pst-pdf Info: MODE: 1 (pdfTeX mode) on input line 214. + +(/usr/share/texlive/texmf-dist/tex/latex/pstricks/pstricks.sty +Package: pstricks 2013/05/22 v0.57 LaTeX wrapper for `PSTricks' (RN,HV) + +(/usr/share/texlive/texmf-dist/tex/generic/pstricks/pstricks.tex +(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/pst-xkey.tex +File: pst-xkey.tex 2005/11/25 v1.6 PSTricks specialization of xkeyval (HA) +) +(/usr/share/texlive/texmf-dist/tex/generic/pstricks/pst-fp.tex +`pst-fp' v0.05, 2010/01/17 (hv) +\pstFP@xs=\count202 +\pstFP@xia=\count203 +\pstFP@xib=\count204 +\pstFP@xfa=\count205 +\pstFP@xfb=\count206 +\pstFP@rega=\count207 +\pstFP@regb=\count208 +\pstFP@regs=\count209 +\pstFP@times=\count210 +) +(/usr/share/texmf/tex/generic/pgf/utilities/pgfutil-common.tex +\pgfutil@everybye=\toks53 +) +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeys.code.tex) +(/usr/share/texmf/tex/generic/pgf/utilities/pgffor.code.tex +\pgffor@iter=\dimen216 +\pgffor@skip=\dimen217 +\pgffor@stack=\toks54 +\pgffor@toks=\toks55 +) +\psLoopIndex=\count211 + +`PSTricks' v2.45 <2013/07/09> (tvz) +\pst@dima=\dimen218 +\pst@dimb=\dimen219 +\pst@dimc=\dimen220 +\pst@dimd=\dimen221 +\pst@dimg=\dimen222 +\pst@dimh=\dimen223 +\pst@dimm=\dimen224 +\pst@dimn=\dimen225 +\pst@dimo=\dimen226 +\pst@dimp=\dimen227 +\pst@hbox=\box40 +\pst@ibox=\box41 +\pst@boxg=\box42 +\pst@cnta=\count212 +\pst@cntb=\count213 +\pst@cntc=\count214 +\pst@cntd=\count215 +\pst@cntg=\count216 +\pst@cnth=\count217 +\pst@cntm=\count218 +\pst@cntn=\count219 +\pst@cnto=\count220 +\pst@cntp=\count221 +\@zero=\count222 +\pst@toks=\toks56 +(/usr/share/texlive/texmf-dist/tex/generic/pstricks/pstricks.con) +\psunit=\dimen228 +Normal \dimen register pool exhausted, switching to extended pool. +\psxunit=\dimen256 +\psyunit=\dimen257 +\pst@C@@rType=\count223 +\pslinewidth=\dimen258 +\psk@startLW=\dimen259 +\psk@endLW=\dimen260 +\pst@customdefs=\toks57 +\pslinearc=\dimen261 +\pst@symbolStep=\dimen262 +\pst@symbolWidth=\dimen263 +\pst@symbolLinewidth=\dimen264 +\everypsbox=\toks58 +\psframesep=\dimen265 +\pslabelsep=\dimen266 +\sh@wgridXunit=\dimen267 +\sh@wgridYunit=\dimen268 +\pst@shift=\dimen269 +) +File: pstricks.tex 2013/07/09 v2.45 `PSTricks' (tvz,hv) + +(/usr/share/texlive/texmf-dist/tex/generic/pstricks/pst-fp.tex) +File: pst-fp.tex 2013/07/09 v2.45 `PST-fp' (hv) +) +\ppf@output=\toks59 + +(/usr/share/texmf/tex/latex/preview/preview.sty +Package: preview 2010/02/14 11.87 (AUCTeX/preview-latex) +\pr@snippet=\count224 +\pr@box=\box43 +\pr@output=\toks60 +) +(/usr/share/texlive/texmf-dist/tex/latex/graphics/dvips.def +File: dvips.def 1999/02/16 v3.0i Driver-dependant file (DPC,SPQR) +) +(/usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +Package: environ 2013/04/01 v0.3 A new environment syntax + +(/usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +Package: trimspaces 2009/09/17 v1.1 Trim spaces around a token list +)))) +(./gabarit-maitrise.aux (./resume.aux) (./abstract.aux) (./remerciements.aux) +(./introduction.aux) (./chapitre1.aux) (./chapitre2.aux) (./chapitre3.aux) +(./chapitre4.aux) (./chapitre5.aux) (./chapitre6.aux) (./chapitre7.aux) +(./chapitre8.aux) (./chapitre9.aux) (./conclusion.aux) (./annexe1.aux) +(./annexe2.aux) (./annexe3.aux) (./deed.aux)) +\openout1 = `gabarit-maitrise.aux'. + +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 58. +LaTeX Font Info: Try loading font information for TS1+cmr on input line 58. + +(/usr/share/texlive/texmf-dist/tex/latex/base/ts1cmr.fd +File: ts1cmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions +) +LaTeX Font Info: ... okay on input line 58. +LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 58. +LaTeX Font Info: ... okay on input line 58. +LaTeX Info: Redefining \degres on input line 58. + + +Package frenchb.ldf Warning: The definition of \@makecaption has been changed, +(frenchb.ldf) frenchb will NOT customise it; +(frenchb.ldf) reported on input line 58. + +LaTeX Info: Redefining \dots on input line 58. +LaTeX Info: Redefining \up on input line 58. +(/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii +[Loading MPS to PDF converter (version 2006.09.02).] +\scratchcounter=\count225 +\scratchdimen=\dimen270 +\scratchbox=\box44 +\nofMPsegments=\count226 +\nofMParguments=\count227 +\everyMPshowfont=\toks61 +\MPscratchCnt=\count228 +\MPscratchDim=\dimen271 +Normal \count register pool exhausted, switching to extended pool. +\MPnumerator=\count277 +\makeMPintoPDFobject=\count278 +\everyMPtoPDFconversion=\toks62 +) ABD: EveryShipout initializing macros +\AtBeginShipoutBox=\box45 +Package hyperref Info: Link coloring ON on input line 58. + +(/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty +Package: nameref 2012/10/27 v2.43 Cross-referencing by name of section + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty +Package: gettitlestring 2010/12/03 v1.4 Cleanup title references (HO) +) +\c@section@level=\count279 +) +LaTeX Info: Redefining \ref on input line 58. +LaTeX Info: Redefining \pageref on input line 58. +LaTeX Info: Redefining \nameref on input line 58. + +(./gabarit-maitrise.out) (./gabarit-maitrise.out) +\@outlinefile=\write5 +\openout5 = `gabarit-maitrise.out'. + +Package caption Info: Begin \AtBeginDocument code. +Package caption Info: End \AtBeginDocument code. + +Preview: Fontsize 10.95pt +Preview: PDFoutput 1 +\c@theo=\count280 +\c@hypothese=\count281 +LaTeX Font Info: Try loading font information for T1+phv on input line 64. +(/usr/share/texlive/texmf-dist/tex/latex/psnfss/t1phv.fd +File: t1phv.fd 2001/06/04 scalable font definitions for T1/phv. +) +LaTeX Font Info: Font shape `T1/phv/m/n' will be +(Font) scaled to size 13.24796pt on input line 64. + + +File: ul_p.pdf Graphic file (type pdf) + +Package pdftex.def Info: ul_p.pdf used on input line 64. +(pdftex.def) Requested size: 97.10558pt x 40.15015pt. +LaTeX Font Info: Font shape `T1/phv/bx/n' in size <20.74> not available +(Font) Font shape `T1/phv/b/n' tried instead on input line 64. +LaTeX Font Info: Font shape `T1/phv/b/n' will be +(Font) scaled to size 19.08076pt on input line 64. +LaTeX Font Info: Font shape `T1/phv/bx/n' in size <17.28> not available +(Font) Font shape `T1/phv/b/n' tried instead on input line 64. +LaTeX Font Info: Font shape `T1/phv/b/n' will be +(Font) scaled to size 15.89755pt on input line 64. +LaTeX Font Info: Font shape `T1/phv/bx/n' in size <14.4> not available +(Font) Font shape `T1/phv/b/n' tried instead on input line 64. +LaTeX Font Info: Font shape `T1/phv/b/n' will be +(Font) scaled to size 13.24796pt on input line 64. +LaTeX Font Info: Try loading font information for TS1+phv on input line 64. + +(/usr/share/texlive/texmf-dist/tex/latex/psnfss/ts1phv.fd +File: ts1phv.fd 2001/06/04 scalable font definitions for TS1/phv. +) +LaTeX Font Info: Font shape `TS1/phv/m/n' will be +(Font) scaled to size 13.24796pt on input line 64. + [1 + + +Non-PDF special ignored! +Non-PDF special ignored! +Non-PDF special ignored! + + +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} <./ul_p.pdf>] [2 + +] +\openout2 = `resume.aux'. + + (./resume.tex) [3 + + +] +\openout2 = `abstract.aux'. + + (./abstract.tex [4 + + + +] +Package cleveref Info: loaded `english' language definitions on input line 4. +) [5] [6 + + +] +(./gabarit-maitrise.toc +LaTeX Font Info: Try loading font information for U+msa on input line 3. + +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2013/01/14 v3.01 AMS symbols A +) +LaTeX Font Info: Try loading font information for U+msb on input line 3. + +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2013/01/14 v3.01 AMS symbols B +) +Overfull \hbox (0.15294pt too wide) detected at line 7 + \T1/cmr/bx/n/10.95 xiii + [] + + +Overfull \hbox (3.27959pt too wide) detected at line 8 + \T1/cmr/bx/n/10.95 xvii + [] + +[7] [8] +Overfull \hbox (1.88837pt too wide) detected at line 106 + \T1/cmr/bx/n/10.95 113 + [] + + +Overfull \hbox (1.88837pt too wide) detected at line 107 + \T1/cmr/bx/n/10.95 115 + [] + +[9] +Overfull \hbox (1.88837pt too wide) detected at line 129 + \T1/cmr/bx/n/10.95 125 + [] + + +Overfull \hbox (1.88837pt too wide) detected at line 135 + \T1/cmr/bx/n/10.95 129 + [] + + +Overfull \hbox (1.88837pt too wide) detected at line 136 + \T1/cmr/bx/n/10.95 131 + [] + + +Overfull \hbox (1.88837pt too wide) detected at line 137 + \T1/cmr/bx/n/10.95 137 + [] + +) [10] (./gabarit-maitrise.lot) [11 + +] [12 + +] (./gabarit-maitrise.lof) [13] +[14 + +] [15 + +] [16 + +] +\openout2 = `remerciements.aux'. + + (./remerciements.tex) [17 + + +] [18 + + +] +\openout2 = `introduction.aux'. + + (./introduction.tex +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [1 + + +]) +[2] +\openout2 = `chapitre1.aux'. + + (./chapitre1.tex [3 + + + +] [4] [5] +Package hyperref Info: bookmark level for unknown hypothese defaults to 0 on in +put line 169. + [6] [7] [8] +<../varia/mitchell1.pdf, id=964, 505.89pt x 505.89pt> +File: ../varia/mitchell1.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../varia/mitchell1.pdf used on input line 388. +(pdftex.def) Requested size: 379.41655pt x 379.41655pt. + [9] [10 <../varia/mitchell1.pdf>] +Overfull \hbox (0.33621pt too wide) in paragraph at lines 421--423 +[]\T1/cmr/m/n/10.95 Il doit ad-mettre une pro-ba-bi-lité non nulle que plu-sieu +rs chan-ge-ments consé-cu-tifs semblent + [] + + +Underfull \hbox (badness 1424) in paragraph at lines 464--465 +[]|\T1/cmr/m/n/10.95 Défini seule-ment lorsque $\OML/cmm/m/it/10.95 \OMS/cmsy +/m/n/10.95 6\OT1/cmr/m/n/10.95 = 1$\T1/cmr/m/n/10.95 . + [] + +[11] [12] [13] [14] [15] [16]) [17] +\openout2 = `chapitre2.aux'. + + (./chapitre2.tex [18 + + + +] [19] [20] [21] +[22] (../graphiques/increment1.tex +\du=\skip201 +) +<"../graphiques/CH3-SIMGAMMAGAUSS".pdf, id=1158, 505.89pt x 505.89pt> +File: "../graphiques/CH3-SIMGAMMAGAUSS".pdf Graphic file (type pdf) + + +Package pdftex.def Info: "../graphiques/CH3-SIMGAMMAGAUSS".pdf used on input li +ne 233. +(pdftex.def) Requested size: 404.71254pt x 404.71254pt. + [23] +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [24 <../graphiques/CH3-SIMGAMMAGAUSS.pdf>] [25] [26] [27] [28] +Underfull \hbox (badness 1565) in paragraph at lines 537--538 +[]|\T1/cmr/m/n/10.95 Distribution asy-mé-trique à gauche + [] + + +Underfull \hbox (badness 2582) in paragraph at lines 537--538 +\T1/cmr/m/n/10.95 po-si-tif. Dé-place la moyenne dans + [] + +<../contenus/r/code/dGAL-exemples.pdf, id=1230, 505.89pt x 505.89pt> +File: ../contenus/r/code/dGAL-exemples.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/dGAL-exemples.pdf used on input lin +e 551. +(pdftex.def) Requested size: 404.71254pt x 404.71254pt. + [29] [30] [31 <../contenus/r/code/dGAL-exemples.pdf>] [32] +<"../graphiques/CH3-SIMULGAL0121".pdf, id=1282, 505.89pt x 505.89pt> +File: "../graphiques/CH3-SIMULGAL0121".pdf Graphic file (type pdf) + + +Package pdftex.def Info: "../graphiques/CH3-SIMULGAL0121".pdf used on input lin +e 735. +(pdftex.def) Requested size: 404.71254pt x 404.71254pt. + [33] [34 <../graphiques/CH3-SIMULGAL0121.pdf>] +Overfull \hbox (27.4862pt too wide) in paragraph at lines 804--828 + [][] + [] + +[35] [36] + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\begingroup' on input line 930. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\-command' on input line 930. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\NAT@ctype' on input line 930. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\z@' on input line 930. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\let' on input line 930. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\let' on input line 930. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\new@ifnextchar' on input line 930. + + +LaTeX Warning: Command \textendash invalid in math mode on input line 966. + +) [37] +\openout2 = `chapitre3.aux'. + + (./chapitre3.tex [38 + + + +] (../graphiques/probdroite.tex) [39] +(../graphiques/normaletronque.tex) [40] [41] [42] [43] [44] [45] [46]) [47] +\openout2 = `chapitre4.aux'. + + +(./chapitre4.tex [48 + + + +] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] +[59] +Overfull \hbox (22.6247pt too wide) in paragraph at lines 780--789 +[]\T1/cmr/m/n/10.95 Comme le rang de la ma-trice $[]\OML/cmm/m/it/10.95 a\OT1/c +mr/m/n/10.95 (\OML/cmm/m/it/10.95 ^^R[]\OT1/cmr/m/n/10.95 )$ \T1/cmr/m/n/10.95 +est égal au nombre de contraintes $\OML/cmm/m/it/10.95 r$\T1/cmr/m/n/10.95 , al +ors $[]\OML/cmm/m/it/10.95 a\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 ^^R[]\OT1/c +mr/m/n/10.95 )\OMS/cmsy/m/n/10.95 J[][]\OML/cmm/m/it/10.95 a\OT1/cmr/m/n/10.95 +(\OML/cmm/m/it/10.95 ^^R[]\OT1/cmr/m/n/10.95 )[]$ + [] + +[60] [61] [62]) [63] +\openout2 = `chapitre5.aux'. + + (./chapitre5.tex [64 + + + +] [65] [66] [67] +Overfull \hbox (7.84862pt too wide) in paragraph at lines 238--238 +[] + [] + +[68] [69] +Overfull \hbox (26.07318pt too wide) in paragraph at lines 329--329 +[] + [] + +[70]) [71] +\openout2 = `chapitre6.aux'. + + (./chapitre6.tex [72 + + + +] +Underfull \vbox (badness 1354) has occurred while \output is active [] + + [73] +[74] +Overfull \hbox (40.77754pt too wide) in paragraph at lines 153--153 +[] + [] + + +Underfull \vbox (badness 4872) has occurred while \output is active [] + + [75] +Overfull \hbox (7.56583pt too wide) in paragraph at lines 182--186 +[]\T1/cmr/m/n/10.95 On ob-tient la variance-covariance asymp-to-tique des pa-ra +-mètres à par-tir de la variance-covariance + [] + +[76] [77] +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [78]) +[79] +\openout2 = `chapitre7.aux'. + + (./chapitre7.tex [80 + + + +] [81] [82] + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `math shift' on input line 145. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `\chi' on input line 145. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `superscript' on input line 145. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `math shift' on input line 145. + +[83] + +Package amsmath Warning: Foreign command \over; +(amsmath) \frac or \genfrac should be used instead +(amsmath) on input line 226. + +[84] [85]) [86] +\openout2 = `chapitre8.aux'. + + (./chapitre8.tex [87 + + + +] [88] +Overfull \hbox (8.09332pt too wide) in paragraph at lines 158--160 +[]\T1/cmr/m/n/10.95 La dis-tri-bu-tion neutre au risque des ren-de-ments est do +nc $\OML/cmm/m/it/10.95 R[] \OMS/cmsy/m/n/10.95 ^^X \OML/cmm/m/it/10.95 GAL\OT1 +/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 r \OT1/cmr/m/n/10.95 + \OML/cmm/m/it/10.95 +^^\ []\OT1/cmr/m/n/10.95 (1 \OMS/cmsy/m/n/10.95 ^^@ \OML/cmm/m/it/10.95 ^^V \OM +S/cmsy/m/n/10.95 ^^@ \OML/cmm/m/it/10.95 ^^[[]=\OT1/cmr/m/n/10.95 2)\OML/cmm/m/ +it/10.95 ; ^^[; ^^V; ^^\\OT1/cmr/m/n/10.95 )$\T1/cmr/m/n/10.95 . + [] + +[89] [90] [91] [92] [93] [94]) [95] +\openout2 = `chapitre9.aux'. + + (./chapitre9.tex [96 + + + +] +<../contenus/r/code/ABBEYN-chronologie.pdf, id=2151, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-chronologie.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-chronologie.pdf used on inpu +t line 21. +(pdftex.def) Requested size: 289.09473pt x 289.09473pt. + +<../contenus/r/code/ABBEYN-histogramme.pdf, id=2155, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-histogramme.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-histogramme.pdf used on inpu +t line 72. +(pdftex.def) Requested size: 289.09473pt x 289.09473pt. + [97] +<../contenus/r/code/ABBEYN-qq.pdf, id=2168, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-qq.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-qq.pdf used on input line 11 +2. +(pdftex.def) Requested size: 289.09473pt x 289.09473pt. + [98 <../contenus/r/code/ABBEYN-chronologie.pdf>] [99 <../contenus/r/code/ABBEY +N-histogramme.pdf>] [100 <../contenus/r/code/ABBEYN-qq.pdf>] +Underfull \hbox (badness 10000) in paragraph at lines 200--211 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 200--211 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 212--223 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 212--223 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 224--235 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 224--235 + + [] + +[101] +Underfull \hbox (badness 10000) in paragraph at lines 318--329 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 318--329 + + [] + +[102] +Underfull \hbox (badness 10000) in paragraph at lines 329--340 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 329--340 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 340--351 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 340--351 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 351--362 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 351--362 + + [] + + +Underfull \vbox (badness 2189) has occurred while \output is active [] + + [103] +<../contenus/maxima/pointdeselleGMM.png, id=2239, 501.875pt x 301.125pt> +File: ../contenus/maxima/pointdeselleGMM.png Graphic file (type png) + + +Package pdftex.def Info: ../contenus/maxima/pointdeselleGMM.png used on input l +ine 398. +(pdftex.def) Requested size: 250.93687pt x 150.56212pt. + +Underfull \vbox (badness 1642) has occurred while \output is active [] + + [104 <../contenus/maxima/pointdeselleGMM.png>] +<../contenus/r/code/ABBEYN-densiteGALmu-7.pdf, id=2255, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-densiteGALmu-7.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-densiteGALmu-7.pdf used on i +nput line 516. +(pdftex.def) Requested size: 433.63878pt x 433.63878pt. + +<../contenus/r/code/ABBEYN-densiteGALmu-5.pdf, id=2256, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-densiteGALmu-5.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-densiteGALmu-5.pdf used on i +nput line 525. +(pdftex.def) Requested size: 433.63878pt x 433.63878pt. + [105] [106 <../contenus/r/code/ABBEYN-densiteGALmu-7.pdf>] [107 <../contenus/r +/code/ABBEYN-densiteGALmu-5.pdf>] [108 + +] +<../contenus/r/code/ABBEYN-callGAL-7.pdf, id=2301, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-callGAL-7.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-callGAL-7.pdf used on input +line 684. +(pdftex.def) Requested size: 433.63878pt x 433.63878pt. + [109] +<../contenus/r/code/ABBEYN-callGAL-5.pdf, id=2307, 433.62pt x 433.62pt> +File: ../contenus/r/code/ABBEYN-callGAL-5.pdf Graphic file (type pdf) + + +Package pdftex.def Info: ../contenus/r/code/ABBEYN-callGAL-5.pdf used on input +line 693. +(pdftex.def) Requested size: 433.63878pt x 433.63878pt. +) [110 <../contenus/r/code/ABBEYN-callGAL-7.pdf>] [111 <../contenus/r/code/ABBE +YN-callGAL-5.pdf>] +\openout2 = `conclusion.aux'. + + +(./conclusion.tex [112 + + + +]) [113] +\openout2 = `annexe1.aux'. + + (./annexe1.tex [114 + + + +] [115] [116] [117] [118] +[119] [120] [121] [122] [123]) [124] +\openout2 = `annexe2.aux'. + + (./annexe2.tex +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [125 + + + +] +[126]) [127] +\openout2 = `annexe3.aux'. + + (./annexe3.tex [128 + + + +]) [129] (./gabarit-maitrise.bbl [130 + + +] +LaTeX Font Info: Try loading font information for T1+cmtt on input line 31. + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1cmtt.fd +File: t1cmtt.fd 1999/05/25 v2.5h Standard LaTeX font definitions +) [131] [132] [133] +[134]) [135] +\openout2 = `deed.aux'. + + (./deed.tex [136 + + +] +Overfull \hbox (30.35516pt too wide) in paragraph at lines 5--9 +\T1/cmr/m/it/10.95 Condi-tions 4.0 In-ter-na-tio-nal \T1/cmr/m/n/10.95 de Crea- +tive Com-mons dis-po-nible à l'adresse [][]$\T1/cmtt/m/n/10.95 http : / / creat +ivecommons . + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 18--20 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 21--22 + + [] + + + +File: by.pdf Graphic file (type pdf) + + +Package pdftex.def Info: by.pdf used on input line 33. +(pdftex.def) Requested size: 31.29811pt x 31.29802pt. +File: sa.pdf Graphic file (type pdf) + +Package pdftex.def Info: sa.pdf used on input line 33. +(pdftex.def) Requested size: 31.29811pt x 31.29802pt. + +Underfull \hbox (badness 10000) in paragraph at lines 33--38 + + [] + + +File: by-sa.pdf Graphic file (type pdf) + +Package pdftex.def Info: by-sa.pdf used on input line 40. +(pdftex.def) Requested size: 56.04544pt x 19.91692pt. +) [137 <./by.pdf> <./sa.pdf> <./by-sa.pdf>] +\tf@toc=\write6 +\openout6 = `gabarit-maitrise.toc'. + +\tf@lot=\write7 +\openout7 = `gabarit-maitrise.lot'. + +\tf@lof=\write8 +\openout8 = `gabarit-maitrise.lof'. + +Package atveryend Info: Empty hook `BeforeClearDocument' on input line 111. +Package atveryend Info: Empty hook `AfterLastShipout' on input line 111. + (./gabarit-maitrise.aux (./resume.aux) +(./abstract.aux) (./remerciements.aux) (./introduction.aux) (./chapitre1.aux) +(./chapitre2.aux) (./chapitre3.aux) (./chapitre4.aux) (./chapitre5.aux) +(./chapitre6.aux) (./chapitre7.aux) (./chapitre8.aux) (./chapitre9.aux) +(./conclusion.aux) (./annexe1.aux) (./annexe2.aux) (./annexe3.aux) (./deed.aux) +) +Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 111. +Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 111. +Package rerunfilecheck Info: File `gabarit-maitrise.out' has not changed. +(rerunfilecheck) Checksum: AF4E0EC0F1038C50F1FBB2296D912C5B;10760. +Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 111. + ) +Here is how much of TeX's memory you used: + 27782 strings out of 493307 + 473537 string characters out of 6139904 + 657511 words of memory out of 5000000 + 29742 multiletter control sequences out of 15000+600000 + 40296 words of font info for 84 fonts, out of 8000000 for 9000 + 957 hyphenation exceptions out of 8191 + 65i,23n,77p,590b,691s stack positions out of 5000i,500n,10000p,200000b,80000s +{/usr/share/texmf/fonts/enc/dvips/cm-super/cm-super-t1.enc}{/usr/share/texliv +e/texmf-dist/fonts/enc/dvips/base/8r.enc} +Output written on gabarit-maitrise.pdf (155 pages, 1022221 bytes). +PDF statistics: + 2858 PDF objects out of 2984 (max. 8388607) + 2604 compressed objects within 27 object streams + 830 named destinations out of 1000 (max. 500000) + 1161 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/memoire/gabarit-maitrise.lot b/memoire/gabarit-maitrise.lot new file mode 100644 index 0000000..754ddc0 --- /dev/null +++ b/memoire/gabarit-maitrise.lot @@ -0,0 +1,38 @@ +\select@language {french} +\select@language {french} +\addvspace {10pt} +\contentsline {table}{\numberline {1.1}{\ignorespaces Domaine et r\IeC {\^o}le des param\IeC {\`e}tres de la distribution L stable de Mandelbrot\relax }}{12}{table.caption.14} +\contentsline {table}{\numberline {1.2}{\ignorespaces Respect des conditions \IeC {\'e}mises par Madan et Seneta pour les diff\IeC {\'e}rents mod\IeC {\`e}les pr\IeC {\'e}sent\IeC {\'e}s\relax }}{17}{table.caption.15} +\addvspace {10pt} +\contentsline {table}{\numberline {2.1}{\ignorespaces Domaine et r\IeC {\^o}le des param\IeC {\`e}tres de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{30}{table.caption.18} +\contentsline {table}{\numberline {2.2}{\ignorespaces Cas sp\IeC {\'e}ciaux de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{35}{table.caption.21} +\contentsline {table}{\numberline {2.3}{\ignorespaces Changements de param\IeC {\'e}trisation\relax }}{37}{table.caption.22} +\addvspace {10pt} +\addvspace {10pt} +\contentsline {table}{\numberline {4.1}{\ignorespaces Tests d'hypoth\IeC {\`e}se param\IeC {\'e}triques pour la m\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e\relax }}{63}{table.caption.25} +\addvspace {10pt} +\contentsline {table}{\numberline {5.1}{\ignorespaces M\IeC {\'e}thodes d'estimation repr\IeC {\'e}sentables par la forme quadratique \textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {eq:generalquad}\unskip \@@italiccorr )}} \relax }}{67}{table.caption.26} +\addvspace {10pt} +\contentsline {table}{\numberline {6.1}{\ignorespaces Contraintes lin\IeC {\'e}aires pour les cas particuliers de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e $GAL(\theta ,\sigma ,\mu ,\tau )$\relax }}{77}{table.caption.27} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\contentsline {table}{\numberline {9.1}{\ignorespaces Statistiques d'ordre de l'\IeC {\'e}chantillon $R_1$\relax }}{97}{table.caption.29} +\contentsline {table}{\numberline {9.2}{\ignorespaces Valeurs relatives aux premiers moments de l'\IeC {\'e}chantillon $R_1$\relax }}{98}{table.caption.30} +\contentsline {table}{\numberline {9.3}{\ignorespaces Test de normalit\IeC {\'e} d'Epps-Pulley pour $R_1$\relax }}{98}{table.caption.32} +\contentsline {table}{\numberline {9.4}{\ignorespaces Param\IeC {\`e}tres $\theta _1$ de la premi\IeC {\`e}re optimisation\relax }}{100}{table.caption.34} +\contentsline {table}{\numberline {9.5}{\ignorespaces Param\IeC {\`e}tres $\theta _1$ de la premi\IeC {\`e}re optimisation\relax }}{102}{table.caption.35} +\contentsline {table}{\numberline {9.6}{\ignorespaces Param\IeC {\`e}tres des donn\IeC {\'e}es $R_1$\relax }}{103}{table.caption.36} +\contentsline {table}{\numberline {9.7}{\ignorespaces Approximation de la densit\IeC {\'e} de $R_1$\relax }}{104}{table.caption.38} +\contentsline {table}{\numberline {9.8}{\ignorespaces Approximation de la fonction de r\IeC {\'e}partition de $R_1$\relax }}{105}{table.caption.39} +\contentsline {table}{\numberline {9.9}{\ignorespaces Courbes de densit\IeC {\'e}\relax }}{105}{table.caption.40} +\contentsline {table}{\numberline {9.10}{\ignorespaces Valeur de l'int\IeC {\'e}grale de l'approximation de la densit\IeC {\'e} par la m\IeC {\'e}thode du point de selle \relax }}{105}{table.caption.41} +\contentsline {table}{\numberline {9.11}{\ignorespaces Test du $\chi ^2$\relax }}{108}{table.caption.44} +\contentsline {table}{\numberline {9.12}{\ignorespaces Test de Kolmogorov-Smirnov\relax }}{108}{table.caption.45} +\contentsline {table}{\numberline {9.13}{\ignorespaces Test de distance minimale bas\IeC {\'e} sur la fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments\relax }}{109}{table.caption.46} +\contentsline {table}{\numberline {9.14}{\ignorespaces Caract\IeC {\'e}ristiques de l'option\relax }}{109}{table.caption.47} +\contentsline {table}{\numberline {9.15}{\ignorespaces Param\IeC {\`e}tres neutres au risque\relax }}{109}{table.caption.48} +\addvspace {10pt} +\addvspace {10pt} +\addvspace {10pt} +\contentsline {table}{\numberline {C.1}{\ignorespaces Prix du titre Abbey National (penny sterling) du 31 juillet au 8 octobre 1991\relax }}{129}{table.caption.60} diff --git a/memoire/gabarit-maitrise.out b/memoire/gabarit-maitrise.out new file mode 100644 index 0000000..fd914bf --- /dev/null +++ b/memoire/gabarit-maitrise.out @@ -0,0 +1,135 @@ +\BOOKMARK [0][-]{section*.2}{R\351sum\351}{}% 1 +\BOOKMARK [0][-]{section*.4}{Abstract}{}% 2 +\BOOKMARK [0][-]{section*.5}{Table des mati\350res}{}% 3 +\BOOKMARK [0][-]{section*.6}{Liste des tableaux}{}% 4 +\BOOKMARK [0][-]{section*.7}{Liste des figures}{}% 5 +\BOOKMARK [0][-]{section*.9}{Remerciements}{}% 6 +\BOOKMARK [0][-]{section*.11}{Introduction}{}% 7 +\BOOKMARK [0][-]{chapter.1}{Les mod\350les de rendements financiers}{}% 8 +\BOOKMARK [1][-]{section.1.1}{L'utilisation de mod\350les en finance}{chapter.1}% 9 +\BOOKMARK [2][-]{subsection.1.1.1}{Diff\351rents types de mod\350les}{section.1.1}% 10 +\BOOKMARK [2][-]{subsection.1.1.2}{Le risque de mod\351lisation}{section.1.1}% 11 +\BOOKMARK [1][-]{section.1.2}{Les rendements financiers}{chapter.1}% 12 +\BOOKMARK [2][-]{subsection.1.2.1}{D\351finitions et notations}{section.1.2}% 13 +\BOOKMARK [2][-]{subsection.1.2.2}{Rendements cumul\351s}{section.1.2}% 14 +\BOOKMARK [2][-]{subsection.1.2.3}{Donn\351es disponibles}{section.1.2}% 15 +\BOOKMARK [1][-]{section.1.3}{Les premiers mod\350les}{chapter.1}% 16 +\BOOKMARK [2][-]{subsection.1.3.1}{Le mod\350le de Bachelier}{section.1.3}% 17 +\BOOKMARK [2][-]{subsection.1.3.2}{Proposition de Mandelbrot}{section.1.3}% 18 +\BOOKMARK [2][-]{subsection.1.3.3}{Le mod\350le de Press}{section.1.3}% 19 +\BOOKMARK [2][-]{subsection.1.3.4}{Le mod\350le de Praetz}{section.1.3}% 20 +\BOOKMARK [1][-]{section.1.4}{Conditions essentielles de Madan et Seneta}{chapter.1}% 21 +\BOOKMARK [0][-]{chapter.2}{La distribution de Laplace asym\351trique g\351n\351ralis\351e}{}% 22 +\BOOKMARK [1][-]{section.2.1}{Le processus de Laplace}{chapter.2}% 23 +\BOOKMARK [2][-]{subsection.2.1.1}{Le processus gamma}{section.2.1}% 24 +\BOOKMARK [2][-]{subsection.2.1.2}{Le processus de Wiener}{section.2.1}% 25 +\BOOKMARK [2][-]{subsection.2.1.3}{Le processus de Laplace est un processus subordonn\351}{section.2.1}% 26 +\BOOKMARK [1][-]{section.2.2}{Distribution de Laplace asym\351trique g\351n\351ralis\351e}{chapter.2}% 27 +\BOOKMARK [2][-]{subsection.2.2.1}{Fonction caract\351ristique}{section.2.2}% 28 +\BOOKMARK [2][-]{subsection.2.2.2}{Invariance d'\351chelle}{section.2.2}% 29 +\BOOKMARK [2][-]{subsection.2.2.3}{Fonctions g\351n\351ratrices}{section.2.2}% 30 +\BOOKMARK [2][-]{subsection.2.2.4}{Moments et r\364le des param\350tres}{section.2.2}% 31 +\BOOKMARK [2][-]{subsection.2.2.5}{Changement d'\351chelle et de localisation}{section.2.2}% 32 +\BOOKMARK [2][-]{subsection.2.2.6}{Repr\351sentation alternative et simulation}{section.2.2}% 33 +\BOOKMARK [2][-]{subsection.2.2.7}{Fonction de Bessel et densit\351}{section.2.2}% 34 +\BOOKMARK [1][-]{section.2.3}{Cas particuliers}{chapter.2}% 35 +\BOOKMARK [2][-]{subsection.2.3.1}{Distribution de Laplace asym\351trique}{section.2.3}% 36 +\BOOKMARK [1][-]{section.2.4}{Relation avec le mod\350le de madan1990variance}{chapter.2}% 37 +\BOOKMARK [0][-]{chapter.3}{Approximation de la densit\351 et de la fonction de r\351partition}{}% 38 +\BOOKMARK [1][-]{section.3.1}{L'approximation de Laplace}{chapter.3}% 39 +\BOOKMARK [1][-]{section.3.2}{L'approximation de Temme}{chapter.3}% 40 +\BOOKMARK [1][-]{section.3.3}{La m\351thode du point de selle}{chapter.3}% 41 +\BOOKMARK [2][-]{subsection.3.3.1}{Approximation de la densit\351}{section.3.3}% 42 +\BOOKMARK [2][-]{subsection.3.3.2}{Unicit\351 du point de selle}{section.3.3}% 43 +\BOOKMARK [2][-]{subsection.3.3.3}{Approximation de la fonction de r\351partition}{section.3.3}% 44 +\BOOKMARK [2][-]{subsection.3.3.4}{Quelques propri\351t\351s des approximations}{section.3.3}% 45 +\BOOKMARK [1][-]{section.3.4}{Application de la m\351thode du point de selle}{chapter.3}% 46 +\BOOKMARK [2][-]{subsection.3.4.1}{Approximation de la densit\351}{section.3.4}% 47 +\BOOKMARK [2][-]{subsection.3.4.2}{Approximation de la fonction de r\351partition}{section.3.4}% 48 +\BOOKMARK [0][-]{chapter.4}{M\351thode des moments g\351n\351ralis\351e}{}% 49 +\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4}% 50 +\BOOKMARK [2][-]{subsection.4.1.1}{M\351thode classique des moments}{section.4.1}% 51 +\BOOKMARK [1][-]{section.4.2}{M\351thode des moments g\351n\351ralis\351e}{chapter.4}% 52 +\BOOKMARK [2][-]{subsection.4.2.1}{D\351finition}{section.4.2}% 53 +\BOOKMARK [2][-]{subsection.4.2.2}{Convergence}{section.4.2}% 54 +\BOOKMARK [2][-]{subsection.4.2.3}{Matrice de pond\351ration optimale}{section.4.2}% 55 +\BOOKMARK [2][-]{subsection.4.2.4}{M\351thode des moments g\351n\351ralis\351e it\351rative}{section.4.2}% 56 +\BOOKMARK [2][-]{subsection.4.2.5}{Distribution asymptotique des estimateurs}{section.4.2}% 57 +\BOOKMARK [1][-]{section.4.3}{Estimation sous contraintes}{chapter.4}% 58 +\BOOKMARK [2][-]{subsection.4.3.1}{Distribution asymptotique des estimateurs contraints}{section.4.3}% 59 +\BOOKMARK [1][-]{section.4.4}{Tests d'hypoth\350ses param\351triques}{chapter.4}% 60 +\BOOKMARK [2][-]{subsection.4.4.1}{Test de Wald}{section.4.4}% 61 +\BOOKMARK [2][-]{subsection.4.4.2}{Test du multiplicateur de Lagrange}{section.4.4}% 62 +\BOOKMARK [2][-]{subsection.4.4.3}{Test bas\351 sur la statistique de m\351trique de distance}{section.4.4}% 63 +\BOOKMARK [2][-]{subsection.4.4.4}{En r\351sum\351}{section.4.4}% 64 +\BOOKMARK [0][-]{chapter.5}{M\351thode de l'\351quation d'estimation optimale}{}% 65 +\BOOKMARK [1][-]{section.5.1}{\311quation d'estimation optimale}{chapter.5}% 66 +\BOOKMARK [1][-]{section.5.2}{\311quation d'estimation optimale modifi\351e}{chapter.5}% 67 +\BOOKMARK [0][-]{chapter.6}{Estimation des param\350tres de la distribution de Laplace asym\351trique g\351n\351ralis\351e}{}% 68 +\BOOKMARK [1][-]{section.6.1}{Vecteur de param\350tres initiaux}{chapter.6}% 69 +\BOOKMARK [1][-]{section.6.2}{M\351thode des moments g\351n\351ralis\351e}{chapter.6}% 70 +\BOOKMARK [2][-]{subsection.6.2.1}{Matrice de pond\351ration optimale}{section.6.2}% 71 +\BOOKMARK [2][-]{subsection.6.2.2}{Variance-covariance des param\350tres}{section.6.2}% 72 +\BOOKMARK [2][-]{subsection.6.2.3}{Contraintes lin\351aires}{section.6.2}% 73 +\BOOKMARK [1][-]{section.6.3}{M\351thode de l'\351quation d'estimation optimale}{chapter.6}% 74 +\BOOKMARK [0][-]{chapter.7}{Tests statistiques}{}% 75 +\BOOKMARK [1][-]{section.7.1}{Test de normalit\351}{chapter.7}% 76 +\BOOKMARK [2][-]{subsection.7.1.1}{Test de Shapiro-Wilk}{section.7.1}% 77 +\BOOKMARK [2][-]{subsection.7.1.2}{Test d\220Epps-Pulley}{section.7.1}% 78 +\BOOKMARK [1][-]{section.7.2}{Tests d'ad\351quation}{chapter.7}% 79 +\BOOKMARK [2][-]{subsection.7.2.1}{Test 2 de Pearson}{section.7.2}% 80 +\BOOKMARK [2][-]{subsection.7.2.2}{Test de Kolmogorov-Smirnov}{section.7.2}% 81 +\BOOKMARK [2][-]{subsection.7.2.3}{Test de distance minimale bas\351 sur la fonction g\351n\351ratrice des moments}{section.7.2}% 82 +\BOOKMARK [0][-]{chapter.8}{\311valuation d'options}{}% 83 +\BOOKMARK [1][-]{section.8.1}{D\351finitions}{chapter.8}% 84 +\BOOKMARK [2][-]{subsection.8.1.1}{\311quation martingale}{section.8.1}% 85 +\BOOKMARK [2][-]{subsection.8.1.2}{Param\350tres neutres au risque}{section.8.1}% 86 +\BOOKMARK [1][-]{section.8.2}{Aper\347u du mod\350le de Black-Scholes}{chapter.8}% 87 +\BOOKMARK [1][-]{section.8.3}{M\351thodes d'\351valuation pour options europ\351ennes}{chapter.8}% 88 +\BOOKMARK [2][-]{subsection.8.3.1}{M\351thode de Heston}{section.8.3}% 89 +\BOOKMARK [2][-]{subsection.8.3.2}{M\351thode de Carr et Madan}{section.8.3}% 90 +\BOOKMARK [2][-]{subsection.8.3.3}{Prix d'exercice hors du cours}{section.8.3}% 91 +\BOOKMARK [2][-]{subsection.8.3.4}{Critique de la m\351thode de Carr-Madan}{section.8.3}% 92 +\BOOKMARK [2][-]{subsection.8.3.5}{M\351thode d\220Epps}{section.8.3}% 93 +\BOOKMARK [1][-]{section.8.4}{Particularit\351s}{chapter.8}% 94 +\BOOKMARK [2][-]{subsection.8.4.1}{Option sur actions avec dividendes}{section.8.4}% 95 +\BOOKMARK [2][-]{subsection.8.4.2}{Options sur contrats \340 terme et taux de change}{section.8.4}% 96 +\BOOKMARK [0][-]{chapter.9}{Exemple d'application}{}% 97 +\BOOKMARK [1][-]{section.9.1}{Description des donn\351es}{chapter.9}% 98 +\BOOKMARK [1][-]{section.9.2}{Estimation}{chapter.9}% 99 +\BOOKMARK [1][-]{section.9.3}{Approximation}{chapter.9}% 100 +\BOOKMARK [1][-]{section.9.4}{Graphiques}{chapter.9}% 101 +\BOOKMARK [1][-]{section.9.5}{Tests statistiques}{chapter.9}% 102 +\BOOKMARK [1][-]{section.9.6}{\311valuation d'options}{chapter.9}% 103 +\BOOKMARK [0][-]{section*.52}{Conclusion}{}% 104 +\BOOKMARK [0][-]{appendix.A}{\311l\351ments de th\351orie des probabilit\351s}{}% 105 +\BOOKMARK [1][-]{section.A.1}{D\351finitions de base}{appendix.A}% 106 +\BOOKMARK [1][-]{section.A.2}{Transform\351es d'une variable al\351atoire}{appendix.A}% 107 +\BOOKMARK [2][-]{subsection.A.2.1}{La fonction caract\351ristique}{section.A.2}% 108 +\BOOKMARK [3][-]{section*.53}{Transform\351e de Fourier}{subsection.A.2.1}% 109 +\BOOKMARK [3][-]{section*.54}{D\351finition}{subsection.A.2.1}% 110 +\BOOKMARK [3][-]{section*.55}{Les moments}{subsection.A.2.1}% 111 +\BOOKMARK [2][-]{subsection.A.2.2}{Inversion de la fonction caract\351ristique}{section.A.2}% 112 +\BOOKMARK [3][-]{section*.56}{La densit\351}{subsection.A.2.2}% 113 +\BOOKMARK [3][-]{section*.57}{La fonction de r\351partition}{subsection.A.2.2}% 114 +\BOOKMARK [2][-]{subsection.A.2.3}{La fonction g\351n\351ratrice des moments}{section.A.2}% 115 +\BOOKMARK [2][-]{subsection.A.2.4}{La fonction g\351n\351ratrice des cumulants}{section.A.2}% 116 +\BOOKMARK [2][-]{subsection.A.2.5}{La transform\351e d'Esscher}{section.A.2}% 117 +\BOOKMARK [1][-]{section.A.3}{La transform\351e de Fourier rapide}{appendix.A}% 118 +\BOOKMARK [1][-]{section.A.4}{Processus de L\351vy}{appendix.A}% 119 +\BOOKMARK [2][-]{subsection.A.4.1}{D\351finition et propri\351t\351s}{section.A.4}% 120 +\BOOKMARK [3][-]{section*.58}{Repr\351sentation de L\351vy-Khintchine}{subsection.A.4.1}% 121 +\BOOKMARK [3][-]{section*.59}{Repr\351sentation de L\351vy-It\364}{subsection.A.4.1}% 122 +\BOOKMARK [2][-]{subsection.A.4.2}{Processus subordonn\351}{section.A.4}% 123 +\BOOKMARK [1][-]{section.A.5}{Th\351or\350mes d'int\351gration}{appendix.A}% 124 +\BOOKMARK [2][-]{subsection.A.5.1}{Th\351or\350me de convergence domin\351e de Lebesgue}{section.A.5}% 125 +\BOOKMARK [2][-]{subsection.A.5.2}{Th\351or\350me de Fubini}{section.A.5}% 126 +\BOOKMARK [0][-]{appendix.B}{\311l\351ments de statistique math\351matique}{}% 127 +\BOOKMARK [1][-]{section.B.1}{Loi faible des grands nombres}{appendix.B}% 128 +\BOOKMARK [1][-]{section.B.2}{Th\351or\350me central limite}{appendix.B}% 129 +\BOOKMARK [2][-]{subsection.B.2.1}{Cas univari\351}{section.B.2}% 130 +\BOOKMARK [2][-]{subsection.B.2.2}{Cas multivari\351}{section.B.2}% 131 +\BOOKMARK [1][-]{section.B.3}{M\351thode delta multivari\351e}{appendix.B}% 132 +\BOOKMARK [0][-]{appendix.C}{Donn\351es}{}% 133 +\BOOKMARK [0][-]{section*.62}{Bibliographie}{}% 134 +\BOOKMARK [0][-]{section*.64}{Contrat de partage}{}% 135 diff --git a/memoire/gabarit-maitrise.pdf b/memoire/gabarit-maitrise.pdf new file mode 100644 index 0000000..ca21342 Binary files /dev/null and b/memoire/gabarit-maitrise.pdf differ diff --git a/memoire/gabarit-maitrise.synctex.gz b/memoire/gabarit-maitrise.synctex.gz new file mode 100644 index 0000000..720f2fc Binary files /dev/null and b/memoire/gabarit-maitrise.synctex.gz differ diff --git a/memoire/gabarit-maitrise.tex b/memoire/gabarit-maitrise.tex new file mode 100644 index 0000000..5b6236c --- /dev/null +++ b/memoire/gabarit-maitrise.tex @@ -0,0 +1,111 @@ +%% GABARIT POUR MÉMOIRE STANDARD +%% +%% Consulter la documentation de la classe ulthese pour une +%% description détaillée de la classe, de ce gabarit et des options +%% disponibles. +%% +%% [Ne pas hésiter à supprimer les commentaires après les avoir lus.] +%% +%% Déclaration de la classe avec les langues les plus courantes. Le +%% français sera la langue par défaut du document. +\documentclass[11pt,english,francais]{ulthese} \usepackage{scrtime} +\usepackage{natbib} +\usepackage{tikz} \usepackage{multirow} +%% Encodage utilisé pour les caractères accentués dans les fichiers +%% source du document. Les gabarits sont encodés en UTF-8. +\usepackage[utf8]{inputenc} +%% Charger ici les autres paquetages nécessaires pour le document. +%% Quelques exemples; décommenter au besoin. +\usepackage{amsmath} % recommandé pour les mathématiques +\numberwithin{equation}{section} % pour numéroter les équations selon les sections +\usepackage{amsfonts} \usepackage{amsthm} \usepackage{thmtools} +% \usepackage{mathpazo} % texte en Palatino +% \usepackage{mathptmx} % texte en Times +%% Gestion des hyperliens dans le document. S'assurer que hyperref est +%% le dernier paquetage chargé. +\usepackage{hyperref} \hypersetup{colorlinks,allcolors=ULlinkcolor} +\usepackage{cleveref} +\usepackage[hypcap]{caption} % permet de faire le lien au dessus des figures au lieu d en dessous +\usepackage[off]{auto-pst-pdf} +%% Pour utiliser Computer Modern comme police sans empattements +%% (serif) plutôt que Helvetica, décommenter la ligne ci-dessous. +% \renewcommand{\sfdefault}{cmss} + +%% Options de mise en forme du mode français de babel. Consulter la +%% documentation du paquetage babel pour les options disponibles. +\frenchbsetup{% + CompactItemize=false, % ne pas compacter les listes + ThinSpaceInFrenchNumbers=true % espace fine dans les nombres +} \DeclareMathOperator{\sgn}{sgn} +%% Style de la bibliographie. +\bibliographystyle{plainnatmod} +%% Déclarations de la page titre. Remplacer les éléments entre < >. +%% Supprimer les caractères < >. Couper un long titre ou un long +%% sous-titre manuellement avec \\. Terminer une éventuelle deuxième +%% ligne de titre ou de sous-titre avec \vspace*{-\baselineskip}. +\titre{Modélisation des rendements financiers \\ à l'aide de la + distribution de Laplace \\ asymétrique généralisée + \vspace*{-\baselineskip}} +% \titre{Ceci est un exemple de long titre \\ +% avec saut de ligne manuel \vspace*{-\baselineskip}} +% \soustitre{Sous-titre le cas échéant} +% \soustitre{Ceci est un exemple de long-titre \\ +% avec saut de ligne manuel \vspace*{-\baselineskip}} +\auteur{François Pelletier} \programme{Maîtrise en actuariat} +\annee{2013} +\MSc % ou l'un de \LLM, \MA, \MMus, \MServSoc, \MScGeogr, \MATDR +\setcounter{tocdepth}{4} +\begin{document} +\newtheorem{theo}{Théorème}[subsection] +\newtheorem{hypothese}{Hypothèse}[chapter] + +\frontmatter % pages liminaires + +\pagetitre % production de la page titre + +\include{resume} % résumé français +\include{abstract} % résumé anglais +\cleardoublepage + +\tableofcontents % production de la TdM +\cleardoublepage + +\listoftables % production de la liste des tableaux +\cleardoublepage + +\listoffigures % production de la liste des figures +\cleardoublepage + +%\dedicace{Dédicace si désiré} +%\cleardoublepage + +\epigraphe{Dieu ne se soucie pas de nos difficultés mathématiques. Il intègre empiriquement.}{Albert Einstein} +\cleardoublepage + +\include{remerciements} % remerciements + +\mainmatter % corps du document + +\include{introduction} % introduction +\include{chapitre1} % chapitre 1 +\include{chapitre2} % chapitre 2, etc. +\include{chapitre3} +\include{chapitre4} +\include{chapitre5} +\include{chapitre6} +\include{chapitre7} +\include{chapitre8} +\include{chapitre9} +\include{conclusion} % conclusion + +\appendix % annexes le cas échéant + +\include{annexe1} % Annexe 1 +\include{annexe2} % Annexe 2 +\include{annexe3} % Annexe 3 + +\bibliography{memoire} % production de la bibliographie + +\include{deed} + +\end{document} diff --git a/memoire/gabarit-maitrise.toc b/memoire/gabarit-maitrise.toc new file mode 100644 index 0000000..68e7b42 --- /dev/null +++ b/memoire/gabarit-maitrise.toc @@ -0,0 +1,137 @@ +\select@language {french} +\select@language {french} +\contentsline {chapter}{R\IeC {\'e}sum\IeC {\'e}}{iii}{section*.2} +\contentsline {chapter}{Abstract}{v}{section*.4} +\contentsline {chapter}{Table des mati{\`e}res}{vii}{section*.5} +\contentsline {chapter}{Liste des tableaux}{xi}{section*.6} +\contentsline {chapter}{Liste des figures}{xiii}{section*.7} +\contentsline {chapter}{Remerciements}{xvii}{section*.9} +\contentsline {chapter}{Introduction}{1}{section*.11} +\contentsline {chapter}{\chapternumberline {1}Les mod\IeC {\`e}les de rendements financiers}{3}{chapter.1} +\contentsline {section}{\numberline {1.1}L'utilisation de mod\IeC {\`e}les en finance}{3}{section.1.1} +\contentsline {subsection}{\numberline {1.1.1}Diff\IeC {\'e}rents types de mod\IeC {\`e}les}{3}{subsection.1.1.1} +\contentsline {subsection}{\numberline {1.1.2}Le risque de mod\IeC {\'e}lisation}{4}{subsection.1.1.2} +\contentsline {section}{\numberline {1.2}Les rendements financiers}{5}{section.1.2} +\contentsline {subsection}{\numberline {1.2.1}D\IeC {\'e}finitions et notations}{5}{subsection.1.2.1} +\contentsline {subsection}{\numberline {1.2.2}Rendements cumul\IeC {\'e}s}{7}{subsection.1.2.2} +\contentsline {subsection}{\numberline {1.2.3}Donn\IeC {\'e}es disponibles}{8}{subsection.1.2.3} +\contentsline {section}{\numberline {1.3}Les premiers mod\IeC {\`e}les}{8}{section.1.3} +\contentsline {subsection}{\numberline {1.3.1}Le mod\IeC {\`e}le de Bachelier}{8}{subsection.1.3.1} +\contentsline {subsection}{\numberline {1.3.2}Proposition de Mandelbrot}{11}{subsection.1.3.2} +\contentsline {subsection}{\numberline {1.3.3}Le mod\IeC {\`e}le de Press}{12}{subsection.1.3.3} +\contentsline {subsection}{\numberline {1.3.4}Le mod\IeC {\`e}le de Praetz}{15}{subsection.1.3.4} +\contentsline {section}{\numberline {1.4}Conditions essentielles de Madan et Seneta}{16}{section.1.4} +\contentsline {chapter}{\chapternumberline {2}La distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{19}{chapter.2} +\contentsline {section}{\numberline {2.1}Le processus de Laplace}{19}{section.2.1} +\contentsline {subsection}{\numberline {2.1.1}Le processus gamma}{20}{subsection.2.1.1} +\contentsline {subsection}{\numberline {2.1.2}Le processus de Wiener}{21}{subsection.2.1.2} +\contentsline {subsection}{\numberline {2.1.3}Le processus de Laplace est un processus subordonn\IeC {\'e}}{22}{subsection.2.1.3} +\contentsline {section}{\numberline {2.2}Distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{25}{section.2.2} +\contentsline {subsection}{\numberline {2.2.1}Fonction caract\IeC {\'e}ristique}{25}{subsection.2.2.1} +\contentsline {subsection}{\numberline {2.2.2}Invariance d'\IeC {\'e}chelle}{26}{subsection.2.2.2} +\contentsline {subsection}{\numberline {2.2.3}Fonctions g\IeC {\'e}n\IeC {\'e}ratrices}{28}{subsection.2.2.3} +\contentsline {subsection}{\numberline {2.2.4}Moments et r\IeC {\^o}le des param\IeC {\`e}tres}{28}{subsection.2.2.4} +\contentsline {subsection}{\numberline {2.2.5}Changement d'\IeC {\'e}chelle et de localisation}{30}{subsection.2.2.5} +\contentsline {subsection}{\numberline {2.2.6}Repr\IeC {\'e}sentation alternative et simulation}{32}{subsection.2.2.6} +\contentsline {subsection}{\numberline {2.2.7}Fonction de Bessel et densit\IeC {\'e}}{33}{subsection.2.2.7} +\contentsline {section}{\numberline {2.3}Cas particuliers}{35}{section.2.3} +\contentsline {subsection}{\numberline {2.3.1}Distribution de Laplace asym\IeC {\'e}trique}{35}{subsection.2.3.1} +\contentsline {section}{\numberline {2.4}Relation avec le mod\IeC {\`e}le de \cite {madan1990variance}}{37}{section.2.4} +\contentsline {chapter}{\chapternumberline {3}Approximation de la densit\IeC {\'e} et de la fonction de r\IeC {\'e}partition}{39}{chapter.3} +\contentsline {section}{\numberline {3.1}L'approximation de Laplace}{39}{section.3.1} +\contentsline {section}{\numberline {3.2}L'approximation de Temme}{41}{section.3.2} +\contentsline {section}{\numberline {3.3}La m\IeC {\'e}thode du point de selle}{42}{section.3.3} +\contentsline {subsection}{\numberline {3.3.1}Approximation de la densit\IeC {\'e}}{42}{subsection.3.3.1} +\contentsline {subsection}{\numberline {3.3.2}Unicit\IeC {\'e} du point de selle}{44}{subsection.3.3.2} +\contentsline {subsection}{\numberline {3.3.3}Approximation de la fonction de r\IeC {\'e}partition}{44}{subsection.3.3.3} +\contentsline {subsection}{\numberline {3.3.4}Quelques propri\IeC {\'e}t\IeC {\'e}s des approximations}{45}{subsection.3.3.4} +\contentsline {section}{\numberline {3.4}Application de la m\IeC {\'e}thode du point de selle}{46}{section.3.4} +\contentsline {subsection}{\numberline {3.4.1}Approximation de la densit\IeC {\'e}}{46}{subsection.3.4.1} +\contentsline {subsection}{\numberline {3.4.2}Approximation de la fonction de r\IeC {\'e}partition}{47}{subsection.3.4.2} +\contentsline {chapter}{\chapternumberline {4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{49}{chapter.4} +\contentsline {section}{\numberline {4.1}Introduction}{49}{section.4.1} +\contentsline {subsection}{\numberline {4.1.1}M\IeC {\'e}thode classique des moments}{50}{subsection.4.1.1} +\contentsline {section}{\numberline {4.2}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{50}{section.4.2} +\contentsline {subsection}{\numberline {4.2.1}D\IeC {\'e}finition}{51}{subsection.4.2.1} +\contentsline {subsection}{\numberline {4.2.2}Convergence}{52}{subsection.4.2.2} +\contentsline {subsection}{\numberline {4.2.3}Matrice de pond\IeC {\'e}ration optimale}{53}{subsection.4.2.3} +\contentsline {subsection}{\numberline {4.2.4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e it\IeC {\'e}rative}{55}{subsection.4.2.4} +\contentsline {subsection}{\numberline {4.2.5}Distribution asymptotique des estimateurs}{55}{subsection.4.2.5} +\contentsline {section}{\numberline {4.3}Estimation sous contraintes}{58}{section.4.3} +\contentsline {subsection}{\numberline {4.3.1}Distribution asymptotique des estimateurs contraints}{59}{subsection.4.3.1} +\contentsline {section}{\numberline {4.4}Tests d'hypoth\IeC {\`e}ses param\IeC {\'e}triques}{61}{section.4.4} +\contentsline {subsection}{\numberline {4.4.1}Test de Wald}{61}{subsection.4.4.1} +\contentsline {subsection}{\numberline {4.4.2}Test du multiplicateur de Lagrange}{62}{subsection.4.4.2} +\contentsline {subsection}{\numberline {4.4.3}Test bas\IeC {\'e} sur la statistique de m\IeC {\'e}trique de distance}{63}{subsection.4.4.3} +\contentsline {subsection}{\numberline {4.4.4}En r\IeC {\'e}sum\IeC {\'e}}{63}{subsection.4.4.4} +\contentsline {chapter}{\chapternumberline {5}M\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale}{65}{chapter.5} +\contentsline {section}{\numberline {5.1}\IeC {\'E}quation d'estimation optimale}{67}{section.5.1} +\contentsline {section}{\numberline {5.2}\IeC {\'E}quation d'estimation optimale modifi\IeC {\'e}e}{70}{section.5.2} +\contentsline {chapter}{\chapternumberline {6}Estimation des param\IeC {\`e}tres de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{73}{chapter.6} +\contentsline {section}{\numberline {6.1}Vecteur de param\IeC {\`e}tres initiaux}{73}{section.6.1} +\contentsline {section}{\numberline {6.2}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{74}{section.6.2} +\contentsline {subsection}{\numberline {6.2.1}Matrice de pond\IeC {\'e}ration optimale}{75}{subsection.6.2.1} +\contentsline {subsection}{\numberline {6.2.2}Variance-covariance des param\IeC {\`e}tres}{76}{subsection.6.2.2} +\contentsline {subsection}{\numberline {6.2.3}Contraintes lin\IeC {\'e}aires}{77}{subsection.6.2.3} +\contentsline {section}{\numberline {6.3}M\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale}{78}{section.6.3} +\contentsline {chapter}{\chapternumberline {7}Tests statistiques}{81}{chapter.7} +\contentsline {section}{\numberline {7.1}Test de normalit\IeC {\'e}}{81}{section.7.1} +\contentsline {subsection}{\numberline {7.1.1}Test de Shapiro-Wilk}{81}{subsection.7.1.1} +\contentsline {subsection}{\numberline {7.1.2}Test d\IeC {\textquoteright }Epps-Pulley}{82}{subsection.7.1.2} +\contentsline {section}{\numberline {7.2}Tests d'ad\IeC {\'e}quation}{82}{section.7.2} +\contentsline {subsection}{\numberline {7.2.1}Test $\chi ^2$ de Pearson}{83}{subsection.7.2.1} +\contentsline {subsection}{\numberline {7.2.2}Test de Kolmogorov-Smirnov}{84}{subsection.7.2.2} +\contentsline {subsection}{\numberline {7.2.3}Test de distance minimale bas\IeC {\'e} sur la fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments}{85}{subsection.7.2.3} +\contentsline {chapter}{\chapternumberline {8}\IeC {\'E}valuation d'options}{87}{chapter.8} +\contentsline {section}{\numberline {8.1}D\IeC {\'e}finitions}{87}{section.8.1} +\contentsline {subsection}{\numberline {8.1.1}\IeC {\'E}quation martingale}{88}{subsection.8.1.1} +\contentsline {subsection}{\numberline {8.1.2}Param\IeC {\`e}tres neutres au risque}{89}{subsection.8.1.2} +\contentsline {section}{\numberline {8.2}Aper\IeC {\c c}u du mod\IeC {\`e}le de Black-Scholes}{90}{section.8.2} +\contentsline {section}{\numberline {8.3}M\IeC {\'e}thodes d'\IeC {\'e}valuation pour options europ\IeC {\'e}ennes}{90}{section.8.3} +\contentsline {subsection}{\numberline {8.3.1}M\IeC {\'e}thode de Heston}{91}{subsection.8.3.1} +\contentsline {subsection}{\numberline {8.3.2}M\IeC {\'e}thode de Carr et Madan}{91}{subsection.8.3.2} +\contentsline {subsection}{\numberline {8.3.3}Prix d'exercice hors du cours}{93}{subsection.8.3.3} +\contentsline {subsection}{\numberline {8.3.4}Critique de la m\IeC {\'e}thode de Carr-Madan}{93}{subsection.8.3.4} +\contentsline {subsection}{\numberline {8.3.5}M\IeC {\'e}thode d\IeC {\textquoteright }Epps}{93}{subsection.8.3.5} +\contentsline {section}{\numberline {8.4}Particularit\IeC {\'e}s}{95}{section.8.4} +\contentsline {subsection}{\numberline {8.4.1}Option sur actions avec dividendes}{95}{subsection.8.4.1} +\contentsline {subsection}{\numberline {8.4.2}Options sur contrats \IeC {\`a} terme et taux de change}{95}{subsection.8.4.2} +\contentsline {chapter}{\chapternumberline {9}Exemple d'application}{97}{chapter.9} +\contentsline {section}{\numberline {9.1}Description des donn\IeC {\'e}es}{97}{section.9.1} +\contentsline {section}{\numberline {9.2}Estimation}{99}{section.9.2} +\contentsline {section}{\numberline {9.3}Approximation}{104}{section.9.3} +\contentsline {section}{\numberline {9.4}Graphiques}{105}{section.9.4} +\contentsline {section}{\numberline {9.5}Tests statistiques}{108}{section.9.5} +\contentsline {section}{\numberline {9.6}\IeC {\'E}valuation d'options}{109}{section.9.6} +\contentsline {chapter}{Conclusion}{113}{section*.52} +\contentsline {appendix}{\chapternumberline {A}\IeC {\'E}l\IeC {\'e}ments de th\IeC {\'e}orie des probabilit\IeC {\'e}s}{115}{appendix.A} +\contentsline {section}{\numberline {A.1}D\IeC {\'e}finitions de base}{115}{section.A.1} +\contentsline {section}{\numberline {A.2}Transform\IeC {\'e}es d'une variable al\IeC {\'e}atoire}{116}{section.A.2} +\contentsline {subsection}{\numberline {A.2.1}La fonction caract\IeC {\'e}ristique}{116}{subsection.A.2.1} +\contentsline {subsubsection}{Transform\IeC {\'e}e de Fourier}{116}{section*.53} +\contentsline {subsubsection}{D\IeC {\'e}finition}{116}{section*.54} +\contentsline {subsubsection}{Les moments}{117}{section*.55} +\contentsline {subsection}{\numberline {A.2.2}Inversion de la fonction caract\IeC {\'e}ristique}{117}{subsection.A.2.2} +\contentsline {subsubsection}{La densit\IeC {\'e}}{117}{section*.56} +\contentsline {subsubsection}{La fonction de r\IeC {\'e}partition}{117}{section*.57} +\contentsline {subsection}{\numberline {A.2.3}La fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments}{118}{subsection.A.2.3} +\contentsline {subsection}{\numberline {A.2.4}La fonction g\IeC {\'e}n\IeC {\'e}ratrice des cumulants}{119}{subsection.A.2.4} +\contentsline {subsection}{\numberline {A.2.5}La transform\IeC {\'e}e d'Esscher}{120}{subsection.A.2.5} +\contentsline {section}{\numberline {A.3}La transform\IeC {\'e}e de Fourier rapide}{120}{section.A.3} +\contentsline {section}{\numberline {A.4}Processus de L\IeC {\'e}vy}{121}{section.A.4} +\contentsline {subsection}{\numberline {A.4.1}D\IeC {\'e}finition et propri\IeC {\'e}t\IeC {\'e}s}{121}{subsection.A.4.1} +\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-Khintchine}{122}{section*.58} +\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-It\IeC {\^o}}{122}{section*.59} +\contentsline {subsection}{\numberline {A.4.2}Processus subordonn\IeC {\'e}}{123}{subsection.A.4.2} +\contentsline {section}{\numberline {A.5}Th\IeC {\'e}or\IeC {\`e}mes d'int\IeC {\'e}gration}{123}{section.A.5} +\contentsline {subsection}{\numberline {A.5.1}Th\IeC {\'e}or\IeC {\`e}me de convergence domin\IeC {\'e}e de Lebesgue}{124}{subsection.A.5.1} +\contentsline {subsection}{\numberline {A.5.2}Th\IeC {\'e}or\IeC {\`e}me de Fubini}{124}{subsection.A.5.2} +\contentsline {appendix}{\chapternumberline {B}\IeC {\'E}l\IeC {\'e}ments de statistique math\IeC {\'e}matique}{125}{appendix.B} +\contentsline {section}{\numberline {B.1}Loi faible des grands nombres}{125}{section.B.1} +\contentsline {section}{\numberline {B.2}Th\IeC {\'e}or\IeC {\`e}me central limite}{125}{section.B.2} +\contentsline {subsection}{\numberline {B.2.1}Cas univari\IeC {\'e}}{126}{subsection.B.2.1} +\contentsline {subsection}{\numberline {B.2.2}Cas multivari\IeC {\'e}}{126}{subsection.B.2.2} +\contentsline {section}{\numberline {B.3}M\IeC {\'e}thode delta multivari\IeC {\'e}e}{127}{section.B.3} +\contentsline {appendix}{\chapternumberline {C}Donn\IeC {\'e}es}{129}{appendix.C} +\contentsline {chapter}{Bibliographie}{131}{section*.62} +\contentsline {chapter}{Contrat de partage}{137}{section*.64} diff --git a/memoire/introduction.aux b/memoire/introduction.aux new file mode 100644 index 0000000..9835c49 --- /dev/null +++ b/memoire/introduction.aux @@ -0,0 +1,80 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{toc}{\contentsline {chapter}{Introduction}{1}{section*.11}} +\@setckpt{introduction}{ +\setcounter{page}{3} +\setcounter{equation}{0} +\setcounter{enumi}{0} +\setcounter{enumii}{0} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{0} +\setcounter{section}{0} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{21} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{0} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{0} +\setcounter{Item}{0} +\setcounter{Hfootnote}{0} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{7} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/introduction.log b/memoire/introduction.log new file mode 100644 index 0000000..7947a54 --- /dev/null +++ b/memoire/introduction.log @@ -0,0 +1,3338 @@ +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.7.6) 27 AUG 2013 15:21 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**\input introduction.tex +(./introduction.tex +! Undefined control sequence. +l.1 \chapter + *{Introduction} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1 \chapter* + {Introduction} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no * in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +! Undefined control sequence. +l.2 \phantomsection + \addcontentsline{toc}{chapter}{Introduction} % inclure da... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 1--3 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.4 L + 'utilisation de la distribution de Laplace asymétrique généralisée ... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no L in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 4--5 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.6 A + u chapitre 1, on présente les différents types de modèles financiers... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 6--7 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.8 A + u chapitre 2, on introduit les processus gamma et de Wiener qui sont le... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 8--9 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.10 A + u chapitre 3, on présente la méthode du point de selle qui permet d'... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 10--11 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.12 A + u chapitre 4, on présente la méthode des moments généralisée ains... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 12--13 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.14 A + u chapitre 5, on présente la méthode des équations d'estimation opt... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 14--15 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.16 A + u chapitre 6, on applique les méthodes d'estimation des deux chapitre... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 6 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 16--17 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.18 A + u chapitre 7, on présente les tests de normalité de Shapiro-Wilk et ... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no â in font nullfont! +Missing character: There is no € in font nullfont! +Missing character: There is no ™ in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 18. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 18. +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 18--19 +[] + [] + + +Overfull \hbox (10.74304pt too wide) in paragraph at lines 18--19 +\OML/cmm/m/it/10 ^^_[]$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.20 A + u chapitre 8, on présente différentes méthodes pour l'évaluation d... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 20--21 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.22 A + u chapitre 9, on présente un exemple d'application des outils dévelo... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 22--23 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.24 E + nfin, on présente en annexe certaines notions de théorie des probabi... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 24--25 +[] + [] + +) +! Emergency stop. +<*> \input introduction.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 7 strings out of 493485 + 181 string characters out of 3143527 + 47818 words of memory out of 3000000 + 3414 multiletter control sequences out of 15000+200000 + 3640 words of font info for 14 fonts, out of 3000000 for 9000 + 957 hyphenation exceptions out of 8191 + 18i,1n,12p,1019b,56s stack positions out of 5000i,500n,10000p,200000b,50000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/memoire/introduction.tex b/memoire/introduction.tex new file mode 100644 index 0000000..4bcd4a0 --- /dev/null +++ b/memoire/introduction.tex @@ -0,0 +1,75 @@ +\chapter*{Introduction} +\phantomsection\addcontentsline{toc}{chapter}{Introduction} % inclure dans TdM + +L'utilisation de la distribution de Laplace asymétrique généralisée +dans le cadre de la modélisation des rendements financiers est de plus +en plus courante dans les milieux académiques et pratiques. Entre +autres, elle présente une alternative intéressante au modèle de +Black-Scholes pour l'évaluation des options de type européennes. De +plus, elle peut être utilisée pour modéliser les rendements +obligataires à long terme et les variations du taux de +change. Cependant, la littérature actuelle présente peu d'outils qui +facilitent l'utilisation en pratique de cette distribution. La +principale motivation derrière le travail de recherche et de synthèse +présenté dans ce texte a été de développer certains outils qui +permettent l'estimation des paramètres de la distribution et +l'approximation de celle-ci. De plus, plusieurs tests statistiques +sont présentés afin d'évaluer la validité du modèle estimé et de +certaines contraintes linéaires qu'on pourrait lui appliquer. + +Au chapitre 1, on présente les différents types de modèles financiers +ainsi que le risque associé à la modélisation. On introduit aussi +différents concepts théoriques entourant les rendements +financiers. Puis, on dresse un historique des modèles financiers ayant +mené à l'utilisation de la distribution de Laplace asymétrique +généralisée. + +Au chapitre 2, on introduit les processus gamma et de Wiener qui sont +les deux composantes essentielles du processus de Laplace. Puis, on +présente les principales caractéristiques de la distribution de +Laplace asymétrique généralisée. Enfin, on présente quelques cas +particuliers de celle-ci et on fait le lien avec le modèle +variance-gamma de Madan et Seneta. + +Au chapitre 3, on présente la méthode du point de selle qui permet +d'effectuer l'approximation de la densité et de la fonction de +répartition lorsqu'elles n'ont pas de forme analytique. On applique +ensuite cette méthode à la distribution de Laplace asymétrique +généralisée. + +Au chapitre 4, on présente la méthode des moments généralisée ainsi +que son utilisation dans le cadre de l'estimation sous contraintes. On +présente ensuite certains tests d'hypothèse paramétriques. + +Au chapitre 5, on présente la méthode des équations d'estimation +optimales ainsi qu'une version légèrement modifiée qui diminue la +quantité de calcul nécessaire. + +Au chapitre 6, on applique les méthodes d'estimation des deux +chapitres précédents à la distribution de Laplace asymétrique +généralisée. On présente en premier lieu une méthode qui permet +d'obtenir un point de départ pour l'algorithme d'optimisation. Puis, +on détaille les résultats obtenus. + +Au chapitre 7, on présente les tests de normalité de Shapiro-Wilk et +d'Epps-Pulley, puis ceux d'adéquation du $\chi^2$ de Pearson, celui de +Kolmogorov-Smirnov ainsi qu'un autre basé sur la fonction +génératrice des moments. + +Au chapitre 8, on présente différentes méthodes pour l'évaluation +d'options. On présente d'abord quelques notions liées aux produits +dérivés, puis on détaille trois méthodes pouvant être utilisées avec +la distribution de Laplace asymétrique généralisée. Enfin, on présente +quelques particularités liées à certains titres financiers. + +Au chapitre 9, on présente un exemple d'application des outils +développés aux chapitres 3 à 8 avec un ensemble de données. + +Enfin, on présente en annexe certaines notions de théorie des +probabilités et de statistique. + + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/memoire.bib b/memoire/memoire.bib new file mode 100644 index 0000000..7e536df --- /dev/null +++ b/memoire/memoire.bib @@ -0,0 +1,697 @@ +@article{KOUTROUVELIS01011980, + author = {Koutrouvelis, I. A.}, + title = {A goodness-of-fit test of simple hypotheses based on + the empirical characteristic function}, + volume = 67, + number = 1, + pages = {238-240}, + year = 1980, + doi = {10.1093/biomet/67.1.238}, + abstract = {SUMMARY The empirical characteristic function φn(t) + and its asymptotic distribution are utilized to find + a chi-squared goodness-of-fit test for simple null + hypotheses. In addition the optimum number and + location of points t at which φn(t) has to be + evaluated is investigated for specified alternative + hypotheses.}, + URL = + {http://biomet.oxfordjournals.org/content/67/1/238.abstract}, + eprint = + {http://biomet.oxfordjournals.org/content/67/1/238.full.pdf+html}, + journal = {Biometrika} +} + +@Manual{RpackageVarianceGamma, + title = {VarianceGamma: The Variance Gamma Distribution}, + author = {David Scott and Christine Yang Dong}, + year = 2012, + note = {R package version 0.3-1}, + url = {http://CRAN.R-project.org/package=VarianceGamma}, +} + +@Manual{Rsoftware, + title = {R: A Language and Environment for Statistical + Computing}, + author = {{R Core Team}}, + organization = {R Foundation for Statistical Computing}, + address = {Vienna, Austria}, + year = 2012, + note = {{ISBN} 3-900051-07-0}, + url = {http://www.R-project.org/}, +} + +@ARTICLE{Singer:2009, + AUTHOR = {Singer, S. and Nelder, J. }, + TITLE = {Nelder-Mead algorithm}, + YEAR = 2009, + JOURNAL = {Scholarpedia}, + VOLUME = 4, + NUMBER = 7, + PAGES = 2928, + URL = {http://dx.doi.org/10.4249/scholarpedia.2928} +} + +@book{abramowitz1965handbook, + title = {Handbook of mathematical functions: with formulas, + graphs, and mathematical tables}, + author = {Abramowitz, Milton and Stegun, Irene A}, + volume = 55, + year = 1965, + publisher = {Dover publications} +} + +@article{applebaum2004levy, + title = {L{\'e}vy processes: From probability to finance and + quantum groups}, + author = {Applebaum, David}, + journal = {Notices of the AMS}, + volume = 51, + number = 11, + pages = {1336--1347}, + year = 2004 +} + +@book{bachelier1900theorie, + title = {Th{\'e}orie de la sp{\'e}culation}, + author = {Bachelier, Louis}, + year = 1900, + publisher = {Gauthier-Villars} +} + +@book{barndorff2001levy, + title = {L{\'e}vy Processes: Theory and Applications}, + author = {Barndorff-Nielsen, O.E.E. and Mikosch, T.E. and + Resnick, S.I.E.}, + isbn = 9780817641672, + lccn = {lc00068064}, + url = {http://books.google.ca/books?id=ExpTdTauXMwC}, + year = 2001, + publisher = {Birkhäuser} +} + +@article{berkson1980minimum, + title = {Minimum chi-square, not maximum likelihood!}, + author = {Berkson, Joseph}, + journal = {The Annals of Statistics}, + pages = {457--487}, + year = 1980, + publisher = {JSTOR} +} + +@book{bingham2004risk, + title = {Risk-neutral valuation: Pricing and hedging of + financial derivatives}, + author = {Bingham, Nicholas H and Kiesel, R{\"u}diger}, + year = 2004, + publisher = {Springer} +} + +@article{black1973pricing, + title = {The pricing of options and corporate liabilities}, + author = {Black, Fischer and Scholes, Myron}, + journal = {The journal of political economy}, + pages = {637--654}, + year = 1973, + publisher = {JSTOR} +} + +@article{black1976pricing, + title = {The pricing of commodity contracts}, + author = {Black, Fischer}, + journal = {Journal of financial economics}, + volume = 3, + number = 1, + pages = {167--179}, + year = 1976, + publisher = {Elsevier} +} + +@article{buckle1995bayesian, + title = {Bayesian inference for stable distributions}, + author = {Buckle, DJ}, + journal = {Journal of the American Statistical Association}, + volume = 90, + number = 430, + pages = {605--613}, + year = 1995, + publisher = {Taylor \& Francis Group} +} + +@book{butler2007saddlepoint, + title = {Saddlepoint approximations with applications}, + author = {Butler, Ronald W}, + volume = 22, + year = 2007, + publisher = {Cambridge University Press} +} + +@article{carr1999option, + title = {Option valuation using the fast Fourier transform}, + author = {Carr, Peter and Madan, Dilip}, + journal = {Journal of Computational Finance}, + volume = 2, + number = 4, + pages = {61--73}, + year = 1999 +} + +@article{crowder1986consistency, + title = {On consistency and inconsistency of estimating + equations}, + author = {Crowder, Martin}, + journal = {Econometric Theory}, + pages = {305--330}, + year = 1986, + publisher = {JSTOR} +} + +@article{crowder1987linear, + title = {On linear and quadratic estimating functions}, + author = {Crowder, Martin}, + journal = {Biometrika}, + volume = 74, + number = 3, + pages = {591--597}, + year = 1987, + publisher = {Biometrika Trust} +} + +@article{daniels1954saddlepoint, + title = {Saddlepoint approximations in statistics}, + author = {Daniels, Henry E}, + journal = {The Annals of Mathematical Statistics}, + pages = {631--650}, + year = 1954, + publisher = {JSTOR} +} + +@article{daniels1987tail, + title = {Tail probability approximations}, + author = {Daniels, Henry E}, + journal = {International Statistical Review/Revue + Internationale de Statistique}, + pages = {37--48}, + year = 1987, + publisher = {JSTOR} +} + +@TechReport{derman1996modelrisk, + author = {Derman, Emanuel}, + title = {Model Risk}, + institution = {Goldman Sachs}, + year = 1996 +} + +@book{dodge2004statistique, + title = {Statistique: dictionnaire encyclop{\'e}dique}, + author = {Dodge, Yadolah}, + year = 2004, + publisher = {Springer Verlag France} +} + +@article{epps1983test, + title = {A test for normality based on the empirical + characteristic function}, + author = {Epps, Thomas W and Pulley, Lawrence B}, + journal = {Biometrika}, + volume = 70, + number = 3, + pages = {723--726}, + year = 1983, + publisher = {Biometrika Trust} +} + +@book{epps2007pricing, + title = {Pricing derivative securities}, + author = {Epps, Thomas W}, + year = 2007, + publisher = {World Scientific Publishing Company Incorporated} +} + +@book{everitt2006cambridge, + title = {The Cambridge dictionary of statistics}, + author = {Everitt, Brian and Skrondal, Anders}, + volume = 4, + year = 2006, + publisher = {Cambridge University Press Cambridge} +} + +@article{feuerverger1981efficiency, + title = {On the efficiency of empirical characteristic + function procedures}, + author = {Feuerverger, Andrey and McDunnough, Philip}, + journal = {Journal of the Royal Statistical Society. Series B + (Methodological)}, + pages = {20--27}, + year = 1981, + publisher = {JSTOR} +} + +@article{fox1986large, + title = {Large-sample properties of parameter estimates for + strongly dependent stationary Gaussian time series}, + author = {Fox, Robert and Taqqu, Murad S}, + journal = {The Annals of Statistics}, + volume = 14, + number = 2, + pages = {517--532}, + year = 1986, + publisher = {Institute of Mathematical Statistics} +} + +@article{gil1951note, + title = {Note on the inversion theorem}, + author = {Gil-Pelaez, J}, + journal = {Biometrika}, + volume = 38, + number = {3-4}, + pages = {481--482}, + year = 1951, + publisher = {Biometrika Trust} +} + +@book{gourieroux1989statistique, + title = {Statistique et mod{\`e}les {\'e}conom{\'e}triques: + Notions g{\'e}n{\'e}rales, estimation, + pr{\'e}vision, algorithmes}, + author = {Gourieroux, Christian and Monfort, Alain}, + volume = 1, + year = 1989, + publisher = {Economica} +} + +@book{hall2005generalized, + title = {Generalized method of moments}, + author = {Hall, Alastair R}, + year = 2005, + publisher = {Oxford University Press Oxford} +} + +@book{hamilton1994time, + title = {Time series analysis}, + author = {Hamilton, James Douglas}, + volume = 2, + year = 1994, + publisher = {Cambridge Univ Press} +} + +@article{hansen1982large, + title = {Large sample properties of generalized method of + moments estimators}, + author = {Hansen, Lars Peter}, + journal = {Econometrica: Journal of the Econometric Society}, + pages = {1029--1054}, + year = 1982, + publisher = {JSTOR} +} + +@article{henze1990approximation, + title = {An approximation to the limit distribution of the + Epps-Pulley test statistic for normality}, + author = {Henze, N}, + journal = {Metrika}, + volume = 37, + number = 1, + pages = {7--18}, + year = 1990, + publisher = {Springer} +} + +@article{heston1993closed, + title = {A closed-form solution for options with stochastic + volatility with applications to bond and currency + options}, + author = {Heston, Steven L}, + journal = {Review of financial studies}, + volume = 6, + number = 2, + pages = {327--343}, + year = 1993, + publisher = {Soc Financial Studies} +} + +@article{hinkley1977estimation, + title = {Estimation of the Pareto law from underreported + data: A further analysis}, + author = {Hinkley, David V and Revankar, Nagesh S}, + journal = {Journal of Econometrics}, + volume = 5, + number = 1, + pages = {1--11}, + year = 1977, + publisher = {Elsevier} +} + +@book{hogg1978introduction, + title = {Introduction to mathematical statistics}, + author = {Hogg, R.V. and Craig, A.T.}, + isbn = 9780029789902, + lccn = 77002884, + url = {http://books.google.ca/books?id=OZYQAQAAIAAJ}, + year = 1978, + publisher = {Macmillan} +} + +@book{hull1999options, + title = {Options, futures, and other derivatives}, + author = {Hull, John C}, + year = 1999, + publisher = {Pearson Education India} +} + +@article{itkin2005pricing, + title = {Pricing options with VG model using FFT}, + author = {Itkin, Andrey}, + journal = {arXiv preprint physics/0503137}, + year = 2005 +} + +@book{kotz2001laplace, + title = {The Laplace Distribution and Generalizations: A + Revisit With Applications to Communications, + Exonomics, Engineering, and Finance}, + author = {Kotz, S. and Kozubowski, T.J. and Podg{\'o}rski, K.}, + isbn = 9780817641665, + lccn = 00068900, + series = {Progress in Mathematics Series}, + url = {http://books.google.ca/books?id=cb8B07hwULUC}, + year = 2001, + publisher = {Birkh{\"a}user} +} + +@article{kozubowski1999class, + title = {A class of asymmetric distributions}, + author = {Kozubowski, Tomasz J and Podg{\'o}rski, Krzysztof}, + journal = {Actuarial Research Clearing House}, + volume = 1, + pages = {113--134}, + year = 1999, + publisher = {Citeseer} +} + +@article{kozubowski2001asymmetric, + title = {Asymmetric Laplace laws and modeling financial data}, + author = {Kozubowski, Tomasz J and Podg{\'o}rski, Krzysztof}, + journal = {Mathematical and computer modelling}, + volume = 34, + number = 9, + pages = {1003--1021}, + year = 2001, + publisher = {Elsevier} +} + +@book{kyprianou2007introductory, + title = {Introductory lectures on fluctuations of L{\'e}vy + processes with applications}, + author = {Kyprianou, Andreas E}, + year = 2007, + publisher = {Springer} +} + +@article{lugannani1980saddle, + title = {Saddle point approximation for the distribution of + the sum of independent random variables}, + author = {Lugannani, Robert and Rice, Stephen}, + journal = {Advances in applied probability}, + pages = {475--490}, + year = 1980, + publisher = {JSTOR} +} + +@book{lukacs1960characteristic, + title = {Characteristic functions}, + author = {Lukacs, Eugene}, + volume = 4, + year = 1960, + publisher = {Griffin London} +} + +@article{luong1987minimum, + title = {Minimum-distance methods based on quadratic + distances for transforms}, + author = {Luong, A and Thompson, ME}, + journal = {Canadian Journal of Statistics}, + volume = 15, + number = 3, + pages = {239--251}, + year = 1987, + publisher = {Wiley Online Library} +} + +@article{madan1990variance, + title = {The variance gamma (VG) model for share market + returns}, + author = {Madan, Dilip B and Seneta, Eugene}, + journal = {Journal of business}, + pages = {511--524}, + year = 1990, + publisher = {JSTOR} +} + +@article{madan1998variance, + title = {The variance gamma process and option pricing}, + author = {Madan, Dilip B and Carr, Peter P and Chang, Eric C}, + journal = {European Finance Review}, + volume = 2, + number = 1, + pages = {79--105}, + year = 1998, + publisher = {Kluwer Academic Publishers} +} + +@article{mandelbrot1963variation, + title = {The variation of certain speculative prices}, + author = {Mandelbrot, B.}, + journal = {Journal of business}, + pages = {394--419}, + year = 1963, + publisher = {University of Chicago Press} +} + +@article{merton1976option, + title = {Option pricing when underlying stock returns are + discontinuous}, + author = {Merton, Robert C}, + journal = {Journal of financial economics}, + volume = 3, + number = 1, + pages = {125--144}, + year = 1976, + publisher = {Elsevier} +} + +@article{mitchell1916critique, + title = {A critique of index numbers of the prices of stocks}, + author = {Mitchell, Wesley C}, + journal = {The Journal of Political Economy}, + pages = {625--693}, + year = 1916, + publisher = {JSTOR} +} + +@book{musiela2005martingale, + title = {Martingale methods in financial modelling}, + author = {Musiela, Marek and Rutkowski, Marek}, + volume = 36, + year = 2005, + publisher = {Springer} +} + +@article{newey1987hypothesis, + title = {Hypothesis testing with efficient method of moments + estimation}, + author = {Newey, Whitney K and West, Kenneth D}, + journal = {International Economic Review}, + volume = 28, + number = 3, + pages = {777--787}, + year = 1987, + publisher = {JSTOR} +} + +@article{newey1994large, + title = {Large sample estimation and hypothesis testing}, + author = {Newey, Whitney K and McFadden, Daniel}, + journal = {Handbook of econometrics}, + volume = 4, + pages = {2111--2245}, + year = 1994, + publisher = {Elsevier} +} + +@article{praetz1972distribution, + title = {The distribution of share price changes}, + author = {Praetz, Peter D}, + journal = {Journal of business}, + pages = {49--55}, + year = 1972, + publisher = {JSTOR} +} + +@article{press1967compound, + title = {A compound events model for security prices}, + author = {Press, S James}, + journal = {Journal of Business}, + pages = {317--335}, + year = 1967, + publisher = {JSTOR} +} + +@article{randal2004non, + title = {Non-parametric estimation of historical volatility}, + author = {Randal, John A and Thomson, Peter J and Lally, + Martin T}, + journal = {Quantitative Finance}, + volume = 4, + number = 4, + pages = {427--440}, + year = 2004, + publisher = {Taylor \& Francis} +} + +@book{sato1999levy, + title = {L{\'e}vy Processes and Infinitely Divisible + Distributions}, + author = {Sato, K.}, + isbn = 9780521553025, + lccn = 99015232, + series = {Cambridge Studies in Advanced Mathematics}, + url = {http://books.google.ca/books?id=CwT5BNG0-owC}, + year = 1999, + publisher = {Cambridge University Press} +} + +@book{schoutens2003levy, + title = {L{\'e}vy processes in Finance}, + author = {Schoutens, Wim}, + year = 2003, + publisher = {Wiley} +} + +@article{seneta2004fitting, + title = {Fitting the variance-gamma model to financial data}, + author = {Seneta, Eugene}, + journal = {Journal of Applied Probability}, + pages = {177--187}, + year = 2004, + publisher = {JSTOR} +} + +@article{shapiro1965analysis, + title = {An analysis of variance test for normality (complete + samples)}, + author = {Shapiro, Samuel Sanford and Wilk, Martin B}, + journal = {Biometrika}, + volume = 52, + number = {3/4}, + pages = {591--611}, + year = 1965, + publisher = {JSTOR} +} + +@article{shephard1991characteristic, + title = {From characteristic function to distribution + function: a simple framework for the theory}, + author = {Shephard, Neil G}, + journal = {Econometric Theory}, + volume = 7, + number = 04, + pages = {519--529}, + year = 1991, + publisher = {Cambridge Univ Press} +} + +@book{spiegel1999schaum, + title = {Schaum's mathematical handbook of formulas and + tables}, + author = {Spiegel, Murray R and Liu, John}, + volume = 1000, + year = 1999, + publisher = {McGraw-Hill} +} + +@book{stuart1987kendall, + title = {Kendall’s advanced theory of statistics, Vol. 1}, + author = {Stuart, Alan and Ord, J Keith}, + year = 1987, + publisher = {Oxford University Press, New York} +} + +@Manual{thevolskew, + title = {The Volatility Skew}, + author = {Alan Verga}, + year = 2013, + url = {http://www.thevolatilityskew.com} +} + +@phdthesis{torczon1989multi, + title = {Multi-directional search: a direct search algorithm + for parallel machines}, + author = {Torczon, Virginia Joanne}, + year = 1989, + school = {Citeseer} +} + +@article{wang2003evaluating, + title = {Evaluating Kolmogorov's distribution}, + author = {Wang, Jingbo and Tsang, Wai Wan and Marsaglia, + George}, + journal = {Journal of Statistical Software}, + volume = 8, + number = 18, + pages = {1--4}, + year = 2003, + publisher = {American Statistical Association} +} + +@article{wendel1961non, + title = {The non-absolute convergence of Gil-Pelaez'inversion + integral}, + author = {Wendel, JG}, + journal = {The Annals of Mathematical Statistics}, + volume = 32, + number = 1, + pages = {338--339}, + year = 1961, + publisher = {Institute of Mathematical Statistics} +} + +@article{teschl2004topics, + title={Topics in Real and Functional analysis}, + author={Teschl, Gerald}, + journal={unpublished, available online at \url{http://www.mat.univie.ac.at/\~gerald}}, + year={2004} +} + +@article{wolfowitz1957minimum, + title = {The minimum distance method}, + author = {Wolfowitz, Jacob}, + journal = {The Annals of Mathematical Statistics}, + pages = {75--88}, + year = 1957, + publisher = {JSTOR} +} + +@book{wooldridge2001econometric, + title = {Econometric analysis of cross section and panel + data}, + author = {Wooldridge, Jeffrey M}, + year = 2001, + publisher = {MIT press} +} + +@article{fama1993common, + title={Common risk factors in the returns on stocks and bonds}, + author={Fama, Eugene F and French, Kenneth R}, + journal={Journal of financial economics}, + volume={33}, + number={1}, + pages={3--56}, + year={1993}, + publisher={Elsevier} +} + +@inproceedings{walterlevy, + title={Levy-stability-under-addition and fractal structure of markets: imlications for the actuaries and emphasized axamination of MATIF national contract}, + author={Walter, Christian} +} \ No newline at end of file diff --git a/memoire/plainnatmod.bst b/memoire/plainnatmod.bst new file mode 100644 index 0000000..cc74f1f --- /dev/null +++ b/memoire/plainnatmod.bst @@ -0,0 +1,1436 @@ +%% File: `plainnat.bst' +%% A modification of `plain.bst' for use with natbib package +%% +%% Copyright 1993-2007 Patrick W Daly +%% Max-Planck-Institut f\"ur Sonnensystemforschung +%% Max-Planck-Str. 2 +%% D-37191 Katlenburg-Lindau +%% Germany +%% E-mail: daly@mps.mpg.de +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN +%% archives in directory macros/latex/base/lppl.txt; either +%% version 1 of the License, or any later version. +%% + % Version and source file information: + % \ProvidesFile{natbst.mbs}[2007/11/26 1.93 (PWD)] + % + % BibTeX `plainnat' family + % version 0.99b for BibTeX versions 0.99a or later, + % for LaTeX versions 2.09 and 2e. + % + % For use with the `natbib.sty' package; emulates the corresponding + % member of the `plain' family, but with author-year citations. + % + % With version 6.0 of `natbib.sty', it may also be used for numerical + % citations, while retaining the commands \citeauthor, \citefullauthor, + % and \citeyear to print the corresponding information. + % + % For version 7.0 of `natbib.sty', the KEY field replaces missing + % authors/editors, and the date is left blank in \bibitem. + % + % Includes field EID for the sequence/citation number of electronic journals + % which is used instead of page numbers. + % + % Includes fields ISBN and ISSN. + % + % Includes field URL for Internet addresses. + % + % Includes field DOI for Digital Object Idenfifiers. + % + % Works best with the url.sty package of Donald Arseneau. + % + % Works with identical authors and year are further sorted by + % citation key, to preserve any natural sequence. + % +ENTRY + { address + author + booktitle + chapter + doi + eid + edition + editor + howpublished + institution + isbn + issn + journal + key + month + note + number + organization + pages + publisher + school + series + title + type + url + volume + year + } + {} + { label extra.label sort.label short.list } + +INTEGERS { output.state before.all mid.sentence after.sentence after.block } + +FUNCTION {init.state.consts} +{ #0 'before.all := + #1 'mid.sentence := + #2 'after.sentence := + #3 'after.block := +} + +STRINGS { s t } + +FUNCTION {output.nonnull} +{ 's := + output.state mid.sentence = + { ", " * write$ } + { output.state after.block = + { add.period$ write$ + newline$ + "\newblock " write$ + } + { output.state before.all = + 'write$ + { add.period$ " " * write$ } + if$ + } + if$ + mid.sentence 'output.state := + } + if$ + s +} + +FUNCTION {output} +{ duplicate$ empty$ + 'pop$ + 'output.nonnull + if$ +} + +FUNCTION {output.check} +{ 't := + duplicate$ empty$ + { pop$ "empty " t * " in " * cite$ * warning$ } + 'output.nonnull + if$ +} + +FUNCTION {fin.entry} +{ add.period$ + write$ + newline$ +} + +FUNCTION {new.block} +{ output.state before.all = + 'skip$ + { after.block 'output.state := } + if$ +} + +FUNCTION {new.sentence} +{ output.state after.block = + 'skip$ + { output.state before.all = + 'skip$ + { after.sentence 'output.state := } + if$ + } + if$ +} + +FUNCTION {not} +{ { #0 } + { #1 } + if$ +} + +FUNCTION {and} +{ 'skip$ + { pop$ #0 } + if$ +} + +FUNCTION {or} +{ { pop$ #1 } + 'skip$ + if$ +} + +FUNCTION {new.block.checka} +{ empty$ + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.block.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.sentence.checka} +{ empty$ + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {new.sentence.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {field.or.null} +{ duplicate$ empty$ + { pop$ "" } + 'skip$ + if$ +} + +FUNCTION {emphasize} +{ duplicate$ empty$ + { pop$ "" } + { "\emph{" swap$ * "}" * } + if$ +} + +INTEGERS { nameptr namesleft numnames } + +FUNCTION {format.names} +{ 's := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr "{ff~}{vv~}{ll}{, jj}" format.name$ 't := + nameptr #1 > + { namesleft #1 > + { ", " * t * } + { numnames #2 > + { "," * } + 'skip$ + if$ + t "others" = + { " et~al." * } + { " et " * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {format.key} +{ empty$ + { key field.or.null } + { "" } + if$ +} + +FUNCTION {format.authors} +{ author empty$ + { "" } + { author format.names } + if$ +} + +FUNCTION {format.editors} +{ editor empty$ + { "" } + { editor format.names + editor num.names$ #1 > + { ", editors" * } + { ", editor" * } + if$ + } + if$ +} + +FUNCTION {format.isbn} +{ isbn empty$ + { "" } + { new.block "ISBN " isbn * } + if$ +} + +FUNCTION {format.issn} +{ issn empty$ + { "" } + { new.block "ISSN " issn * } + if$ +} + +FUNCTION {format.url} +{ url empty$ + { "" } + { new.block "URL \url{" url * "}" * } + if$ +} + +FUNCTION {format.doi} +{ doi empty$ + { "" } + { new.block "\doi{" doi * "}" * } + if$ +} + +FUNCTION {format.title} +{ title empty$ + { "" } + { title "t" change.case$ } + if$ +} + +FUNCTION {format.full.names} +{'s := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr + "{vv~}{ll}" format.name$ 't := + nameptr #1 > + { + namesleft #1 > + { ", " * t * } + { + numnames #2 > + { "," * } + 'skip$ + if$ + t "others" = + { " et~al." * } + { " et " * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {author.editor.full} +{ author empty$ + { editor empty$ + { "" } + { editor format.full.names } + if$ + } + { author format.full.names } + if$ +} + +FUNCTION {author.full} +{ author empty$ + { "" } + { author format.full.names } + if$ +} + +FUNCTION {editor.full} +{ editor empty$ + { "" } + { editor format.full.names } + if$ +} + +FUNCTION {make.full.names} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.full + { type$ "proceedings" = + 'editor.full + 'author.full + if$ + } + if$ +} + +FUNCTION {output.bibitem} +{ newline$ + "\bibitem[" write$ + label write$ + ")" make.full.names duplicate$ short.list = + { pop$ } + { * } + if$ + "]{" * write$ + cite$ write$ + "}" write$ + newline$ + "" + before.all 'output.state := +} + +FUNCTION {n.dashify} +{ 't := + "" + { t empty$ not } + { t #1 #1 substring$ "-" = + { t #1 #2 substring$ "--" = not + { "--" * + t #2 global.max$ substring$ 't := + } + { { t #1 #1 substring$ "-" = } + { "-" * + t #2 global.max$ substring$ 't := + } + while$ + } + if$ + } + { t #1 #1 substring$ * + t #2 global.max$ substring$ 't := + } + if$ + } + while$ +} + +FUNCTION {format.date} +{ year duplicate$ empty$ + { "empty year in " cite$ * warning$ + pop$ "" } + 'skip$ + if$ + month empty$ + 'skip$ + { month + " " * swap$ * + } + if$ + extra.label * +} + +FUNCTION {format.btitle} +{ title emphasize +} + +FUNCTION {tie.or.space.connect} +{ duplicate$ text.length$ #3 < + { "~" } + { " " } + if$ + swap$ * * +} + +FUNCTION {either.or.check} +{ empty$ + 'pop$ + { "can't use both " swap$ * " fields in " * cite$ * warning$ } + if$ +} + +FUNCTION {format.bvolume} +{ volume empty$ + { "" } + { "volume" volume tie.or.space.connect + series empty$ + 'skip$ + { " of " * series emphasize * } + if$ + "volume and number" number either.or.check + } + if$ +} + +FUNCTION {format.number.series} +{ volume empty$ + { number empty$ + { series field.or.null } + { output.state mid.sentence = + { "number" } + { "Number" } + if$ + number tie.or.space.connect + series empty$ + { "there's a number but no series in " cite$ * warning$ } + { " in " * series * } + if$ + } + if$ + } + { "" } + if$ +} + +FUNCTION {format.edition} +{ edition empty$ + { "" } + { output.state mid.sentence = + { edition "l" change.case$ " edition" * } + { edition "t" change.case$ " edition" * } + if$ + } + if$ +} + +INTEGERS { multiresult } + +FUNCTION {multi.page.check} +{ 't := + #0 'multiresult := + { multiresult not + t empty$ not + and + } + { t #1 #1 substring$ + duplicate$ "-" = + swap$ duplicate$ "," = + swap$ "+" = + or or + { #1 'multiresult := } + { t #2 global.max$ substring$ 't := } + if$ + } + while$ + multiresult +} + +FUNCTION {format.pages} +{ pages empty$ + { "" } + { pages multi.page.check + { "pages" pages n.dashify tie.or.space.connect } + { "page" pages tie.or.space.connect } + if$ + } + if$ +} + +FUNCTION {format.eid} +{ eid empty$ + { "" } + { "art." eid tie.or.space.connect } + if$ +} + +FUNCTION {format.vol.num.pages} +{ volume field.or.null + number empty$ + 'skip$ + { "\penalty0 (" number * ")" * * + volume empty$ + { "there's a number but no volume in " cite$ * warning$ } + 'skip$ + if$ + } + if$ + pages empty$ + 'skip$ + { duplicate$ empty$ + { pop$ format.pages } + { ":\penalty0 " * pages n.dashify * } + if$ + } + if$ +} + +FUNCTION {format.vol.num.eid} +{ volume field.or.null + number empty$ + 'skip$ + { "\penalty0 (" number * ")" * * + volume empty$ + { "there's a number but no volume in " cite$ * warning$ } + 'skip$ + if$ + } + if$ + eid empty$ + 'skip$ + { duplicate$ empty$ + { pop$ format.eid } + { ":\penalty0 " * eid * } + if$ + } + if$ +} + +FUNCTION {format.chapter.pages} +{ chapter empty$ + 'format.pages + { type empty$ + { "chapter" } + { type "l" change.case$ } + if$ + chapter tie.or.space.connect + pages empty$ + 'skip$ + { ", " * format.pages * } + if$ + } + if$ +} + +FUNCTION {format.in.ed.booktitle} +{ booktitle empty$ + { "" } + { editor empty$ + { "In " booktitle emphasize * } + { "In " format.editors * ", " * booktitle emphasize * } + if$ + } + if$ +} + +FUNCTION {empty.misc.check} +{ author empty$ title empty$ howpublished empty$ + month empty$ year empty$ note empty$ + and and and and and + key empty$ not and + { "all relevant fields are empty in " cite$ * warning$ } + 'skip$ + if$ +} + +FUNCTION {format.thesis.type} +{ type empty$ + 'skip$ + { pop$ + type "t" change.case$ + } + if$ +} + +FUNCTION {format.tr.number} +{ type empty$ + { "Technical Report" } + 'type + if$ + number empty$ + { "t" change.case$ } + { number tie.or.space.connect } + if$ +} + +FUNCTION {format.article.crossref} +{ key empty$ + { journal empty$ + { "need key or journal for " cite$ * " to crossref " * crossref * + warning$ + "" + } + { "In \emph{" journal * "}" * } + if$ + } + { "In " } + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {format.book.crossref} +{ volume empty$ + { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ + "In " + } + { "Volume" volume tie.or.space.connect + " of " * + } + if$ + editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { series empty$ + { "need editor, key, or series for " cite$ * " to crossref " * + crossref * warning$ + "" * + } + { "\emph{" * series * "}" * } + if$ + } + 'skip$ + if$ + } + 'skip$ + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {format.incoll.inproc.crossref} +{ editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { booktitle empty$ + { "need editor, key, or booktitle for " cite$ * " to crossref " * + crossref * warning$ + "" + } + { "In \emph{" booktitle * "}" * } + if$ + } + { "In " } + if$ + } + { "In " } + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {article} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { journal emphasize "journal" output.check + eid empty$ + { format.vol.num.pages output } + { format.vol.num.eid output } + if$ + format.date "year" output.check + } + { format.article.crossref output.nonnull + eid empty$ + { format.pages output } + { format.eid output } + if$ + } + if$ + format.issn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {book} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {booklet} +{ output.bibitem + format.authors output + author format.key output + new.block + format.title "title" output.check + howpublished address new.block.checkb + howpublished output + address output + format.date output + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inbook} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + format.chapter.pages "chapter and pages" output.check + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { format.chapter.pages "chapter and pages" output.check + new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {incollection} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.chapter.pages output + new.sentence + publisher "publisher" output.check + address output + format.edition output + format.date "year" output.check + } + { format.incoll.inproc.crossref output.nonnull + format.chapter.pages output + } + if$ + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inproceedings} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.pages output + address empty$ + { organization publisher new.sentence.checkb + organization output + publisher output + format.date "year" output.check + } + { address output.nonnull + format.date "year" output.check + new.sentence + organization output + publisher output + } + if$ + } + { format.incoll.inproc.crossref output.nonnull + format.pages output + } + if$ + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {conference} { inproceedings } + +FUNCTION {manual} +{ output.bibitem + format.authors output + author format.key output + new.block + format.btitle "title" output.check + organization address new.block.checkb + organization output + address output + format.edition output + format.date output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {mastersthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + "Master's thesis" format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {misc} +{ output.bibitem + format.authors output + author format.key output + title howpublished new.block.checkb + format.title output + howpublished new.block.checka + howpublished output + format.date output + format.issn output + format.url output + new.block + note output + fin.entry + empty.misc.check +} + +FUNCTION {phdthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.btitle "title" output.check + new.block + "PhD thesis" format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {proceedings} +{ output.bibitem + format.editors output + editor format.key output + new.block + format.btitle "title" output.check + format.bvolume output + format.number.series output + address output + format.date "year" output.check + new.sentence + organization output + publisher output + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {techreport} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + format.tr.number output.nonnull + institution "institution" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {unpublished} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + note "note" output.check + format.date output + format.url output + fin.entry +} + +FUNCTION {default.type} { misc } + + +MACRO {jan} {"January"} + +MACRO {feb} {"February"} + +MACRO {mar} {"March"} + +MACRO {apr} {"April"} + +MACRO {may} {"May"} + +MACRO {jun} {"June"} + +MACRO {jul} {"July"} + +MACRO {aug} {"August"} + +MACRO {sep} {"September"} + +MACRO {oct} {"October"} + +MACRO {nov} {"November"} + +MACRO {dec} {"December"} + + + +MACRO {acmcs} {"ACM Computing Surveys"} + +MACRO {acta} {"Acta Informatica"} + +MACRO {cacm} {"Communications of the ACM"} + +MACRO {ibmjrd} {"IBM Journal of Research and Development"} + +MACRO {ibmsj} {"IBM Systems Journal"} + +MACRO {ieeese} {"IEEE Transactions on Software Engineering"} + +MACRO {ieeetc} {"IEEE Transactions on Computers"} + +MACRO {ieeetcad} + {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"} + +MACRO {ipl} {"Information Processing Letters"} + +MACRO {jacm} {"Journal of the ACM"} + +MACRO {jcss} {"Journal of Computer and System Sciences"} + +MACRO {scp} {"Science of Computer Programming"} + +MACRO {sicomp} {"SIAM Journal on Computing"} + +MACRO {tocs} {"ACM Transactions on Computer Systems"} + +MACRO {tods} {"ACM Transactions on Database Systems"} + +MACRO {tog} {"ACM Transactions on Graphics"} + +MACRO {toms} {"ACM Transactions on Mathematical Software"} + +MACRO {toois} {"ACM Transactions on Office Information Systems"} + +MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"} + +MACRO {tcs} {"Theoretical Computer Science"} + + +READ + +FUNCTION {sortify} +{ purify$ + "l" change.case$ +} + +INTEGERS { len } + +FUNCTION {chop.word} +{ 's := + 'len := + s #1 len substring$ = + { s len #1 + global.max$ substring$ } + 's + if$ +} + +FUNCTION {format.lab.names} +{ 's := + s #1 "{vv~}{ll}" format.name$ + s num.names$ duplicate$ + #2 > + { pop$ " et~al." * } + { #2 < + 'skip$ + { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = + { " et~al." * } + { " et " * s #2 "{vv~}{ll}" format.name$ * } + if$ + } + if$ + } + if$ +} + +FUNCTION {author.key.label} +{ author empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.editor.key.label} +{ author empty$ + { editor empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { editor format.lab.names } + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.key.organization.label} +{ author empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "The " #4 organization chop.word #3 text.prefix$ } + if$ + } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {editor.key.organization.label} +{ editor empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "The " #4 organization chop.word #3 text.prefix$ } + if$ + } + 'key + if$ + } + { editor format.lab.names } + if$ +} + +FUNCTION {calc.short.authors} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.key.label + { type$ "proceedings" = + 'editor.key.organization.label + { type$ "manual" = + 'author.key.organization.label + 'author.key.label + if$ + } + if$ + } + if$ + 'short.list := +} + +FUNCTION {calc.label} +{ calc.short.authors + short.list + "(" + * + year duplicate$ empty$ + short.list key field.or.null = or + { pop$ "" } + 'skip$ + if$ + * + 'label := +} + +FUNCTION {sort.format.names} +{ 's := + #1 'nameptr := + "" + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { + s nameptr "{vv{ } }{ll{ }}{ ff{ }}{ jj{ }}" format.name$ 't := + nameptr #1 > + { + " " * + namesleft #1 = t "others" = and + { "zzzzz" * } + { numnames #2 > nameptr #2 = and + { "zz" * year field.or.null * " " * } + 'skip$ + if$ + t sortify * + } + if$ + } + { t sortify * } + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {sort.format.title} +{ 't := + "A " #2 + "An " #3 + "The " #4 t chop.word + chop.word + chop.word + sortify + #1 global.max$ substring$ +} + +FUNCTION {author.sort} +{ author empty$ + { key empty$ + { "to sort, need author or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.editor.sort} +{ author empty$ + { editor empty$ + { key empty$ + { "to sort, need author, editor, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { editor sort.format.names } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.organization.sort} +{ author empty$ + { organization empty$ + { key empty$ + { "to sort, need author, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "The " #4 organization chop.word sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {editor.organization.sort} +{ editor empty$ + { organization empty$ + { key empty$ + { "to sort, need editor, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "The " #4 organization chop.word sortify } + if$ + } + { editor sort.format.names } + if$ +} + + +FUNCTION {presort} +{ calc.label + label sortify + " " + * + type$ "book" = + type$ "inbook" = + or + 'author.editor.sort + { type$ "proceedings" = + 'editor.organization.sort + { type$ "manual" = + 'author.organization.sort + 'author.sort + if$ + } + if$ + } + if$ + " " + * + year field.or.null sortify + * + " " + * + cite$ + * + #1 entry.max$ substring$ + 'sort.label := + sort.label * + #1 entry.max$ substring$ + 'sort.key$ := +} + +ITERATE {presort} + +SORT + +STRINGS { longest.label last.label next.extra } + +INTEGERS { longest.label.width last.extra.num number.label } + +FUNCTION {initialize.longest.label} +{ "" 'longest.label := + #0 int.to.chr$ 'last.label := + "" 'next.extra := + #0 'longest.label.width := + #0 'last.extra.num := + #0 'number.label := +} + +FUNCTION {forward.pass} +{ last.label label = + { last.extra.num #1 + 'last.extra.num := + last.extra.num int.to.chr$ 'extra.label := + } + { "a" chr.to.int$ 'last.extra.num := + "" 'extra.label := + label 'last.label := + } + if$ + number.label #1 + 'number.label := +} + +FUNCTION {reverse.pass} +{ next.extra "b" = + { "a" 'extra.label := } + 'skip$ + if$ + extra.label 'next.extra := + extra.label + duplicate$ empty$ + 'skip$ + { "{\natexlab{" swap$ * "}}" * } + if$ + 'extra.label := + label extra.label * 'label := +} + +EXECUTE {initialize.longest.label} + +ITERATE {forward.pass} + +REVERSE {reverse.pass} + +FUNCTION {bib.sort.order} +{ sort.label 'sort.key$ := +} + +ITERATE {bib.sort.order} + +SORT + +FUNCTION {begin.bib} +{ preamble$ empty$ + 'skip$ + { preamble$ write$ newline$ } + if$ + "\begin{thebibliography}{" number.label int.to.str$ * "}" * + write$ newline$ + "\providecommand{\natexlab}[1]{#1}" + write$ newline$ + "\providecommand{\url}[1]{\texttt{#1}}" + write$ newline$ + "\expandafter\ifx\csname urlstyle\endcsname\relax" + write$ newline$ + " \providecommand{\doi}[1]{doi: #1}\else" + write$ newline$ + " \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi" + write$ newline$ +} + +EXECUTE {begin.bib} + +EXECUTE {init.state.consts} + +ITERATE {call.type$} + +FUNCTION {end.bib} +{ newline$ + "\end{thebibliography}" write$ newline$ +} + +EXECUTE {end.bib} diff --git a/memoire/remerciements.aux b/memoire/remerciements.aux new file mode 100644 index 0000000..a8d4360 --- /dev/null +++ b/memoire/remerciements.aux @@ -0,0 +1,80 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{toc}{\contentsline {chapter}{Remerciements}{xvii}{section*.9}} +\@setckpt{remerciements}{ +\setcounter{page}{18} +\setcounter{equation}{0} +\setcounter{enumi}{0} +\setcounter{enumii}{0} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{0} +\setcounter{section}{0} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{18} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{0} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{0} +\setcounter{Item}{0} +\setcounter{Hfootnote}{0} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{6} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/remerciements.log b/memoire/remerciements.log new file mode 100644 index 0000000..392a08a --- /dev/null +++ b/memoire/remerciements.log @@ -0,0 +1,1050 @@ +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.7.6) 24 SEP 2013 21:14 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**\input remerciements.tex +(./remerciements.tex +! Undefined control sequence. +l.1 \chapter + *{Remerciements} % ne pas numéroter +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1 \chapter* + {Remerciements} % ne pas numéroter +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no * in font nullfont! +Missing character: There is no R in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +! Undefined control sequence. +l.2 \phantomsection + \addcontentsline{toc}{chapter}{Remerciements} % inclure d... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 1--3 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.4 J + e dois avouer que je ne savais pas trop à quoi m'attendre au moment +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no J in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¹ in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no J in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ‰ in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¨ in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no C in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no R in font nullfont! +Missing character: There is no R in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no J in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ® in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no J in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ¢ in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no J in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no  in font nullfont! +Missing character: There is no « in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no © in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no  in font nullfont! +Missing character: There is no » in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no ' in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no   in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ª in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no . in font nullfont! +) +! Emergency stop. +<*> \input remerciements.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 5 strings out of 493485 + 157 string characters out of 3143527 + 47818 words of memory out of 3000000 + 3412 multiletter control sequences out of 15000+200000 + 3640 words of font info for 14 fonts, out of 3000000 for 9000 + 957 hyphenation exceptions out of 8191 + 5i,0n,4p,106b,17s stack positions out of 5000i,500n,10000p,200000b,50000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/memoire/remerciements.tex b/memoire/remerciements.tex new file mode 100644 index 0000000..99b5697 --- /dev/null +++ b/memoire/remerciements.tex @@ -0,0 +1,17 @@ +\chapter*{Remerciements} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Remerciements} % inclure dans TdM + +Je dois avouer que je ne savais pas trop à quoi m'attendre au moment où j'ai commencé la rédaction de ce mémoire. En essayant de ne pas oublier personne, j'aimerais tout d'abord remercier celui qui m'a accompagné durant ces deux dernières années et qui m'a énormément appris, le professeur Andrew Luong. + +J'aimerais aussi remercier mes évaluateurs, les professeurs Claire Bilodeau et Julien Trufin. J'aimerais aussi remercier le professeur Vincent Goulet, qui m'a fortement encouragé de m'inscrire à la maîtrise, et pour les nombreuses références qu'il a pu me fournir, pour la rédaction entre autres, qui m'ont grandement facilité la tâche. Je remercie tout autant le professeur Ghislain Léveillé pour son soutien et pour m'avoir offert la possibilité de présenter mes résultats. + +J'aimerais aussi remercier l'ensemble des étudiants gradués, des professeurs et du personnel de l'École d'Actuariat de l'Université Laval, ainsi que mes collègues de travail de la CARRA, pour leur soutien et leurs encouragements. + +Je suis reconnaissant envers l'École d'Actuariat et la Chaire d'Actuariat de l'Université Laval pour le soutien financier en m'offrant des postes d'auxiliaires d'enseignement et de recherche. + +Enfin, j'aimerais remercier mes parents, ma soeur et l'ensemble de ma famille et de mes amis pour avoir accepté que je passe plus de temps à «écrire des formules bizarres sur l'ordinateur» qu'à être avec eux. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/resume.aux b/memoire/resume.aux new file mode 100644 index 0000000..6e09dee --- /dev/null +++ b/memoire/resume.aux @@ -0,0 +1,80 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{toc}{\contentsline {chapter}{R\IeC {\'e}sum\IeC {\'e}}{iii}{section*.2}} +\@setckpt{resume}{ +\setcounter{page}{4} +\setcounter{equation}{0} +\setcounter{enumi}{0} +\setcounter{enumii}{0} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{@memmarkcntra}{0} +\setcounter{storedpagenumber}{1} +\setcounter{book}{0} +\setcounter{part}{0} +\setcounter{chapter}{0} +\setcounter{section}{0} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{@ppsavesec}{0} +\setcounter{@ppsaveapp}{0} +\setcounter{vslineno}{0} +\setcounter{poemline}{0} +\setcounter{modulo@vs}{0} +\setcounter{memfvsline}{0} +\setcounter{verse}{0} +\setcounter{chrsinstr}{0} +\setcounter{poem}{0} +\setcounter{newflo@tctr}{4} +\setcounter{@contsubnum}{0} +\setcounter{maxsecnumdepth}{2} +\setcounter{sidefootnote}{0} +\setcounter{pagenote}{0} +\setcounter{pagenoteshadow}{0} +\setcounter{memfbvline}{0} +\setcounter{bvlinectr}{0} +\setcounter{cp@cntr}{0} +\setcounter{ism@mctr}{0} +\setcounter{xsm@mctr}{0} +\setcounter{csm@mctr}{0} +\setcounter{ksm@mctr}{0} +\setcounter{xksm@mctr}{0} +\setcounter{cksm@mctr}{0} +\setcounter{msm@mctr}{0} +\setcounter{xmsm@mctr}{0} +\setcounter{cmsm@mctr}{0} +\setcounter{bsm@mctr}{0} +\setcounter{workm@mctr}{0} +\setcounter{sheetsequence}{4} +\setcounter{lastsheet}{155} +\setcounter{lastpage}{137} +\setcounter{figure}{0} +\setcounter{lofdepth}{1} +\setcounter{table}{0} +\setcounter{lotdepth}{1} +\setcounter{NAT@ctr}{0} +\setcounter{nprt@mantissa@digitsbefore}{0} +\setcounter{nprt@mantissa@digitsafter}{0} +\setcounter{nprt@exponent@digitsbefore}{0} +\setcounter{nprt@exponent@digitsafter}{0} +\setcounter{nprt@digitsfirstblock}{0} +\setcounter{nprt@blockcnt}{0} +\setcounter{nprt@cntprint}{0} +\setcounter{parentequation}{0} +\setcounter{thmt@dummyctr}{0} +\setcounter{Item}{0} +\setcounter{Hfootnote}{0} +\setcounter{memhycontfloat}{0} +\setcounter{Hpagenote}{0} +\setcounter{bookmark@seq@number}{1} +\setcounter{ContinuedFloat}{0} +\setcounter{app@runs}{1} +\setcounter{pspicture}{0} +\setcounter{section@level}{0} +\setcounter{theo}{0} +\setcounter{hypothese}{0} +} diff --git a/memoire/resume.tex b/memoire/resume.tex new file mode 100644 index 0000000..048ba9b --- /dev/null +++ b/memoire/resume.tex @@ -0,0 +1,19 @@ +\chapter*{Résumé} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Résumé} % inclure dans TdM + +\begin{otherlanguage*}{francais} + Les modèles classiques en finance sont basés sur des + hypothèses qui ne sont pas toujours vérifiables + empiriquement. L'objectif de ce mémoire est de présenter + l'utilisation de la distribution de Laplace asymétrique généralisée + comme une alternative intéressante à ces derniers. Pour ce faire, on + utilise ses différentes propriétés afin de développer des méthodes + d'estimation paramétrique, d'approximation et de test, en plus d'élaborer quelques principes d'évaluation de produits dérivés. On + présente enfin un exemple d'application numérique afin d'illustrer ces + différents concepts. +\end{otherlanguage*} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/memoire/sa.pdf b/memoire/sa.pdf new file mode 100644 index 0000000..f8a641f Binary files /dev/null and b/memoire/sa.pdf differ diff --git a/memoire/ul_p.eps b/memoire/ul_p.eps new file mode 100644 index 0000000..627de53 Binary files /dev/null and b/memoire/ul_p.eps differ diff --git a/memoire/ul_p.pdf b/memoire/ul_p.pdf new file mode 100644 index 0000000..6535aad Binary files /dev/null and b/memoire/ul_p.pdf differ diff --git a/memoire/ulthese.cls b/memoire/ulthese.cls new file mode 100644 index 0000000..99ed5a4 --- /dev/null +++ b/memoire/ulthese.cls @@ -0,0 +1,284 @@ +%% +%% This is file `ulthese.cls', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% ulthese.dtx (with options: `class') +%% +%% This is a generated file. +%% +%% Copyright (C) 2012 Universite Laval +%% +%% This file may be distributed and/or modified under the conditions +%% of the LaTeX Project Public License, either version 1.3c of this +%% license or (at your option) any later version. The latest version +%% of this license is in: +%% +%% http://www.latex-project.org/lppl.txt +%% +%% and version 1.3c or later is part of all distributions of LaTeX +%% version 2006/05/20 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Universite Laval +%% . +%% +%% This work consists of the files ulthese.dtx and ulthese.ins and the +%% derived files listed in the README file. +%% +\NeedsTeXFormat{LaTeX2e}[2009/09/24] +\ProvidesClass{ulthese}% + [2012/09/30 v1.0 Classe pour les theses et memoires de l'Universite Laval] +\RequirePackage{ifthen} +\newboolean{UL@natbib} +\setboolean{UL@natbib}{true} +\DeclareOption{nonatbib}{\setboolean{UL@natbib}{false}} +\newcommand*{\UL@ptsize}{} +\DeclareOption{10pt}{% + \PassOptionsToClass{10pt}{memoir} + \renewcommand*{\UL@ptsize}{10}} +\DeclareOption{11pt}{% + \PassOptionsToClass{11pt}{memoir} + \renewcommand*{\UL@ptsize}{11}} +\DeclareOption{12pt}{% + \PassOptionsToClass{12pt}{memoir} + \renewcommand*{\UL@ptsize}{12}} +\DeclareOption*{\PassOptionsToClass{\CurrentOption}{memoir}} +\ExecuteOptions{11pt,letterpaper} +\ProcessOptions +\LoadClass[twoside,openright]{memoir} +\RequirePackage[T1]{fontenc} +\ifthenelse{\boolean{UL@natbib}}{\RequirePackage{natbib}}{} +\RequirePackage{babel} +\RequirePackage[autolanguage]{numprint} +\RequirePackage{graphicx} +\RequirePackage{xcolor} +\RequirePackage{textcomp} +\RequirePackage[scaled=0.92]{helvet} +\definecolor{ULlinkcolor}{rgb}{0,0,0.3} +\setlrmarginsandblock{35mm}{25mm}{*} +\setulmarginsandblock{25mm}{*}{1} +\setheadfoot{\baselineskip}{20mm} +\checkandfixthelayout[lines] +\OnehalfSpacing +\setlength{\parskip}{0.5\baselineskip} +\setlength{\parindent}{0em} +\renewcommand{\tocheadstart}{\SingleSpacing\chapterheadstart} +\renewcommand{\lotheadstart}{\SingleSpacing\chapterheadstart} +\renewcommand{\lofheadstart}{\SingleSpacing\chapterheadstart} +\makepagestyle{ul} + \makeevenfoot{ul}{\thepage}{}{} + \makeoddfoot{ul}{}{}{\thepage} +\aliaspagestyle{plain}{ul} +\pagestyle{ul} +\ifthenelse{\UL@ptsize=10}{% + \newcommand*{\UL@fonttitle}{\normalfont\Huge\bfseries\sffamily} + \newcommand*{\UL@fontsubtitle}{\normalfont\huge\bfseries\sffamily} + \newcommand*{\UL@fontauthor}{\normalfont\LARGE\bfseries\sffamily} + \newcommand*{\UL@fontprogram}{\UL@fontauthor} + \newcommand*{\UL@fontbase}{\normalfont\LARGE\sffamily}}{} +\ifthenelse{\UL@ptsize=11}{% + \newcommand*{\UL@fonttitle}{\normalfont\huge\bfseries\sffamily} + \newcommand*{\UL@fontsubtitle}{\normalfont\LARGE\bfseries\sffamily} + \newcommand*{\UL@fontauthor}{\normalfont\Large\bfseries\sffamily} + \newcommand*{\UL@fontprogram}{\UL@fontauthor} + \newcommand*{\UL@fontbase}{\normalfont\Large\sffamily}}{} +\ifthenelse{\UL@ptsize=12}{% + \newcommand*{\UL@fonttitle}{\normalfont\LARGE\bfseries\sffamily} + \newcommand*{\UL@fontsubtitle}{\normalfont\Large\bfseries\sffamily} + \newcommand*{\UL@fontauthor}{\normalfont\large\bfseries\sffamily} + \newcommand*{\UL@fontprogram}{\UL@fontauthor} + \newcommand*{\UL@fontbase}{\normalfont\large\sffamily}}{} +\newboolean{UL@hassubtitle} +\newboolean{UL@isprogmasc} +\newcommand{\UL@maintitle}{} +\newcommand{\UL@subtitle}{} +\newcommand*{\UL@author}{} +\newcommand*{\UL@program}{} +\newcommand*{\UL@year}{} +\newcommand*{\UL@typeofdoc}{} +\newcommand*{\UL@degree}{} +\newcommand*{\UL@nameother}{} +\newcommand*{\UL@degreeother}{} +\newcommand*{\UL@facUL}{} +\newcommand*{\UL@facother}{} +\newcommand*{\UL@extensionat}{} +\newcommand*{\UL@extensionloc}{} +\newcommand{\titre}[1]{\renewcommand{\UL@maintitle}{#1}} +\newcommand{\soustitre}[1]{% + \setboolean{UL@hassubtitle}{true} + \renewcommand{\UL@subtitle}{#1}} +\newcommand*{\auteur}[1]{\renewcommand*{\UL@author}{#1}} +\newcommand*{\annee}[1]{\renewcommand*{\UL@year}{#1}} +\newcommand*{\programme}[1]{\renewcommand*{\UL@program}{#1}} +\newcommand*{\LLD}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Docteur en droit (L.L.D.)}} +\newcommand*{\DPsy}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Docteur en psychologie (D.Psy.)}} +\newcommand*{\DThP}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Docteur en th\'eologie pratique (D.Th.P.)}} +\newcommand*{\PhD}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Philosophi{\ae} doctor (Ph.D.)}} +\newcommand*{\LLM}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en droit (L.L.M.)}} +\newcommand*{\MA}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre \`es arts (M.A.)}} +\newcommand*{\MMus}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en musique (M.Mus.)}} +\newcommand*{\MSc}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre \`es sciences (M.Sc.)}} +\newcommand*{\MServSoc}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en service social (M.Serv.Soc.)}} +\newcommand*{\MScGeogr}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en sciences géographiques (M.Sc.G\'eogr.)}} +\newcommand*{\MATDR}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en am\'enagement du territoire et d\'eveloppement r\'egional (M.ATDR)}} +\newcommand*{\multifacultaire}{\renewcommand*{\UL@typenum}{1}} +\newcommand*{\cotutelle}{% + \renewcommand*{\UL@typenum}{2} + \edef\UL@typeofdoc{\UL@typeofdoc\ en cotutelle}} +\newcommand*{\univcotutelle}[1]{\renewcommand*{\UL@nameother}{#1}} +\newcommand*{\gradecotutelle}[1]{\renewcommand*{\UL@degreeother}{#1}} +\newcommand*{\extensionUdeS}{% + \renewcommand*{\UL@typenum}{3} + \renewcommand*{\UL@extensionat}{Universit\'e de Sherbrooke} + \renewcommand*{\UL@extensionloc}{Sherbrooke, Qu\'ebec}} +\newcommand*{\extensionUQO}{% + \renewcommand*{\UL@typenum}{3} + \renewcommand*{\UL@extensionat}{Universit\'e du Qu\'ebec en Outaouais} + \renewcommand*{\UL@extensionloc}{Gatineau, Qu\'ebec}} +\newcommand*{\extensionUQAC}{% + \renewcommand*{\UL@typenum}{3} + \renewcommand*{\UL@extensionat}{Universit\'e du Qu\'ebec \`a Chicoutimi} + \renewcommand*{\UL@extensionloc}{Chicoutimi, Qu\'ebec}} +\newcommand{\faculteUL}[1]{\renewcommand*{\UL@facUL}{#1}} +\newcommand*{\faculteUdeS}[1]{\renewcommand*{\UL@facother}{#1}} +\newcommand*{\faculteUQO}[1]{\renewcommand*{\UL@facother}{#1}} +\newcommand*{\faculteUQAC}[1]{\renewcommand*{\UL@facother}{#1}} +\newcommand{\UL@title}{% + {\UL@fonttitle\UL@maintitle\par} + {\UL@fontsubtitle + \ifthenelse{\boolean{UL@hassubtitle}}{\UL@subtitle}{% + \vspace*{\baselineskip}}\par}} +\newcommand{\UL@docid}{% + {\UL@fontprogram\UL@typeofdoc\par + \ifnum\UL@typenum=2 \UL@program\par \fi}} +\newcommand{\UL@details}{% + \ifcase\UL@typenum\relax% 0 standard + \vspace{96pt} + {\UL@fontprogram\UL@program}\par + \UL@degree\par + \vspace{112pt} + Qu\'ebec, Canada\par + \or% 1 multifacultaire + \vspace{96pt} + {\UL@fontprogram\UL@program}\par + \UL@degree\par + \vspace{36pt} + \UL@facUL\par + \vspace{48pt} + Qu\'ebec, Canada\par + \or% 2 cotutelle + \vspace{72pt} + Universit\'e Laval\par Qu\'ebec, Canada\par + \UL@degree\par + \vspace{\baselineskip} et\par \vspace{\baselineskip} + \UL@nameother\par + \UL@degreeother\par + \or% 3 extension + \vspace{48pt} + {\UL@fontprogram\UL@program\ de l'Universit\'e Laval\par + \ifthenelse{\boolean{UL@isprogmasc}}{offert}{offerte} + en extension \`a l'\UL@extensionat}\par + \vspace{36pt} + \UL@degree\par + \vspace{36pt} + \UL@facother\par \UL@extensionat\par \UL@extensionloc\par + \vspace{\baselineskip} + \UL@facUL\par Universit\'e Laval\par Qu\'ebec, Canada\par + \fi} +\newcommand{\pagetitre}{{% + \clearpage + \thispagestyle{empty} + \SingleSpacing\setlength{\parskip}{0pt} + \centering + \UL@fontbase + \ifthenelse{\UL@typenum > 1}{\vspace*{0pt}\par}{% + \includegraphics[height=40px,keepaspectratio=true]{ul_p}\par} + \vspace{72pt} + \UL@title + \vspace{48pt} + \UL@docid + \vspace{72pt} + {\UL@fontauthor\UL@author}\par + \UL@details + \vfill + {\textcopyright} \UL@author, \UL@year\par + \cleardoublepage + }} +\addto\captionsfrench{\renewcommand{\listfigurename}{Liste des figures}} +\newcommand{\dedicace}[1]{{% + \clearpage + \pagestyle{empty} + \setlength{\beforeepigraphskip}{10\baselineskip} + \setlength{\epigraphrule}{0pt} + \epigraphtextposition{flushright} + \mbox{}\epigraph{\itshape #1}{} + }} +\newcommand{\epigraphe}[2]{{% + \clearpage + \pagestyle{empty} + \setlength{\beforeepigraphskip}{10\baselineskip} + \mbox{}\epigraph{#1}{#2} + }} +\renewenvironment{quote}{% + \list{}{\rightmargin 10mm \leftmargin 10mm}% + \item[]}{\endlist} +\renewenvironment{quotation}{% + \list{}{% + \SingleSpacing + \listparindent 0em + \itemindent \listparindent + \leftmargin 10mm + \rightmargin \leftmargin + \parsep 6\p@ \@plus\p@}% + \item[]}{\endlist} +\setsecnumdepth{subsection} +\endinput +%% +%% End of file `ulthese.cls'. diff --git a/memoire/ulthese.dtx b/memoire/ulthese.dtx new file mode 100644 index 0000000..2e88725 --- /dev/null +++ b/memoire/ulthese.dtx @@ -0,0 +1,1608 @@ +% \iffalse meta-comment +% +% Copyright (C) 2012 Universite Laval +% +% This file may be distributed and/or modified under the conditions +% of the LaTeX Project Public License, either version 1.3c of this +% license or (at your option) any later version. The latest version +% of this license is in: +% +% http://www.latex-project.org/lppl.txt +% +% and version 1.3c or later is part of all distributions of LaTeX +% version 2006/05/20 or later. +% +% This work has the LPPL maintenance status `maintained'. +% +% The Current Maintainer of this work is Universite Laval +% . +% +% This work consists of the files ulthese.dtx and ulthese.ins +% and the derived files listed in the README file. +% +% \fi +% +% \iffalse +%<*dtx> +\ProvidesFile{ulthese.dtx} +% +%\NeedsTeXFormat{LaTeX2e}[2009/09/24] +%\ProvidesClass{ulthese}% +%<*class> + [2012/09/30 v1.0 Classe pour les theses et memoires de l'Universite Laval] +% +%<*driver> +\documentclass[11pt]{ltxdoc} + \usepackage[utf8]{inputenc} + \usepackage[T1]{fontenc} + \usepackage{natbib} + \usepackage[francais]{babel} + \usepackage[autolanguage]{numprint} + \usepackage{mathpazo} + \usepackage[scaled=0.92]{helvet} + \usepackage{color,booktabs} + \EnableCrossrefs + \CodelineIndex + \RecordChanges + + \definecolor{doclinkcolor}{rgb}{0,0,0.5} + + \usepackage{hyperref} + \hypersetup{colorlinks,allcolors=doclinkcolor} + + \frenchbsetup{% + CompactItemize=false, % ne pas compacter les listes + ThinSpaceInFrenchNumbers=true % espace fine dans les nombres + } +\begin{document} + \DocInput{ulthese.dtx} +\end{document} +% +% \fi +% \CheckSum{568} +% \DoNotIndex{\',\^,\`,\ ,\ae} +% \DoNotIndex{\RequirePackage,\ExecuteOptions,\ifthenelse,\ProcessOptions} +% \DoNotIndex{\newcommand,\newcommand*,\newboolean} +% \DoNotIndex{\setlength,\setboolean} +% \changes{1.0}{2012-09-30}{Version initiale} +% \GetFileInfo{ulthese.dtx} +% \title{\textsf{ulthese}: une classe pour les thèses et mémoires de +% l'Université Laval\thanks{Ce document décrit la classe +% \textsf{ulthese}~\fileversion, datée du \filedate.}} +% \author{Faculté des études supérieures et postdoctorales\thanks{% +% Cette classe et sa documentation ont été rédigés par Vincent +% Goulet~(Faculté des sciences et de génie) avec la collaboration de +% Koassi D'Almeida~(Faculté des études supérieures et postdoctorales) et +% Pierre Lasou~(Bibliothèque).}} +% \maketitle +% \tableofcontents +% +% \section{Introduction} +% +% La classe \textsf{ulthese} permet de composer avec {\LaTeX} des thèses +% et mémoires immédiatement conformes aux règles générales de +% présentation matérielle de la Faculté des études supérieures et +% postdoctorales (FESP) de l'Université Laval. Ces règles définissent +% principalement la présentation de la page titre des thèses et mémoires +% ainsi que la disposition du texte sur la page. La classe en elle-même +% est donc relativement simple. +% +% Cependant, la classe \textsf{ulthese} est basée sur la classe +% \textsf{memoir}, une extension de la classe standard \textsf{book} +% facilitant à plusieurs égards la préparation de documents d'allure +% professionnelle dans {\LaTeX}. La classe \textsf{memoir} est très +% configurable et incorpore d'office plus de 30 des paquetages +% (\emph{packages}) les plus populaires\footnote{% +% Consulter la section~18.24 de la documentation de \textsf{memoir} +% pour la liste ou encore le fichier journal (\emph{log}) de la +% compilation d'un document utilisant la classe \textsf{ulthese}.}. % +% L'intégralité des fonctionnalités de \textsf{memoir} est disponible +% dans \textsf{ulthese}. +% +% La classe \textsf{memoir} fait maintenant partie des distributions +% {\LaTeX} modernes; elle devrait donc être installée et disponible sur +% votre système. La classe est livrée avec une documentation exhaustive: +% le manuel d'instructions fait près de 600~pages! Il peut être utile de +% s'y référer de temps à autre pour réaliser une mise en page +% particulière. Rechercher sur votre système le fichier |memman.pdf| ou +% le % +% \href{http://mirrors.ctan.org/macros/latex/contrib/memoir/memman.pdf}{% +% consulter en ligne} % +% sur le site % +% \href{http://www.ctan.org/}{% +% \emph{Comprehensive R Archive Network} (CTAN)}. +% +% +% \section{Installation} +% +% L'installation de la classe consiste à créer le fichier +% |ulthese.cls| et plusieurs gabarits |.tex| à partir du code source +% documenté se trouvant dans le fichier |ulthese.dtx|. Il est +% recommandé de simplement créer ces fichiers dans le dossier de +% travail de la thèse ou du mémoire. +% +% Pour procéder à l'installation, décompresser l'archive |ulthese.zip| +% dans le dossier de travail, puis compiler avec {\LaTeX} le fichier +% |ulthese.ins| en exécutant +% \begin{quote} +% |latex ulthese.ins| +% \end{quote} +% depuis une invite de commande. Si l'on est peu familier avec +% l'invite de commande, on peut aussi procéder comme avec tout +% document {\LaTeX}, soit ouvrir le fichier |ulthese.ins| dans son +% éditeur de texte favori et lancer la compilation avec {\LaTeX}, +% pdf{\TeX} ou pdf{\LaTeX} depuis celui-ci. +% +% \section{Utilisation} +% +% La classe est chargée avec la commande +% \begin{quote} +% |\documentclass[|\meta{options}|]{ulthese}| +% \end{quote} +% Les marges, l'interligne et la numérotation des pages sont adaptées +% aux règles de présentation matérielle de la FESP. Les options et les +% commandes définies par la classe sont décrites dans les sections +% suivantes. +% +% \subsection{Options de la classe} +% \label{sec:utilisation:options} +% +% Les \meta{options} que l'on peut spécifier au chargement de la classe +% sont les suivantes: +% \begin{description} +% \item[|10pt|] sélectionne une taille de police de 10~points; +% \item[|11pt|] sélectionne une taille de police de 11~points; +% \item[|12pt|] sélectionne une taille de police de 12~points; +% \item[|nonatbib|] empêche de charger le paquetage \textsf{natbib} (voir +% la section~\ref{sec:bibliographystyle}); +% \item[|english, francais, ...|] langues utilisées dans le document; +% \item[\mdseries\meta{options memoir}] autres options du paquetage +% \textsf{memoir}. +% \end{description} +% +% Si aucune taille de police n'est précisée, la classe utilisera une +% taille par défaut de 11~points. La taille des polices de la page +% titre n'est pas affectée par les options |10pt|, |11pt| et |12pt|. +% +% Le paquetage \textsf{natbib} est normalement chargé par la classe. +% L'option |nonatbib| permet d'empêcher le chargement en cas de +% conflit avec un autre paquetage de mise en forme de la +% bibliographie. +% +% Les langues sont passées au paquetage \textsf{babel}; voir la +% section~\ref{sec:langues}. Leur libellé devrait donc correspondre +% aux options de ce paquetage. La dernière langue spécifiée est la +% langue active par défaut dans le document. +% +% Toute autre option sera passée à la classe \textsf{memoir} dont, +% entre autres, le format du papier. Le format lettre nord-américain +% (option |letterpaper|) est utilisé par défaut. Si la thèse doit être +% imprimée en format international A4, utiliser l'option |a4paper|. La +% classe \textsf{memoir} est toujours chargée avec les options +% |twoside|, et |openright|. +% +% \subsection{Nouvelles commandes} +% +% La classe \textsf{ulthese} définit quelques nouvelles commandes +% servant principalement à créer la page titre et des éléments des pages +% liminaires. +% +% \subsubsection{Commandes de la page titre} +% \label{sec:commandes-pagetitre} +% +% Les commandes ci-dessous servent à définir les divers éléments de la +% page titre et leur disposition sur la page. +% +% \begin{DescribeMacro}{\titre} +% Titre principal de la thèse ou du mémoire. Ne pas utiliser la +% commande |\title| de {\LaTeX} pour ce faire. +% +% Un titre très long devra être coupé manuellement avec |\\| ou +% |\newline|. Dans un tel cas, il est également nécessaire de +% retrancher de l'espacement vertical pour éviter que les autres +% éléments de la page titre ne soient poussés vers le bas. Pour ce +% faire, ajouter la déclaration +% \begin{quote} +% |\vspace*{-|$n$|\baselineskip}| +% \end{quote} +% à la fin de la \emph{dernière} ligne de titre s'il y a $n + 1$ +% lignes de titre au total. +% +% Par exemple, la déclaration d'un titre d'une seule ligne est: +% \begin{quote} +% |\titre{Ceci est un titre d'une seule ligne}| +% \end{quote} +% Pour un titre de deux lignes, on écrira: +% \begin{quote} +% |\titre{Ceci est la première ligne d'un long titre \\| \\ +% | et ceci est la seconde \vspace*{-1\baselineskip}}| +% \end{quote} +% Pour un titre de trois lignes, c'est alors: +% \begin{quote} +% |\titre{Ceci est la première ligne d'un très long titre \\| \\ +% | pour lequel deux lignes ne suffisent pas \\| \\ +% | alors voici la troisième \vspace*{-2\baselineskip}}| +% \end{quote} +% Ainsi de suite. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\soustitre} +% Sous-titre de la thèse ou du mémoire, le cas échéant. Les remarques +% sur un long titre principal s'appliquent également au sous-titre. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\auteur} +% Nom complet de l'auteur de la thèse ou du mémoire, sous la forme +% |Prénom Nom| avec seulement des majuscules initiales. Ne pas +% utiliser la commande |\author| de {\LaTeX} pour le nom de l'auteur. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\annee} +% Année du dépôt final de la thèse ou du mémoire. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\programme} +% Nom du complet officiel du programme d'études comme «Doctorat en +% informatique» ou «Maîtrise en mathématiques». Si le programme +% comporte une majeure, séparer sa mention de celle du programme +% principal par un tiret demi-quadratin (obtenu avec |--|). +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\PhD,} +% \begin{DescribeMacro}{\MSc,} +% \begin{DescribeMacro}{\MA, \dots} +% Déclaration du type de grade auquel donne droit la thèse ou le +% mémoire. Consulter le tableau~\ref{tab:grades} pour la liste +% complète des commandes et des grades correspondants. +% \end{DescribeMacro} +% \end{DescribeMacro} +% \end{DescribeMacro} +% +% \begin{table} +% \centering +% \begin{tabular}{lp{9cm}} +% \toprule +% Commande & Nom du grade (sigle) \\ +% \midrule +% |\LLD|\index{LLD=\verb+\LLD+} +% & Docteur en droit (L.L.D.) \\ +% |\DPsy|\index{DPsy=\verb+\DPsy+} +% & Docteur en psychologie (D.Psy.) \\ +% |\DThP|\index{DThP=\verb+\DThP+} +% & Docteur en théologie pratique (D.Th.P.) \\ +% |\PhD|\index{PhD=\verb+\PhD+} +% & Philosophi{\ae} doctor (Ph.D.) \\ +% \addlinespace[6pt] +% |\LLM|\index{LLM=\verb+\LLM+} +% & Maître en droit (L.L.M.) \\ +% |\MA|\index{MA=\verb+\MA+} +% & Maître ès arts (M.A.) \\ +% |\MMus|\index{MMus=\verb+\MMus+} +% & Maître en musique (M.Mus.) \\ +% |\MSc|\index{MSc=\verb+\MSc+} +% & Maître ès sciences (M.Sc.) \\ +% |\MServSoc|\index{MServSoc=\verb+\MServSoc+} +% & Maître en service social (M.Serv.Soc.) \\ +% |\MScGeogr|\index{MScGeogr=\verb+\MScGeogr+} +% & Maître en sciences géographiques (M.Sc.G\'eogr.) \\ +% |\MATDR|\index{MATDR=\verb+\MATDR+} +% & Maître en aménagement du territoire et développement régional +% (M.ATDR) \\ +% \bottomrule +% \end{tabular} +% \caption{Commandes de déclaration du grade et libellés correspondants} +% \label{tab:grades} +% \end{table} +% +% \begin{DescribeMacro}{\multifacultaire} +% Déclaration d'une thèse ou d'un mémoire multifacultaire. Requiert +% d'utiliser aussi la commande |\faculteUL| ci-dessous. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\cotutelle} +% Déclaration d'une thèse ou d'un mémoire réalisé en cotutelle avec +% une autre université. Requiert d'utiliser aussi les commandes +% |\univcotutelle| et |\gradecotutelle| ci-dessous. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\univcotutelle} +% Nom, ville et pays de l'université de cotutelle, déclarés sous la +% forme +% \begin{quote} +% |\univcotutelle{Nom de l'université \\ Ville, Pays}| +% \end{quote} +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\gradecotutelle} +% Grade conféré par l'université de cotutelle, déclaré sous la forme +% \begin{quote} +% |\gradecotutelle{Nom du grade (sigle)}| +% \end{quote} +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\extensionUdeS} +% Déclaration que la thèse est réalisée en extension à l'Université +% de Sherbrooke. Requiert d'utiliser aussi les commandes +% |\faculteUL| et |\faculteUdeS| ci-dessous. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\extensionUQO} +% Déclaration que la thèse est réalisée en extension à l'Université du +% Québec en Outaouais. Requiert d'utiliser aussi les commandes +% |\faculteUL| et |\faculteUQO| ci-dessous. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\extensionUQAC} +% Déclaration que le mémoire est réalisé en extension à l'Université +% du Québec à Chicoutimi. Requiert d'utiliser aussi les commandes +% |\faculteUL| et |\faculteUQAC| ci-dessous. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\faculteUL} +% Cette macro a deux usages: +% \begin{enumerate} +% \item noms des facultés pour les thèses et mémoires +% multifacultaires, séparés par des commandes |\\|; +% \item nom de la faculté de l'Université Laval où sont réalisés les +% thèses et mémoires en extension aux universités de Sherbrooke, du +% Québec en Outaouais ou du Québec à Chicoutimi. +% \end{enumerate} +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\faculteUdeS} +% Nom de la faculté de l'Université de Sherbrooke hébergeant la thèse +% en extension. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\faculteUQO} +% Nom de la faculté de l'Université du Québec en Outaouais hébergeant +% la thèse en extension. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\faculteUQAC} +% Nom de la faculté de l'Université du Québec à Chicoutimi hébergeant +% le mémoire en extension. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\pagetitre} +% Déclaration de création de la page titre. Ne pas utiliser la +% commande |\pagetitle| de {\LaTeX} pour ce faire. De toutes les +% commandes ci-dessus, c'est la seule qui doit se trouver dans le +% corps du document plutôt que dans le préambule. +% \end{DescribeMacro} +% +% \subsubsection{Commandes des pages liminaires} +% +% La classe définit deux commandes pour créer des pages liminaires +% prévues aux règles de présentation matérielle. +% +% \begin{DescribeMacro}{\dedicace} +% La commande |\dedicace| ajoute une dédicace («À mes parents», «À +% Camille») à la thèse ou au mémoire. La dédicace est disposée seule +% sur une page recto, à une dizaine de lignes de la marge du haut et +% alignée à droite. Par défaut, elle est composée en italique. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\epigraphe} +% La commande |\epigraph| sert à ajouter une épigraphe au début du +% document. Comme la dédicace, l'épigraphe est disposée seule sur une +% page recto, à une dizaine de lignes de la marge du haut et alignée à +% droite. La commande accepte deux arguments, soit le texte de la +% citation et son auteur ou la source, dans l'ordre. +% \end{DescribeMacro} +% +% Pour ajouter une épigraphe au début d'un ou plusieurs chapitres, +% utiliser directement la commande |\epigraph| de \textsf{memoir}, sur +% laquelle |\dedicace| et |\epigraphe| sont d'ailleurs basées. +% +% \subsection{Citations} +% \label{sec:citations} +% +% \begin{DescribeEnv}{quote} +% {\LaTeX} offre deux environnements pour les citations dans le +% texte: |quote| et |quotation|. Le premier sert pour les citations +% «courtes», quelques lignes au plus. Dans la classe, le texte est +% alors placé en retrait des marges normales de 10~mm à gauche et à +% droite. +% \end{DescribeEnv} +% +% \begin{DescribeEnv}{quotation} +% L'environnement |quotation|, quant à lui, doit être utilisé pour +% les citations «longues», celles qui peuvent s'étendre sur plus de +% cinq lignes ou, surtout, plus d'un paragraphe. Dans la classe, le +% texte est alors toujours placé en retrait de 10~mm, mais également +% à interligne simple. De plus, les paragraphes, le cas échéant, +% sont séparés d'un espace vertical afin de bien les distinguer les +% uns des autres. +% \end{DescribeEnv} +% +% \subsection{Interligne} +% +% \begin{DescribeMacro}{\OnehalfSpacing} +% L'espacement d'un interligne et demi utilisé dans la classe est +% obtenu avec la commande |\OnehalfSpacing| de \textsf{memoir}. +% L'interligne simple est automatiquement rétabli pour la page +% titre, la table des matières, la liste des tableaux, la liste +% des figures et les longues citations (section~\ref{sec:citations}). +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\SingleSpacing} +% Si ce devait être nécessaire ailleurs dans le document, la +% commande |\SingleSpacing| permet de passer à l'interligne simple. +% \end{DescribeMacro} +% +% +% \subsection{Autres paquetages chargés} +% \label{sec:paquetages} +% +% Outre \textsf{memoir} et \textsf{babel}, la classe \textsf{ulthese} +% charge pour son usage quelques paquetages qui peuvent aussi être +% utiles pour l'utilisateur de la classe. Il n'est donc pas nécessaire +% de charger de nouveau les paquetages suivants: +% \begin{description} +% \item[\normalfont\textsf{natbib}] gestion de la bibliographie (si +% l'option |nonatbib| de la classe est absente; voir aussi la +% section~\ref{sec:bibliographystyle}); +% \item[\normalfont\textsf{numprint}] permet de composer +% automatiquement des nombres avec un séparateur toutes les trois +% positions (une espace en français). Requis par la commande +% |\nombre| de \textsf{babel}; +% \item[\normalfont\textsf{graphicx}] support pour l'insertion et la +% manipulation de graphiques; +% \item[\normalfont\textsf{xcolor}] extension du paquetage +% \textsf{color} pour gérer les couleurs dans le texte; +% \item[\normalfont\textsf{textcomp}] multitude de symboles spéciaux, +% dont un beau symbole de copyright, \textcopyright. +% \item[\normalfont\textsf{helvet}] remplacement de la police sans +% empattements (\emph{serifs}) standard par la police Helvetica pour +% composer la page titre. Le paquetage est chargé avec l'option +% |scaled=0.92|. Consulter la section~\ref{sec:police} pour des +% informations additionnelles. +% \end{description} +% La section~\ref{sec:meo} sur la mise en {\oe}uvre de la classe +% fournit plus de détails sur la liste des paquetages chargés et les +% raisons pour lesquelles ils sont requis dans la classe. +% +% \section{Français et autres langues} +% \label{sec:langues} +% +% Une complication additionnelle pour les auteurs rédigeant dans une +% langue autre que l'anglais consiste à adapter {\LaTeX} à leur +% langue, qu'il s'agisse des mots clés, de la typographie ou de la +% césure des mots. La solution à ce problème provient du paquetage +% standard \textsf{babel}. Celui-ci permet de combiner plusieurs +% langues dans un même document et de passer de l'une à l'autre +% facilement. Il est chargé par la classe \textsf{ulthese}. +% +% Aucune langue n'est spécifiée dans la classe. La plupart des auteurs +% auront recours à l'anglais et au français, ne serait-ce que pour les +% deux résumés demandés par la FESP. Les langues utilisées dans le +% document doivent être spécifiées comme options à la classe, tel que +% mentionné à la section~\ref{sec:utilisation:options}. La +% \emph{dernière} langue spécifiée devient par défaut la langue active +% du document. +% +% \begin{DescribeMacro}{\selectlanguage} +% La commande |\selectlanguage| de \textsf{babel} permet de passer de +% la langue courante à la langue spécifiée en argument. +% \end{DescribeMacro} +% +% \begin{DescribeEnv}{otherlanguage} +% L'environnement |otherlanguage| de \textsf{babel} permet de faire +% la même chose que la commande |\selectlanguage|, sauf que le +% changement de langue est local à l'environnement --- utile pour +% les brefs changements de langue. +% \end{DescribeEnv} +% +% Si vous n'êtes pas autrement familier avec le paquetage \textsf{babel}, +% consulter sa documentation, ne serait-ce que les chapitres consacrés +% aux langues utilisées dans votre thèse ou mémoire. Rechercher sur +% votre système le fichier |babel.pdf| ou le % +% \href{http://mirrors.ctan.org/macros/latex/required/babel/babel.pdf}{% +% consulter en ligne} % +% sur CTAN. +% +% \section{Police de caractère du document} +% \label{sec:police} +% +% Les documents {\LaTeX} sont facilement reconnaissables par leur +% police de caractère par défaut, Computer Modern. Avec toute +% distribution {\LaTeX} moderne, il est maintenant simple d'utiliser +% l'une ou l'autre des polices Postscript standards. D'ailleurs la +% classe \textsf{ulthese} utilise la police sans empattements +% Helvetica pour composer la page titre. +% +% La FESP permet l'utilisation des polices Times et +% Palatino\footnote{% +% Police utilisée dans le présent document.} % +% dans les thèse et mémoires {\LaTeX}. Pour utiliser ces polices, +% charger les paquetages \textsf{mathptmx} ou \textsf{mathpazo}, +% respectivement. Pour les détails, consulter la documentation de +% l'ensemble de paquetages PSNFSS. Rechercher sur votre système le +% fichier |psnfss2e.pdf| ou le % +% \href{http://mirrors.ctan.org/macros/latex/required/psnfss/psnfss2e.pdf}{% +% consulter en ligne} % +% sur CTAN. +% +% \section{Gabarits} +% +% Il est recommandé de segmenter tout document d'une certaine ampleur +% dans des fichiers |.tex| distincts pour chaque partie --- +% habituellement un fichier par chapitre. Le document complet est +% composé à l'aide d'un fichier maître qui contient le préambule +% {\LaTeX} et un ensemble de commandes +% |\include|\index{include=\verb+\include+} pour réunir les parties +% dans un tout. +% +% La classe \textsf{ulthese} est livrée avec un ensemble de gabarits sur +% lesquels se baser pour: +% \begin{itemize} +% \item les fichiers maîtres de divers types de thèses et mémoires +% (standard, sur mesure, en cotutelle, en extension, etc.); +% \item les fichiers des parties les plus usuelles (résumés français +% et anglais, avant-propos, introduction, chapitres, conclusion, +% etc.). +% \end{itemize} +% Les noms des fichiers devraient permettre de facilement identifier +% leur contenu (une bonne pratique; |rappels.tex| est plus parlant et +% résiste mieux aux changements à l'ordre des chapitres que +% |chapitre1.tex|). +% +% Pour débuter la rédaction, renommer le gabarit de document maître +% approprié d'après votre numéro de dossier. Par exemple, l'étudiante +% dont le numéro de dossier est 900352789 et qui entame la rédaction +% d'une thèse multifacultaire renommera le fichier +% \begin{quote} +% |gabarit-doctorat-multifacultaire.tex| +% \end{quote} +% en +% \begin{quote} +% |900352789.tex|. +% \end{quote} +% Les autres gabarits de documents maîtres peuvent alors être +% supprimés. +% +% Les gabarits comportent des commentaires succincts pour vous guider +% dans la préparation de votre document. Les sections suivantes +% fournissent des détails additionnels, et ce, dans l'ordre où les +% commandes apparaissent dans les gabarits. +% +% \subsection{Encodage des fichiers} +% +% Taper de longs textes en français en {\LaTeX} devient rapidement +% pénible si on utilise les commandes |\'e|, |\`a| ou |\^e| pour entrer +% les lettres accentuées. Afin de pouvoir plutôt entrer directement |é|, +% |à| ou |ê|, {\LaTeX} doit être configuré pour reconnaître les lettres +% accentuées. C'est le rôle du paquetage \textsf{inputenc}. +% +% Cependant, il existe plusieurs manières différentes d'encoder --- ou +% d'enregistrer --- les lettres accentuées et autres caractères +% spéciaux (comme, par exemple, le symbole de l'euro) dans un +% ordinateur. La méthode la plus répandue et celle standard sur les +% versions récentes des systèmes d'exploitation Linux et OS~X est +% l'UTF-8 de la norme +% \href{http://fr.wikipedia.org/wiki/Unicode}{Unicode}. Les gabarits +% sont livrés dans ce type d'encodage. +% +% La déclaration +% \begin{quote} +% |\usepackage[utf8]{inputenc}| +% \end{quote} +% dans le préambule assure que {\LaTeX} traitera correctement des +% fichiers source encodés en UTF-8. +% +% La norme Unicode n'est pas aussi uniformément supportée par Windows. +% Selon l'éditeur de texte employé et la version du système +% d'exploitation, il peut être nécessaire d'utiliser les normes +% d'encodage % +% \href{http://fr.wikipedia.org/wiki/ISO_8859-1}{ISO~8859-1} % +% (ou Latin-1; option |latin1| de \textsf{inputenc}), % +% \href{http://fr.wikipedia.org/wiki/ISO_8859-15}{ISO~8859-15} % +% (ou Latin-9; option |latin9|) ou % +% \href{http://fr.wikipedia.org/wiki/Windows-1252}{Windows-1252} % +% (options |cp1252| ou |ansinew|). +% +% +% \subsection{Paquetages additionnels} +% +% Tel qu'explicité à la section~\ref{sec:paquetages}, la classe +% charge déjà quelques paquetages. Cependant, il est fort probable que +% vous devrez en charger d'autres pour composer votre document. Les +% gabarits prévoient un endroit pour le chargement de paquetages +% additionnels. Il est recommandé d'inscrire vos commandes +% |\usepackage| à cet endroit afin de respecter un certain ordre de +% chargement; voir la section suivante. +% +% Si vous utilisez un paquetage non standard dans les distributions +% courantes (MiK\TeX, \TeX~Live, Mac\TeX), vous devez le fournir avec +% le code source de votre document lors du dépôt final. +% +% \subsection{Hyperliens} +% \label{sec:hyperref} +% +% Le paquetage \textsf{hyperref} permet de transformer toutes les +% références en hyperliens cliquables lorsque le document {\LaTeX} est +% produit avec |pdflatex|. L'interaction de ce paquetage avec les autres +% est parfois (voire souvent) délicate. Pour cette raison, il est +% habituellement nécessaire de charger \textsf{hyperref} en tout +% dernier. C'est pourquoi il n'est pas chargé dans la classe, mais +% plutôt dans les gabarits. Prendre soin de maintenir le dernier rang de +% chargement lors de l'édition d'un gabarit. +% +% La configuration du paquetage dans les gabarits fait en sorte que +% les liens sont simplement signalés par une couleur de texte +% légèrement contrastante. L'utilisation de couleurs dans un document +% requiert le paquetage |xcolor|, chargé par la classe. La couleur de +% lien par défaut, |ULlinkcolor|, est définie dans la classe; voir la +% section~\ref{sec:couleurs}. +% +% \subsection{Police sans empattements} +% +% Tel que mentionné à la section~\ref{sec:police}, la police sans +% empattements Helvetica est chargée par la classe pour composer la page +% titre. Ainsi, toute utilisation des commandes |\textsf| et |\sffamily| +% sélectionnera cette police. +% +% Si vous composez la thèse ou le mémoire dans la police par défaut de +% {\LaTeX}, vous souhaiterez sans doute que les commandes ci-dessus +% sélectionnent plutôt la version sans empattements de Computer +% Modern; comparer +% \begin{quote} +% \sffamily Texte sans empattements en Helvetica +% \end{quote} +% et +% \begin{quote} +% \fontfamily{cmss}\selectfont Texte sans empattements en Computer Modern +% \end{quote} +% Pour ce faire, décommentez la ligne du gabarit afin que la +% commande +% \begin{quote} +% |\renewcommand{\sfdefault}{cmss}| +% \end{quote} +% apparaisse dans le préambule. +% +% \subsection{Options de \textsf{babel}} +% +% \begin{DescribeMacro}{\frenchbsetup} +% La commande |\frenchbsetup|\index{frenchbsetup=\verb+\frenchbsetup+} +% de \textsf{babel} permet de contrôler certains ajustements +% typographiques apportés par le paquetage en mode français. Consulter +% la documentation de \textsf{babel} pour la liste des options de +% configuration disponibles. +% \end{DescribeMacro} +% +% Les concepteurs de la classe \textsf{ulthese} proposent deux +% ajustements dans les gabarits: +% \begin{enumerate} +% \item l'option |CompactItemize=false| évite que le mode français de +% \textsf{babel} ne diminue l'espacement vertical dans les listes; +% \item avec l'option |ThinSpaceInFrenchNumbers=true|, une espace fine +% sera utilisée comme séparateur des milliers dans les nombres plutôt +% qu'une espace pleine. +% \end{enumerate} +% \begin{DescribeMacro}{\nombre} +% D'ailleurs, à ce sujet, le paquetage \textsf{numprint} étant +% chargé dans la classe, on peut utiliser la commande |\nombre| pour +% formater automatiquement les nombres. Par exemple, le résultat de +% |\nombre{123456789}| est \nombre{123456789}. +% \end{DescribeMacro} +% +% \subsection{Style de la bibliographie} +% \label{sec:bibliographystyle} +% +% \begin{DescribeMacro}{\bibliographystyle} +% Il est fortement recommandé d'utiliser BIB{\TeX} pour la préparation +% de la bibliographie. Le formatage de la bibliographie est contrôlé +% par un style choisi par la commande |\bibliographystyle|. Les styles +% standards de {\LaTeX} sont |plain|, |unsrt|, |alpha| et |abbrv|. +% \end{DescribeMacro} +% +% Pour plus de flexibilité, il est recommandé d'utiliser le paquetage +% \textsf{natbib} pour la gestion des références et des styles de la +% bibliographie. Entre autres choses, ce paquetage supporte le style de +% citation auteur-année fréquemment employé en sciences naturelles, +% plusieurs commandes de citation, un grand nombres de styles de +% bibliographie ainsi que des entrées spécifiques pour les numéros ISBN +% et les URL. Le paquetage fournit des styles de bibliographie +% |plainnat|, |unsrtnat| et |abbrvnat| similaires aux styles standards, +% mais plus complets. Il existe des +% \href{http://mirrors.ctan.org/biblio/bibtex/contrib/bib-fr/}{versions +% francisées} de ces styles (et de quelques autres) dans CTAN. +% +% Afin d'assurer le bon fonctionnement avec \textsf{babel}, le +% paquetage \textsf{natbib} est chargé par la classe \textsf{ulthese} +% (à moins que l'option |nonatbib| ne soit spécifiée). Pour consulter +% sa documentation, rechercher sur votre système le fichier +% |natbib.pdf| ou le % +% \href{http://mirrors.ctan.org/macros/latex/contrib/natbib/natbib.pdf}{% +% consulter en ligne} % +% sur CTAN. +% +% De plus, Vincent Goulet a préparé la feuille de style +% |francais.bst| compatible avec \textsf{natbib}; consulter la +% section \href{http://vgoulet.act.ulaval.ca/latex/}{LaTeX en +% français} de son site Web. La feuille de style permet de composer +% des bibliographies auteur-année respectant les normes de typographie +% française telles qu'édictées dans \cite{Malo:1996}. Pour utiliser ce +% style, on spécifera dans le préambule du document LaTeX +% \begin{quote} +% |\bibliographystyle{francais}| +% \end{quote} +% +% Autrement, la FESP n'a pas d'exigences particulières quant à la +% présentation de la bibliographie (présentation du titre, des auteurs +% et autres informations bibliographiques). +% +% \subsection{Déclarations de la page titre} +% +% Les gabarits comportent toutes les déclarations nécessaires pour +% composer la page titre des divers types de thèse ou de mémoires. +% Vous devez remplacer les éléments se trouvant entre crochets <~> en +% respectant la forme indiquée. Assurez-vous de supprimer les +% caractères < et > afin qu'il n'apparaissent pas sur la page titre de +% votre document. +% +% \subsection{Pages liminaires} +% +% \begin{DescribeMacro}{\frontmatter} +% La commande |\frontmatter| déclare que {\LaTeX} doit considérer le +% matériel qui suit comme des pages liminaires. En pratique, cela +% résulte essentiellement en une numérotation des pages en chiffres +% romains. +% \end{DescribeMacro} +% +% Les normes de présentation de la FESP édictent que les thèses et +% mémoires devraient comporter les pages liminaires suivantes, dans +% l'ordre: +% \begin{enumerate} +% \item la page titre (obligatoire); +% \item un résumé en français (obligatoire); +% \item un résumé en anglais (recommandé mais non obligatoire); +% \item une table des matières (obligatoire); +% \item une liste des tableaux; +% \item une liste des figures; +% \item une liste des abbréviations et des sigles; +% \item une dédicace; +% \item une épigraphe; +% \item des remerciements; +% \item un avant-propos (obligatoire dans le cas d'un mémoire ou d'une +% thèse avec insertion d'articles). +% \end{enumerate} +% +% Les commandes +% \begin{quote} +% |\pagetitre| \\ +% |\tableofcontents| \\ +% |\listoftables| \\ +% |\listoffigures| \\ +% |\dedicace{|\meta{texte}|}| \\ +% |\epigraphe{|\meta{texte}|}{|\meta{auteur}|}| +% \end{quote} +% permettent de générer les pages correspondantes. Seules les deux +% dernières commandes admettent des arguments. +% +% \begin{DescribeMacro}{\chapter*} +% Les résumés, la liste des abbréviations et des sigles, les +% remerciements et l'avant-propos sont composés comme des chapitres +% normaux, mais sans être numérotés. Il faut donc définir ces éléments +% avec la commande |\chapter*|. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\phantomsection} +% \begin{DescribeMacro}{\addcontentsline} +% Les sections declarées avec la commande |\chapter*| +% n'apparaissent pas dans la table des matières. Comme les normes +% de présentation de la FESP exigent que toutes les pages +% liminaires y figurent, on fait suivre les commandes +% |\chapter*{|\meta{Titre}|}| des commandes +% \end{DescribeMacro} +% \end{DescribeMacro} +% \begin{quote} +% |\phantomsection\addcontentsline{toc}{chapter}{|\meta{Titre}|}| +% \end{quote} +% Celles-ci ajoutent à la table des matières (|toc|) une section de +% niveau |chapter| dont le titre est \meta{Titre}. La commande +% |\phantomsection| est rendue nécessaire (ou recommandée) par le +% paquetage \textsf{hyperref}. +% +% \subsection{Corps du document} +% +% \begin{DescribeMacro}{\mainmatter} +% La commande |\mainmatter| délimite le début du corps du document. La +% numérotation des pages passe en chiffres arabes. +% \end{DescribeMacro} +% +% Le corps du document devrait normalement compter une introduction (non +% numérotée), un développement divisé en chapitres (numérotés) et une +% conclusion (non numérotée). +% +% \subsection{Annexes} +% +% \begin{DescribeMacro}{\appendix} +% Si la thèse ou le mémoire comporte une ou plusieurs annexes, +% composer celles-ci comme des chapitres normaux insérés dans le +% document maître après la commande |\appendix|. Cette commande a +% pour effet de passer d'un mode de numération numérique (1, 2, 3, +% \dots) à un mode alphabétique (A, B, C, \dots). +% \end{DescribeMacro} +% +% \subsection{Bibliographie} +% +% \begin{DescribeMacro}{\bibliography} +% Si vous utilisez BIB{\TeX}, la bibliographie est insérée dans le +% document à l'endroit où apparait la commande |\bibliography| dans le +% code source. +% \end{DescribeMacro} +% +% Consulter la section~\ref{sec:bibliographystyle} pour des +% informations additionnelles sur la préparation de la bibliographie. +% +% \section{Aide additionnelle} +% +% Pour obtenir de l'aide additionnelle sur l'utilisation de la classe +% \textsf{ulthese} (et sur celle de {\LaTeX} en général), prière de +% consulter d'abord +% \begin{enumerate} +% \item le \href{http://www.theses.ulaval.ca/wiki/}{WikiThèse} de +% l'Université Laval; +% \item les +% \href{http://listes.ulaval.ca/listserv/archives/ulthese-aide.html}{archives} +% de la liste de distribution \texttt{ulthese-aide}. +% \end{enumerate} +% Si la réponse à votre question ne se trouve ni dans le wiki, ni dans +% les archives, alors écrire à l'adresse +% \href{mailto:ulthese-aide@bibl.ulaval.ca}{\url{ulthese-aide@bibl.ulaval.ca}}. +% +% \StopEventually{\PrintChanges\PrintIndex} +% +% ^^A Début du code de la classe +% +% \appendix +% \section{Mise en {\oe}uvre} +% \label{sec:meo} +% +% Cette section passe en revue le code {\TeX} et {\LaTeX} de la +% classe. Elle ne risque d'intéresser que les personnes qui souhaitent +% explorer comment la classe est programmée. +% +% \subsection{Options de la classe} +% +% Il y a deux options propres à la classe: la taille de la police de +% caractères en points et la possibilité d'empêcher le chargement du +% paquetage \textsf{natbib}. +% +% \begin{macro}{nonatbib} +% Pour des raisons de programmation, on traite l'option |nonatbib| +% en premier. Celle-ci permet d'empêcher la classe de charger le +% paquetage \textsf{natbib} en cas d'incompatibilité avec d'autres +% paquetages spécialisés de mise en forme de la bibliographie. Le +% paquetage \textsf{ifthen} est nécessaire pour définir cette option +% booléenne et pour plusieurs autres usages internes à la classe. +% \begin{macrocode} +%<*class> +\RequirePackage{ifthen} +\newboolean{UL@natbib} +\setboolean{UL@natbib}{true} +\DeclareOption{nonatbib}{\setboolean{UL@natbib}{false}} +% \end{macrocode} +% \end{macro} +% +% Les valeurs possibles pour la taille de la police de caractères sont +% |10pt|, |11pt| et |12pt|. Cette option est gérée au niveau de la +% classe afin de s'assurer que les divers éléments sur la page titre +% sont toujours de la même taille. La taille de la police par défaut +% permet de déterminer si, par exemple, le titre du document doit être +% dans la taille |\Huge|, |\huge| ou |\LARGE| de \textsf{memoir}. +% +% \begin{macro}{\UL@ptsize} +% La taille de la police est passée à \textsf{memoir} et la macro +% |\UL@ptsize| stocke la taille des caractères pour usage futur. +% \begin{macrocode} +\newcommand*{\UL@ptsize}{} +\DeclareOption{10pt}{% + \PassOptionsToClass{10pt}{memoir} + \renewcommand*{\UL@ptsize}{10}} +\DeclareOption{11pt}{% + \PassOptionsToClass{11pt}{memoir} + \renewcommand*{\UL@ptsize}{11}} +\DeclareOption{12pt}{% + \PassOptionsToClass{12pt}{memoir} + \renewcommand*{\UL@ptsize}{12}} +% \end{macrocode} +% \end{macro} +% +% \subsection{Chargement de la classe \textsf{memoir}} +% +% Toutes les options de la classe sont passées à \textsf{memoir}. Le +% format de papier et la taille de police par défaut sont, dans +% l'ordre, |letterpaper| et |11pt|. Les options de \textsf{memoir} +% |twoside| et |openright| sont explicitement déclarées afin d'éviter +% toute tentative de passer outre à ces exigences de la FESP. +% \begin{macrocode} +\DeclareOption*{\PassOptionsToClass{\CurrentOption}{memoir}} +\ExecuteOptions{11pt,letterpaper} +\ProcessOptions +\LoadClass[twoside,openright]{memoir} +% \end{macrocode} +% +% \subsection{Paquetages requis} +% +% La classe s'efforce de charger un minimum de paquetages afin d'éviter +% les conflits potentiels. +% +% Il est aujourd'hui préférable d'utiliser les polices T1. +% \begin{macrocode} +\RequirePackage[T1]{fontenc} +% \end{macrocode} +% +% Le paquetage \textsf{natbib} doit être chargé avant \textsf{babel} +% pour bien fonctionner. C'est pourquoi il est chargé dans la classe, +% à moins que l'option |nonatbib| n'ait été spécifiée au chargement de +% la classe. +% \begin{macrocode} +\ifthenelse{\boolean{UL@natbib}}{\RequirePackage{natbib}}{} +% \end{macrocode} +% +% Le support pour les langues autres que l'anglais est offert par le +% paquetage \textsf{babel}. Les langues sont passées en option de la +% classe, et non du paquetage. Le paquetage \textsf{numprint} est +% requis par \textsf{babel} pour la définition de la commande de mise +% en forme des nombres |\nombre|. +% \begin{macrocode} +\RequirePackage{babel} +\RequirePackage[autolanguage]{numprint} +% \end{macrocode} +% +% L'insertion du logo de l'Université sur la page titre requiert +% \textsf{graphicx}. On définit également une couleur pour les +% hyperliens dans le document (mais \textsf{hyperref} est chargé dans +% les gabarits afin de demeurer le dernier paquetage chargé; voir la +% section~\ref{sec:hyperref}). +% \begin{macrocode} +\RequirePackage{graphicx} +\RequirePackage{xcolor} +% \end{macrocode} +% +% La commande |\textcopyright| utilisée sur la page titre requiert le +% paquetage \textsf{textcomp} pour obtenir un beau signe de copyright. +% \begin{macrocode} +\RequirePackage{textcomp} +% \end{macrocode} +% +% \subsection{Police pour la page titre} +% +% La page titre est composée dans la police Helvetica. Le chargement +% du paquetage \textsf{helvet} a pour effet de redéfinir la commande +% |\sfdefault|. +% \begin{macrocode} +\RequirePackage[scaled=0.92]{helvet} +% \end{macrocode} +% +% \subsection{Couleur des hyperliens} +% \label{sec:couleurs} +% +% La classe définit une couleur standard pour les hyperliens, une +% teinte de bleu assez foncée pour être à fois visible en couleur et +% peu contrastante si le document est imprimé en noir et blanc. +% \begin{macrocode} +\definecolor{ULlinkcolor}{rgb}{0,0,0.3} +% \end{macrocode} +% +% \subsection{Marges} +% +% Les marges exigées par les normes de présentation de la FESP sont de +% 25~mm partout, sauf 35~mm pour la marge de reliure (gauche pour les +% pages impaires, droite pour les pages paires). Le pied de page est +% placé de sorte que le folio de page se retrouve à 10~mm du bas de la +% page. +% \begin{macrocode} +\setlrmarginsandblock{35mm}{25mm}{*} +\setulmarginsandblock{25mm}{*}{1} +\setheadfoot{\baselineskip}{20mm} +\checkandfixthelayout[lines] +% \end{macrocode} +% +% \subsection{Interligne} +% +% L'espacement entre les lignes est d'un interligne et demi. +% L'espacement «double» entre les paragraphes est fixé à +% |0.5\baselineskip| afin d'en arriver à une disposition agréable à +% l'{\oe}il. Le retrait de première ligne est supprimé puisque plus +% nécessaire suite à l'ajout de l'espacement entre les paragraphes. +% \begin{macrocode} +\OnehalfSpacing +\setlength{\parskip}{0.5\baselineskip} +\setlength{\parindent}{0em} +% \end{macrocode} +% +% La table des matières, la liste des tableaux et la liste des figures +% sont composées à interligne simple. +% \begin{macrocode} +\renewcommand{\tocheadstart}{\SingleSpacing\chapterheadstart} +\renewcommand{\lotheadstart}{\SingleSpacing\chapterheadstart} +\renewcommand{\lofheadstart}{\SingleSpacing\chapterheadstart} +% \end{macrocode} +% +% \subsection{Entêtes et pieds de page} +% +% Les règles pour les entêtes et pieds de page sont uniformes pour +% tout le document: aucun entête et folio sur le bord extérieur du +% pied de page. On définit un style de page pour ce faire ainsi qu'un +% alias entre le nouveau style |ul| et le style standard |plain|. +% Raison: les premières pages de chapitres utilisent par défaut le +% style |plain|; avec l'alias c'est le style |ul| qui sera activé. +% \begin{macrocode} +\makepagestyle{ul} + \makeevenfoot{ul}{\thepage}{}{} + \makeoddfoot{ul}{}{}{\thepage} +\aliaspagestyle{plain}{ul} +\pagestyle{ul} +% \end{macrocode} +% +% \subsection{Page titre} +% +% La page titre est composée avec les polices présentées au +% tableau~\ref{tab:polices}. Les commandes sélectionnant ces polices +% sont adaptées selon la taille de police choisie pour le document +% afin d'être toujours identiques. +% +% \begin{table} +% \centering +% \begin{tabular}{ll} +% \toprule +% Élément & Police \\ +% \midrule +% Titre & 20~points gras \\ +% Sous-titre & 17~points gras \\ +% Auteur & 14~points gras \\ +% Nom du programme & 14~points gras \\ +% Autres éléments & 14~points normal \\ +% \bottomrule +% \end{tabular} +% \caption{Taille et type de police des éléments de la page titre} +% \label{tab:polices} +% \end{table} +% +% \begin{macrocode} +\ifthenelse{\UL@ptsize=10}{% + \newcommand*{\UL@fonttitle}{\normalfont\Huge\bfseries\sffamily} + \newcommand*{\UL@fontsubtitle}{\normalfont\huge\bfseries\sffamily} + \newcommand*{\UL@fontauthor}{\normalfont\LARGE\bfseries\sffamily} + \newcommand*{\UL@fontprogram}{\UL@fontauthor} + \newcommand*{\UL@fontbase}{\normalfont\LARGE\sffamily}}{} +\ifthenelse{\UL@ptsize=11}{% + \newcommand*{\UL@fonttitle}{\normalfont\huge\bfseries\sffamily} + \newcommand*{\UL@fontsubtitle}{\normalfont\LARGE\bfseries\sffamily} + \newcommand*{\UL@fontauthor}{\normalfont\Large\bfseries\sffamily} + \newcommand*{\UL@fontprogram}{\UL@fontauthor} + \newcommand*{\UL@fontbase}{\normalfont\Large\sffamily}}{} +\ifthenelse{\UL@ptsize=12}{% + \newcommand*{\UL@fonttitle}{\normalfont\LARGE\bfseries\sffamily} + \newcommand*{\UL@fontsubtitle}{\normalfont\Large\bfseries\sffamily} + \newcommand*{\UL@fontauthor}{\normalfont\large\bfseries\sffamily} + \newcommand*{\UL@fontprogram}{\UL@fontauthor} + \newcommand*{\UL@fontbase}{\normalfont\large\sffamily}}{} +% \end{macrocode} +% +% Définition de deux valeurs booléennes servant à adapter la +% composition de la page titre plus loin: y a-t-il un sous-titre ou +% non et est-ce que le type de programme (doctorat ou maîtrise) est +% masculin ou non? +% \begin{macrocode} +\newboolean{UL@hassubtitle} +\newboolean{UL@isprogmasc} +% \end{macrocode} +% +% Définition des commandes permettant de construire la page titre. +% L'interface utilisateur est basée sur un ensemble de commandes +% internes. On commence par celles-ci. +% \begin{macrocode} +\newcommand{\UL@maintitle}{} +\newcommand{\UL@subtitle}{} +\newcommand*{\UL@author}{} +\newcommand*{\UL@program}{} +\newcommand*{\UL@year}{} +\newcommand*{\UL@typeofdoc}{} +\newcommand*{\UL@degree}{} +\newcommand*{\UL@nameother}{} +\newcommand*{\UL@degreeother}{} +\newcommand*{\UL@facUL}{} +\newcommand*{\UL@facother}{} +\newcommand*{\UL@extensionat}{} +\newcommand*{\UL@extensionloc}{} +% \end{macrocode} +% Puis les commandes visibles pour les utilisateurs, qui redéfinissent +% les commandes internes. Voir la +% section~\ref{sec:commandes-pagetitre} pour leur signification. +% \begin{macrocode} +\newcommand{\titre}[1]{\renewcommand{\UL@maintitle}{#1}} +\newcommand{\soustitre}[1]{% + \setboolean{UL@hassubtitle}{true} + \renewcommand{\UL@subtitle}{#1}} +\newcommand*{\auteur}[1]{\renewcommand*{\UL@author}{#1}} +\newcommand*{\annee}[1]{\renewcommand*{\UL@year}{#1}} +\newcommand*{\programme}[1]{\renewcommand*{\UL@program}{#1}} +\newcommand*{\LLD}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Docteur en droit (L.L.D.)}} +\newcommand*{\DPsy}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Docteur en psychologie (D.Psy.)}} +\newcommand*{\DThP}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Docteur en th\'eologie pratique (D.Th.P.)}} +\newcommand*{\PhD}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Philosophi{\ae} doctor (Ph.D.)}} +\newcommand*{\LLM}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en droit (L.L.M.)}} +\newcommand*{\MA}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre \`es arts (M.A.)}} +\newcommand*{\MMus}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en musique (M.Mus.)}} +\newcommand*{\MSc}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre \`es sciences (M.Sc.)}} +\newcommand*{\MServSoc}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en service social (M.Serv.Soc.)}} +\newcommand*{\MScGeogr}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en sciences géographiques (M.Sc.G\'eogr.)}} +\newcommand*{\MATDR}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en am\'enagement du territoire et d\'eveloppement r\'egional (M.ATDR)}} +\newcommand*{\multifacultaire}{\renewcommand*{\UL@typenum}{1}} +\newcommand*{\cotutelle}{% + \renewcommand*{\UL@typenum}{2} + \edef\UL@typeofdoc{\UL@typeofdoc\ en cotutelle}} +\newcommand*{\univcotutelle}[1]{\renewcommand*{\UL@nameother}{#1}} +\newcommand*{\gradecotutelle}[1]{\renewcommand*{\UL@degreeother}{#1}} +\newcommand*{\extensionUdeS}{% + \renewcommand*{\UL@typenum}{3} + \renewcommand*{\UL@extensionat}{Universit\'e de Sherbrooke} + \renewcommand*{\UL@extensionloc}{Sherbrooke, Qu\'ebec}} +\newcommand*{\extensionUQO}{% + \renewcommand*{\UL@typenum}{3} + \renewcommand*{\UL@extensionat}{Universit\'e du Qu\'ebec en Outaouais} + \renewcommand*{\UL@extensionloc}{Gatineau, Qu\'ebec}} +\newcommand*{\extensionUQAC}{% + \renewcommand*{\UL@typenum}{3} + \renewcommand*{\UL@extensionat}{Universit\'e du Qu\'ebec \`a Chicoutimi} + \renewcommand*{\UL@extensionloc}{Chicoutimi, Qu\'ebec}} +\newcommand{\faculteUL}[1]{\renewcommand*{\UL@facUL}{#1}} +\newcommand*{\faculteUdeS}[1]{\renewcommand*{\UL@facother}{#1}} +\newcommand*{\faculteUQO}[1]{\renewcommand*{\UL@facother}{#1}} +\newcommand*{\faculteUQAC}[1]{\renewcommand*{\UL@facother}{#1}} +% \end{macrocode} +% +% Outre le nom de l'auteur et la notice de copyright en bas de page, +% la page titre peut être divisée en trois grands blocs: +% \begin{enumerate} +% \item le titre et le sous-titre, le cas échéant; +% \item la description du type de document (thèse, thèse en cotutelle, +% mémoire, etc.); +% \item les détails sur la ou les facultés, la ou les universités, +% etc. +% \end{enumerate} +% À ces blocs s'ajoutent quatre grandes catégories de disposition des +% éléments sur la page titre selon le type de thèse ou de mémoire: +% standard, multifacultaire, en cotutelle et en extension. Les +% commandes internes ci-dessous prennent en compte toutes les +% combinaisons possibles. +% +% \begin{macro}{\UL@title} +% La commande |\UL@title| prépare le titre. Si le document ne +% comporte pas de sous-titre, un espace est ajouté pour ne pas +% décaler vers le haut les autres éléments de la page. +% \begin{macrocode} +\newcommand{\UL@title}{% + {\UL@fonttitle\UL@maintitle\par} + {\UL@fontsubtitle + \ifthenelse{\boolean{UL@hassubtitle}}{\UL@subtitle}{% + \vspace*{\baselineskip}}\par}} +% \end{macrocode} +% \end{macro} +% \begin{macro}{\Ul@docid} +% La commande |\Ul@docid| prépare la mention du type de document. La +% thèse ou le mémoire en cotutelle requiert un traitement différent +% puisque le programme d'étude apparaît immédiatement sous la mention. +% \begin{macrocode} +\newcommand{\UL@docid}{% + {\UL@fontprogram\UL@typeofdoc\par + \ifnum\UL@typenum=2 \UL@program\par \fi}} +% \end{macrocode} +% \end{macro} +% \begin{macro}{\Ul@details} +% La commande |\Ul@details| est la plus complexe puisque la +% disposition des informations additionnelles sur le document +% varie beaucoup selon le type de thèse ou de mémoire. +% \begin{macrocode} +\newcommand{\UL@details}{% + \ifcase\UL@typenum\relax% 0 standard + \vspace{96pt} + {\UL@fontprogram\UL@program}\par + \UL@degree\par + \vspace{112pt} + Qu\'ebec, Canada\par + \or% 1 multifacultaire + \vspace{96pt} + {\UL@fontprogram\UL@program}\par + \UL@degree\par + \vspace{36pt} + \UL@facUL\par + \vspace{48pt} + Qu\'ebec, Canada\par + \or% 2 cotutelle + \vspace{72pt} + Universit\'e Laval\par Qu\'ebec, Canada\par + \UL@degree\par + \vspace{\baselineskip} et\par \vspace{\baselineskip} + \UL@nameother\par + \UL@degreeother\par + \or% 3 extension + \vspace{48pt} + {\UL@fontprogram\UL@program\ de l'Universit\'e Laval\par + \ifthenelse{\boolean{UL@isprogmasc}}{offert}{offerte} + en extension \`a l'\UL@extensionat}\par + \vspace{36pt} + \UL@degree\par + \vspace{36pt} + \UL@facother\par \UL@extensionat\par \UL@extensionloc\par + \vspace{\baselineskip} + \UL@facUL\par Universit\'e Laval\par Qu\'ebec, Canada\par + \fi} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\pagetitre} +% Construction de la page titre. Le nom de l'auteur et la notice de +% copyright sont insérées directement dans le code de la commande +% |\pagetitre|. On doit rétablir sur la page titre l'interligne +% simple et l'espacement nul entre les paragraphes (|\parskip|). La +% page titre des thèses et mémoires en cotutelle ou en extension ne +% comporte pas de logo de l'Université Laval. +% \begin{macrocode} +\newcommand{\pagetitre}{{% + \clearpage + \thispagestyle{empty} + \SingleSpacing\setlength{\parskip}{0pt} + \centering + \UL@fontbase + \ifthenelse{\UL@typenum > 1}{\vspace*{0pt}\par}{% + \includegraphics[height=40px,keepaspectratio=true]{ul_p}\par} + \vspace{72pt} + \UL@title + \vspace{48pt} + \UL@docid + \vspace{72pt} + {\UL@fontauthor\UL@author}\par + \UL@details + \vfill + {\textcopyright} \UL@author, \UL@year\par + \cleardoublepage + }} +% \end{macrocode} +% \end{macro} +% +% \subsection{Listes des figures et des tableaux} +% +% Le paquetage \textsf{babel} définit comme titre pour la liste des +% figures «Table des figures», alors que la liste des tableaux est +% «Liste des tableaux». Pour une plus grande symétrie, la classe +% redéfinit le titre correspondant à |\listoffigures|. La commande +% |\addto| est nécessaire pour éviter que \textsf{babel} redéfinisse +% le titre à |\begin{document}|. +% \begin{macrocode} +\addto\captionsfrench{\renewcommand{\listfigurename}{Liste des figures}} +% \end{macrocode} +% +% \subsection{Dédicace et épigraphe} +% +% La dédicace et l'épigraphe sont mises en forme avec la commande +% |\epigraph| de \textsf{memoir}. +% \begin{macro}{\dedicace} +% La dédicace est une épigraphe simplifiée placée seule sur une +% page, alignée à droite à une dizaine de lignes de la marge +% supérieure, sans auteur ou source et sans ligne de démarcation. +% \begin{macrocode} +\newcommand{\dedicace}[1]{{% + \clearpage + \pagestyle{empty} + \setlength{\beforeepigraphskip}{10\baselineskip} + \setlength{\epigraphrule}{0pt} + \epigraphtextposition{flushright} + \mbox{}\epigraph{\itshape #1}{} + }} +% \end{macrocode} +% \end{macro} +% \begin{macro}{\epigraphe} +% L'épigraphe de début de document est placée seule sur une page à +% une dizaine de lignes de la marge supérieure. Pour le reste, on +% s'en remet à la commande |\epigraph| de \textsf{memoir}. +% \begin{macrocode} +\newcommand{\epigraphe}[2]{{% + \clearpage + \pagestyle{empty} + \setlength{\beforeepigraphskip}{10\baselineskip} + \mbox{}\epigraph{#1}{#2} + }} +% \end{macrocode} +% \end{macro} +% +% \subsection{Citations} +% +% \begin{environment}{quote} +% La classe redéfinit l'environnement |quote| de \textsf{memoir} +% afin que le texte des citations se trouve en retrait de 10~mm à +% gauche et à droite, conformément aux règles de présentation de la +% FESP. +% \begin{macrocode} +\renewenvironment{quote}{% + \list{}{\rightmargin 10mm \leftmargin 10mm}% + \item[]}{\endlist} +% \end{macrocode} +% \end{environment} +% +% \begin{environment}{quotation} +% Il en va de même de l'environnement |quotation|. Cependant, cet +% environnement passe également à l'interligne simple et la classe +% ajuste l'espacement vertical entre les paragraphes afin que +% ceux-ci soient bien distincts les uns des autres tout en demeurant +% raisonnablement compacts. Cet espacement est ici fixé à 6~points. +% \begin{macrocode} +\renewenvironment{quotation}{% + \list{}{% + \SingleSpacing + \listparindent 0em + \itemindent \listparindent + \leftmargin 10mm + \rightmargin \leftmargin + \parsep 6\p@ \@plus\p@}% + \item[]}{\endlist} +% \end{macrocode} +% \end{environment} +% +% \subsection{Numérotation des divisions du document} +% +% Par défaut, \textsf{memoir} numérote les divisions du document +% seulement jusqu'au niveau des sections. La classe étend la +% numérotation aux sous-sections. +% \begin{macrocode} +\setsecnumdepth{subsection} +% +% \end{macrocode} +% ^^A Fin du code de la classe +% +% \begin{thebibliography}{1} +% \providecommand{\bibnamefont}[1]{#1} +% \providecommand{\selectlanguage}[1]{\relax} +% \bibitem[{Malo(1996)}]{Malo:1996} +% \bibnamefont{Malo}, M. 1996, {\selectlanguage{francais}\emph{Guide de la +% communication écrite au cégep, à l'université et en entreprise}}, Québec +% Amérique. +% \end{thebibliography} +% +% \Finale +% +% \iffalse +% ^^A Gabarits du document maître +%<*gabarit> +%%% GABARIT POUR THÈSE STANDARD +%%% GABARIT POUR THÈSE SUR MESURE +%%% GABARIT POUR THÈSE MULTIFACULTAIRE +%%% GABARIT POUR THÈSE EN COTUTELLE +%%% GABARIT POUR THÈSE EN EXTENSION À L'UNIVERSITÉ DE SHERBROOKE +%%% GABARIT POUR THÈSE EN EXTENSION À L'UQO +%%% GABARIT POUR MÉMOIRE STANDARD +%%% GABARIT POUR MÉMOIRE SUR MESURE +%%% GABARIT POUR MÉMOIRE EN EXTENSION À L'UQAC +%% +%% Consulter la documentation de la classe ulthese pour une +%% description détaillée de la classe, de ce gabarit et des options +%% disponibles. +%% +%% [Ne pas hésiter à supprimer les commentaires après les avoir lus.] +%% +%% Déclaration de la classe avec les langues les plus courantes. Le +%% français sera la langue par défaut du document. +\documentclass[11pt,english,francais]{ulthese} + %% Encodage utilisé pour les caractères accentués dans les fichiers + %% source du document. Les gabarits sont encodés en UTF-8. + \usepackage[utf8]{inputenc} + + %% Charger ici les autres paquetages nécessaires pour le document. + %% Quelques exemples; décommenter au besoin. + %\usepackage{amsmath} % recommandé pour les mathématiques + %\usepackage{mathpazo} % texte en Palatino + %\usepackage{mathptmx} % texte en Times + + %% Gestion des hyperliens dans le document. S'assurer que hyperref + %% est le dernier paquetage chargé. + \usepackage{hyperref} + \hypersetup{colorlinks,allcolors=ULlinkcolor} + + %% Pour utiliser Computer Modern comme police sans empattements + %% (serif) plutôt que Helvetica, décommenter la ligne ci-dessous. + %\renewcommand{\sfdefault}{cmss} + + %% Options de mise en forme du mode français de babel. Consulter la + %% documentation du paquetage babel pour les options disponibles. + \frenchbsetup{% + CompactItemize=false, % ne pas compacter les listes + ThinSpaceInFrenchNumbers=true % espace fine dans les nombres + } + + %% Style de la bibliographie. + \bibliographystyle{} + + %% Déclarations de la page titre. Remplacer les éléments entre < >. + %% Supprimer les caractères < >. Couper un long titre ou un long + %% sous-titre manuellement avec \\. Terminer une éventuelle deuxième + %% ligne de titre ou de sous-titre avec \vspace*{-\baselineskip}. + \titre{} + % \titre{Ceci est un exemple de long titre \\ + % avec saut de ligne manuel \vspace*{-\baselineskip}} + % \soustitre{Sous-titre le cas échéant} + % \soustitre{Ceci est un exemple de long-titre \\ + % avec saut de ligne manuel \vspace*{-\baselineskip}} + \auteur{} +% \programme{Doctorat en -- } +% \programme{Doctorat sur mesure en -- } +% \programme{Doctorat en } +% \programme{Maîtrise en -- } +% \programme{Maîtrise sur mesure en -- } +% \programme{Maîtrise en } + \annee{<20xx>} +% \PhD % ou l'un de \LLD, \DPsy, \DThP +% \MSc % ou l'un de \LLM, \MA, \MMus, \MServSoc, \MScGeogr, \MATDR +% \cotutelle +% \univcotutelle{ \\ , } +% \gradecotutelle{ ()} +% \multifacultaire +% \faculteUL{ \\ } +% \extensionUdeS +% \extensionUQO +% \extensionUQAC +% \faculteUL{} +% \faculteUdeS{} +% \faculteUQO{} +% \faculteUQAC{} + +\begin{document} + +\frontmatter % pages liminaires + +\pagetitre % production de la page titre + +\include{resume} % résumé français +\include{abstract} % résumé anglais +\cleardoublepage + +\tableofcontents % production de la TdM +\cleardoublepage + +\listoftables % production de la liste des tableaux +\cleardoublepage + +\listoffigures % production de la liste des figures +\cleardoublepage + +\dedicace{Dédicace si désiré} +\cleardoublepage + +\epigraphe{Texte de l'épigraphe}{Source ou auteur} +\cleardoublepage + +\include{remerciements} % remerciements +\include{avantpropos} % avant-propos + +\mainmatter % corps du document + +\include{introduction} % introduction +\include{chapitre1} % chapitre 1 +\include{chapitre2} % chapitre 2, etc. +\include{conclusion} % conclusion + +\appendix % annexes le cas échéant + +\include{annexe} % annexe A + +\bibliography{} % production de la bibliographie + +\end{document} +% +% +% ^^A Gabarits des parties du document +%<*resume> +\chapter*{Résumé} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Résumé} % inclure dans TdM + +\begin{otherlanguage*}{francais} + Texte du résumé en français. +\end{otherlanguage*} +% +% +%<*abstract> +\chapter*{Abstract} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Abstract} % inclure dans TdM + +\begin{otherlanguage*}{english} + Text of English abstract. +\end{otherlanguage*} +% +% +%<*remerciements> +\chapter*{Remerciements} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Remerciements} % inclure dans TdM + +Texte des remerciements en prose. +% +% +%<*avantpropos> +\chapter*{Avant-propos} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Avant-propos} % inclure dans TdM + +L'avant-propos est surtout nécessaire pour une thèse par article. +% +% +%<*introduction> +\chapter*{Introduction} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Introduction} % inclure dans TdM + +Une thèse ou un mémoire devrait normalement débuter par une +introduction. Celle-ci est traitée comme un chapitre normal, sauf +qu'elle n'est pas numérotée. +% +% +%<*chapitre> +\chapter{Titre du chapitre} % numéroté + +Texte du chapitre. +% +% +%<*conclusion> +\chapter*{Conclusion} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Conclusion} % dans TdM + +Une thèse ou un mémoire devrait normalement se terminer par une +conclusion, placée avant les annexes, le cas échéant. Celle-ci est +traitée comme un chapitre normal, sauf qu'elle n'est pas numérotée. +% +% +%<*annexe> +\chapter{Titre de l'annexe} % numérotée + +Texte de l'annexe. +% +% Local Variables: +% mode: doctex +% coding: utf-8 +% TeX-master: t +% End: +% \fi diff --git a/memoire/ulthese.ins b/memoire/ulthese.ins new file mode 100644 index 0000000..5599434 --- /dev/null +++ b/memoire/ulthese.ins @@ -0,0 +1,102 @@ +%% +%% Copyright (C) 2012 Universite Laval +%% +%% This file may be distributed and/or modified under the conditions +%% of the LaTeX Project Public License, either version 1.3c of this +%% license or (at your option) any later version. The latest version +%% of this license is in: +%% +%% http://www.latex-project.org/lppl.txt +%% +%% and version 1.3c or later is part of all distributions of LaTeX +%% version 2006/05/20 or later. +%% + +\input docstrip.tex +\keepsilent +\askforoverwritefalse + +\preamble + +This is a generated file. + +Copyright (C) 2012 Universite Laval + +This file may be distributed and/or modified under the conditions +of the LaTeX Project Public License, either version 1.3c of this +license or (at your option) any later version. The latest version +of this license is in: + + http://www.latex-project.org/lppl.txt + +and version 1.3c or later is part of all distributions of LaTeX +version 2006/05/20 or later. + +This work has the LPPL maintenance status `maintained'. + +The Current Maintainer of this work is Universite Laval +. + +This work consists of the files ulthese.dtx and ulthese.ins and the +derived files listed in the README file. + +\endpreamble + +\generate{ + \file{ulthese.cls}{\from{ulthese.dtx}{class}} + \file{gabarit-doctorat.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,standard}} + \file{gabarit-doctorat-mesure.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,mesure}} + \file{gabarit-doctorat-multifacultaire.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,multifac}} + \file{gabarit-doctorat-cotutelle.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,cotutelle}} + \file{gabarit-doctorat-extension-UdeS.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,UdeS}} + \file{gabarit-doctorat-extension-UQO.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,UQO}} + \file{gabarit-maitrise.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,m,standard}} + \file{gabarit-maitrise-mesure.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,m,mesure}} + \file{gabarit-maitrise-extension-UQAC.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,m,UQAC}} + \file{resume.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{resume}} + \file{abstract.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{abstract}} + \file{remerciements.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{remerciements}} + \file{avantpropos.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{avantpropos}} + \file{introduction.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{introduction}} + \file{chapitre1.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{chapitre}} + \file{chapitre2.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{chapitre}} + \file{conclusion.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{conclusion}} + \file{annexe.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{annexe}} +} + +\obeyspaces +\Msg{*************************************************************} +\Msg{* *} +\Msg{* La classe ulthese est maintenant prete a l'emploi. *} +\Msg{* *} +\Msg{* Utiliser un des gabarits comme base pour votre fichier *} +\Msg{* maitre. Vous pouvez effacer les gabarits inutiles. *} +\Msg{* Taper le contenu du document dans des fichiers .tex *} +\Msg{* distincts; quelques gabarits sont fournis avec la classe. *} +\Msg{* *} +\Msg{* Pour produire la documentation de la classe, passer le *} +\Msg{* fichier ulthese.dtx dans LaTeX. *} +\Msg{* *} +\Msg{* Bonne redaction! *} +\Msg{* *} +\Msg{*************************************************************} + +\endbatchfile \ No newline at end of file diff --git a/memoire/ulthese.pdf b/memoire/ulthese.pdf new file mode 100644 index 0000000..72b5432 Binary files /dev/null and b/memoire/ulthese.pdf differ diff --git a/poster scientifique/baposter.cls b/poster scientifique/baposter.cls new file mode 100644 index 0000000..965c4aa --- /dev/null +++ b/poster scientifique/baposter.cls @@ -0,0 +1,1094 @@ +%% +%% This is file `baposter.cls' +%% +%% A relatively comfortable latex class to produce posters with a grid based +%% layout. It comes with a number of combinable styles and is (maybe only for +%% the writer) easy to extend, as all the graphics is based on pgf. +%% +%% It is meant to be used with pdftex, but might also work with pslatex if you +%% are not interested in things like transparency. +%% +%% Copyright (C) 2007-2011 Brian Amberg +%% Copyright (C) 2011 Reinhold Kainhofer +%% +%% 29. September 2011: +%% - Finally fixed confusion with paper size handling and landscape. This required seperate handling of papersizes +%% known to the geometry package and other packages. +%% 26. September 2011: +%% - Reverted drawing of faded borders to manual method, as the current result does not work with evince, +%% and produced spurious colored boxes with okular and acroread. +%% - Added one more example due to popular request +%% 16. September 2011: +%% - Removed nag package, such that it compiles on older systems +%% - Added more examples +%% 15. September 2011: +%% - Merged fork by (rk) back into mainline, essentially taking over all the great improvements that Reinhold Kainhofer coded. +%% We call this version 2, as it breaks the interface in some small points. Essentially watch out for this: +%% - no/yes keys are now false/true +%% - the shade-lr etc. keys have the hypen removed, and are now called shadelr etc. +% - Added more examples, which are all adapted to work with the current version +%% 27. August 2011 (rk): +%% - Completely factored out all drawing commands to handler functions, assigned via choice keys +%% - Added cornerradius, boxheaderheight, boxpadding options +%% - Added missing roundedright +%% 26. August 2011 (rk): +%% - Turned headerbox cmd into posterbox environment (allows verbatim) +%% - pass several options to packages +%% - added debug cls option to reduce default output +%% - rewrote several loops and ifthenelse +%% - Use boolean and choice keys for all options +%% - Changed all choice options to set a global function to +%% \baposter@OPTION@OPTIONVALUE, which are a functions that execute the +%% option's setting (drawing routine, color setting for tikz, etc.) +%% - Add a5paper +%% 07. April 2011: Fixed paper size handling +%% - Fixed Paper Size handling, you can now specify all page sizes using paperwidth=, paperheight= class options. +%% - Added rudimentary documentation of the available keys. +%% 11. December 2010: Fixed "Empty Second Page" bug +%% - A second empty page was introduced when a document ended with +%% \end{poster} +%% \end{document} +%% This was fixed thanks to a suggestion by Martin Aulbach +%% - Added +%% \thispagestyle{empty} +%% to suppress page numbers, which became visible when the margins are large. +%% The underlying problem of page-size handling has not yet been solved. +%% 30. September 2010: Border Bugfixes +%% - Correct background color with textborder=rounded-small, Thanks to Ke Chen for the bugreport and patch +%% - Correctly draw rectangular text boxes, Thanks to abenkst for the bugreport and fix +%% 10. June 2010: Added option to set the number of columns +%% - added a class option to set the number of columns +%% - columns=5: sets the number of columns to 5, possible values are 1..6, default is 3 in portrait and 4 in landscape format +%% 29. April 2009: Incorporated Patches by Arne Henningsen +%% - added some class options +%% - a4shrink: shrink the paper to A4 size (for printing drafts or handouts) +%% - movebody=Xpt: move the text/poster body Xpt to the right +%% (or to the left if Xpt is negative), +%% e.g. for manually centering the poster on the page +%% - showframe: use the "showframe" option of the "geometry" package +%% - a0paper (default): set paper size to A0 +%% - archE: set paper size to Arch E +%% - setting "background" can be "none" now (otherwise the "showframe" +%% option has no effect) +%% - the page number has been removed (it was mostly not visible before) +%% - the "margin=" option works now +%% 04. December 2008 +%% - Mainly an update to the landscape example +%% 14. November 2008 +%% - Actually center the title when eyecatcher is used. +%% 04. November 2008 +%% - Fixed bug with eyecatcher not working. +%% 26. June 2008 +%% - Fixed bug with plain background mode. +%% 14. June 2008 +%% - Support for portrait/landscape switching. +%% - Some smaller bugfixes. +%% 01. June 2007 +%% - First version released. +%% +%% Use this class with pdflatex +%% +%% I have confirmed that this package works with +%% - texlive 2007 and +%% - miktex 2.7 +%% +%% It does not seem to work with +%% - miktex 2.2 +%% - some old versions of tetex +%% +%% +%% TODO: +%% -) Rename backgrounds back to shaded-tb shade-lr +%% -) Rename textborder back to rounded-small (consistent naming needed!) +%% -) Rename headershade back to shade-lr, shade-tb, shade-tb-inverse +%% -) Rename headershape back to small-rounded +%% -) Option value consistency (small-rounded vs. rounded-small; missing ones) +%% -) Rename \baposterHeaderSetShade, \baposterHeaderDrawText to include @ and verb +%% +%% +%% Licence: GPL +\ProvidesClass{baposter}[2011/11/26 v2.0 baposter class] +\NeedsTeXFormat{LaTeX2e}[1995/06/01] +\LoadClass{article} +\typeout{baposter: Brian Amberg, 2007, 2008, 2009, 2010, 2011 | http://www.brian-amberg.de/uni/poster/} +\typeout{baposter: Reinhold Kainhofer, 2011 | http://reinhold.kainhofer.com/} + +%% Define lengths only once on inclusion, such that we can make multiple posters +\newlength{\baposter@basepaperwidth} +\newlength{\baposter@basepaperheight} +\newlength{\baposter@basemargin} +\newlength{\baposter@finalpaperwidth} +\newlength{\baposter@finalpaperheight} +\newlength{\baposter@finalmargin} +\newlength{\headerheight}% +\newlength{\colwidth}% +\newlength{\colheight}% +\newlength{\baposter@@colspacing}% +\newlength{\baposter@box@@cornerradius}% +\newlength{\baposter@box@@boxheaderheight}% +\newlength{\baposter@box@@boxpadding}% +\newlength{\boxstartx}% +\newlength{\boxstarty}% +\newlength{\boxwidth}% +\newlength{\boxheight}% +\newlength{\baposter@titleimage@left@width}% +\newlength{\baposter@titleimage@right@width}% +\newlength{\baposter@titleimage@textwidth}% +\newbox\baposter@box@content% +\newbox\baposter@titleimage@left% +\newbox\baposter@titleimage@title% +\newbox\baposter@titleimage@right% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Packages +%------------------------------------------------------------------------------- +% The only ``weird'' dependency of this package is pgf. All the rest should be +% installed on any decent system. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\typeout{Use Packages} +\RequirePackage{xkeyval} +\RequirePackage{calc} +\RequirePackage[cmyk]{xcolor} +\RequirePackage{tikz} +\RequirePackage{pgf} +\RequirePackage{ifthen} +\RequirePackage[T1]{fontenc} +%\RequirePackage[l2tabu, orthodox]{nag} +\usetikzlibrary{decorations} +\usetikzlibrary{fadings} +\usetikzlibrary{snakes} +\usetikzlibrary{calc} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Settings +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Choose a smaller value for larger fonts +\newcommand{\baposter@fontscale}{0.292} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Zoom +%------------------------------------------------------------------------------- +% We scale the page from fontscale * papersize up to papersize +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +%% Paper sizes +\newif\if@landscape +\newif\if@geometryKnowsThisSize +\DeclareOptionX{landscape}{\@landscapetrue} +\DeclareOptionX{portrait}{} + +\newcommand{\baposter@setfinalpapersize}[2]{% +\if@geometryKnowsThisSize + \setlength{\baposter@finalpaperwidth}{#1}% + \setlength{\baposter@finalpaperheight}{#2}% +\else +\if@landscape +% Transpose length, if geometry does not handle the papersize based on the key + \setlength{\baposter@finalpaperwidth}{#2}% + \setlength{\baposter@finalpaperheight}{#1}% +\else + \setlength{\baposter@finalpaperwidth}{#1}% + \setlength{\baposter@finalpaperheight}{#2}% +\fi +\fi +} + +% Default paperwidth and paperheight = a0paper +\DeclareOptionX{paperwidth}[841mm]{\setlength{\baposter@finalpaperwidth}{#1}} +\DeclareOptionX{paperheight}[1189mm]{\setlength{\baposter@finalpaperheight}{#1}} +\DeclareOptionX{archA} { \baposter@setfinalpapersize{9in}{12in}}% +\DeclareOptionX{archB} { \baposter@setfinalpapersize{12in}{18in}}% +\DeclareOptionX{archC} { \baposter@setfinalpapersize{18in}{24in}}% +\DeclareOptionX{archD} { \baposter@setfinalpapersize{24in}{36in}}% +\DeclareOptionX{archE} { \baposter@setfinalpapersize{36in}{48in}}% +\DeclareOptionX{archE1} { \baposter@setfinalpapersize{30in}{42in}}% +\DeclareOptionX{archE2} { \baposter@setfinalpapersize{26in}{38in}}% +\DeclareOptionX{archE3} { \baposter@setfinalpapersize{27in}{39in}}% +\DeclareOptionX{a0paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{841mm}{1189mm}}%g +\DeclareOptionX{a1paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{594mm}{841mm}}%g +\DeclareOptionX{a2paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{420mm}{594mm}}%g +\DeclareOptionX{a3paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{297mm}{420mm}}%g +\DeclareOptionX{a4paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{210mm}{297mm}}%g +\DeclareOptionX{a5paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{148mm}{210mm}}%g +\DeclareOptionX{a6paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{105mm}{148mm}}%g +\DeclareOptionX{b0paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{1000mm}{1414mm}}%g +\DeclareOptionX{b1paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{707mm}{1000mm}}%g +\DeclareOptionX{b2paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{500mm}{707mm}}%g +\DeclareOptionX{b3paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{353mm}{500mm}}%g +\DeclareOptionX{b4paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{250mm}{353mm}}%g +\DeclareOptionX{b5paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{176mm}{250mm}}%g +\DeclareOptionX{b6paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{125mm}{176mm}}%g +\DeclareOptionX{ansiapaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{8.5in}{11in}}% +\DeclareOptionX{ansibpaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{11in}{17in}}% +\DeclareOptionX{ansicpaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{17in}{22in}}% +\DeclareOptionX{ansidpaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{22in}{34in}}% +\DeclareOptionX{ansiepaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{34in}{44in}}% +\DeclareOptionX{letterpaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{8.5in}{11in}}% +\DeclareOptionX{legalpaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{8.5in}{14in}}% +\DeclareOptionX{executivepaper}{\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{7.25in}{10.5in}}% +\DeclareOptionX{screen} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{225mm}{180mm}}%g + +% Margin +\setlength{\baposter@finalmargin}{1.5cm} +\DeclareOptionX{fontscale}[0.292]{\renewcommand{\baposter@fontscale}{#1}} +\DeclareOptionX{margin} [1.5cm]{\setlength{\baposter@finalmargin}{#1}} + +% move text/poster body to the right (or to the left if negative) +\newlength{\baposter@movebody} +\setlength{\baposter@movebody}{0cm} +\DeclareOptionX{movebody}[0cm]{\setlength{\baposter@movebody}{#1}} + + +\newif\if@debug +\DeclareOptionX{debug}{\@debugtrue} +%% Will be passed on to other packages (xcolor and geometry), still we don't want unused warnings +\DeclareOptionX{table}{} +\DeclareOptionX{showframe}{} + +\ProcessOptionsX + +\if@debug +\newcommand{\debug}[1]{\typeout{#1}} +\else +\newcommand{\debug}[1]{} +\fi + + + +\setlength{\baposter@basepaperwidth} {\baposter@fontscale\baposter@finalpaperwidth } +\setlength{\baposter@basepaperheight}{\baposter@fontscale\baposter@finalpaperheight} +\setlength{\baposter@basemargin} {\baposter@fontscale\baposter@finalmargin} +\newlength{\baposter@basemarginright} +\setlength{\baposter@basemarginright}{\baposter@basemargin} +\addtolength{\baposter@basemarginright}{-\baposter@fontscale\baposter@movebody} +\newlength{\baposter@basemarginleft} +\setlength{\baposter@basemarginleft}{\baposter@basemargin} +\addtolength{\baposter@basemarginleft}{\baposter@fontscale\baposter@movebody} + +\typeout{Paperwidth=\the\baposter@finalpaperwidth} +\typeout{Paperheight=\the\baposter@finalpaperheight} +\typeout{BasePaperwidth=\the\baposter@basepaperwidth} +\typeout{BasePaperheight=\the\baposter@basepaperheight} +\usepackage[ + paperwidth=\baposter@basepaperwidth, + paperheight=\baposter@basepaperheight, + tmargin=\baposter@basemargin, + bmargin=\baposter@basemargin, + lmargin=\baposter@basemarginleft, + rmargin=\baposter@basemarginright, + ]{geometry} + +\usepackage{pgfpages} +\if@landscape +\if@geometryKnowsThisSize +\pgfpagesuselayout{resize to}[physical paper width=\baposter@finalpaperheight,physical paper height=\baposter@finalpaperwidth] +\else +\pgfpagesuselayout{resize to}[physical paper width=\baposter@finalpaperwidth,physical paper height=\baposter@finalpaperheight] +\fi +\else +\pgfpagesuselayout{resize to}[physical paper width=\baposter@finalpaperwidth,physical paper height=\baposter@finalpaperheight] +\fi + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Default functions for borders/backgrounds +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% These functions will all be redefined from the actual option values. In +%% particular, they will be set to \baposter@optionname@optionvalue, which +%% should do the actual work / setting for that particular optionvalue. + +\newcommand{\baposterPosterDrawBackground}[2]{} % Draw poster background + +\newcommand{\baposterBoxGetShape}{} % Returns path of text box shape +\newcommand{\baposterBoxDrawBackground}[2]{} % Draw bg of boxes +\newcommand{\baposterBoxDrawBorder}[1]{} % Draw border of individual boxes + +\newcommand{\baposterHeaderGetShape}{} % Returns path of text box shape +\newcommand{\baposterHeaderSetShade}[3]{} % Set bg style for box headers +\newcommand{\baposterHeaderDrawBackground}[3]{} % Draw background of box header +\newcommand{\baposterHeaderDrawBorder}[1]{} % Draw border of box header +\newcommand{\baposterHeaderDrawText}[1]{} % Draw text inside box header + +\newcommand{\@@previousbox}{notset} % stores the previously processed box for below=auto + +% Function to set a user-defined background +\newcommand{\baposter@backgroundCmd}{\error{No background command defined. Use \background{...} to define background}} +\newcommand{\background}[1]{\renewcommand{\baposter@backgroundCmd}{#1}} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Handle poster and box options +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\debug{Handling keys} + +%% +%% POSTER OPTIONS +%% +%% Store all poster options in variables of the form \baposter@option +%% choose-keys also store the index in \baposter@optionnr +%% choose-keys typically also assign a function +\definecolor{baposter@silver}{cmyk}{0,0,0,0.7} +\define@boolkey[ba]{poster}[baposter@]{grid} [false] {} +\define@boolkey[ba]{poster}[baposter@]{eyecatcher} [true] {} +\define@cmdkey [ba]{poster}[baposter@]{headerheight} [0.1\textheight]{} +\define@cmdkey [ba]{poster}[baposter@]{columns} [{}] {} +\define@cmdkey [ba]{poster}[baposter@]{colspacing} [1em] {} +\define@cmdkey [ba]{poster}[baposter@]{bgColorOne} [baposter@silver]{} +\define@cmdkey [ba]{poster}[baposter@]{bgColorTwo} [green] {} + +% background can be one of: shadeLR, shadeTB, plain, user, none +\define@choicekey*+[ba]{poster}{background}% + [\baposter@background\baposter@backgroundnr]% + {shadeLR, shadeTB, plain, user, none} [plain] {% + \debug{Poster background: \baposter@background} + \renewcommand{\baposterPosterDrawBackground}[2]{ + \csname baposter@background@\baposter@background\endcsname{##1}{##2}} +}{ + \PackageWarning{baposter}{Unknown background `\baposter@background' (use + shadeLR, shadeTB, plain, none, or user). If user is used, you also + have to define \background{...}.} + \renewcommand{\baposterPosterDrawBackground}[2]{\baposter@background@none{##1}{##2}} +} + + +%% +%% BOX OPTIONS +%% +\define@cmdkey[ba]{posterbox}[baposter@box@]{cornerradius} [1em] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{boxheaderheight} [2em] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{boxpadding} [0.5em] {} + + +% textborder can be one of: none, bars, coils, triangles, rectangle, rounded, +% roundedleft, roundedsmall, faded; UNIMPLEMENTED: roundedright +\edef\baposter@box@textborder@validvalues{none,bars,coils,triangles,rectangle,rounded,roundedleft,roundedsmall,faded} +\define@choicekey*+[ba]{posterbox}{textborder}% + [\baposter@box@textborder\baposter@box@textbordernr]% + {none,bars,coils,triangles,rectangle,rounded,roundedleft,roundedright,roundedsmall,faded} [rectangle] {% + \debug{Text border: \baposter@box@textborder} + \renewcommand{\baposterBoxGetShape}{ + \csname baposter@box@boxshape@\baposter@box@textborder\endcsname} + \renewcommand{\baposterBoxDrawBorder}[1]{ + \csname baposter@box@drawborder@\baposter@box@textborder\endcsname{##1}} +}{ + \PackageWarning{baposter}{Unknown text-border style `\baposter@box@textborder'. + Edit your file to choose a valid option (\baposter@box@textborder@validvalues).} + \renewcommand{\baposterBoxGetShape}{\baposter@boxshape@rectangle} + \renewcommand{\baposterBoxDrawBorder}[1]{\baposter@drawborder@rectangle{##1}} +} + +% boxshade can be one of: shadeLR, shadeTB, plain, none +\define@choicekey*+[ba]{posterbox}{boxshade}% + [\baposter@box@boxshade\baposter@box@boxshadenr]% + {shadelr,shadetb,plain,none} [none] {% + \debug{Box shade: \baposter@box@boxshade} + \renewcommand{\baposterBoxDrawBackground}[2]{ + \csname baposter@box@drawbackground@\baposter@box@boxshade\endcsname{##1}{##2}} +}{ + \PackageWarning{baposter}{Unknown boxshade style `\baposter@boxshade'. + Edit your file to choose a valid option.} + \renewcommand{\baposterBoxDrawBackground}[2]{\baposter@box@drawbackground@none{##1}{##2}} +} + +% headershade can be one of: shade-lr, shade-tb, shade-tb-inverse, plain +\define@choicekey*+[ba]{posterbox}{headershade}% + [\baposter@box@headershade\baposter@box@headershadenr]% + {shadelr, shadetb, shadetbinverse, plain} [shadelr] {% + \debug{Header shade: \baposter@box@headershade} + \renewcommand{\baposterHeaderSetShade}[3]{ + \csname baposter@box@headershade@\baposter@box@headershade\endcsname{##1}{##2}{##3}} +}{ + \PackageWarning{baposter}{Unknown headershade style `\baposter@box@headershade'. + Edit your file to choose a valid option.} + \renewcommand{\baposterHeaderSetShade}[3]{\baposter@box@headershade@none{##1}{##2}{##3}} +} + +% headershape can be one of: rectangle, rounded, smallrounded, roundedleft, roundedright +\define@choicekey*+[ba]{posterbox}{headershape}% + [\baposter@box@headershape\baposter@box@headershapenr]% + {rectangle,rounded,smallrounded,roundedleft,roundedright} [roundedright] {% + \debug{Header shape: \baposter@box@headershape} + \renewcommand{\baposterHeaderGetShape}{ + \csname baposter@box@headershape@\baposter@box@headershape\endcsname} + \renewcommand{\baposterHeaderDrawText}[1]{ + \csname baposter@box@headerdrawtext@\baposter@box@headershape\endcsname{##1}} + \renewcommand{\baposterHeaderDrawBorder}[1]{ + \csname baposter@box@headerdrawborder@\baposter@box@headershape\endcsname{##1}} +}{ + \PackageWarning{baposter}{Unknown headershape style `\baposter@headershape'. + Edit your file to choose a valid option.} + \renewcommand{\baposterHeaderGetShape}{\baposter@box@headershape@rectangle} + \renewcommand{\baposterHeaderDrawText}[1]{\baposter@box@headerdrawtext@rectangle{##1}} + \renewcommand{\baposterHeaderDrawBorder}[1]{\baposter@box@headerdrawborder@rectangle{##1}} +} + +% headerborder can be one of: open, closed, none +\define@choicekey*+[ba]{posterbox}{headerborder}% + [\baposter@box@headerborder\baposter@box@headerbordernr]% + {open,closed,none} [open] {% + \debug{Header border: \baposter@box@headerborder} +% \renewcommand{\baposterHeaderBorder}{ +% \csname baposter@headerborder@\baposter@box@headerborder\endcsname} +}{ + \PackageWarning{baposter}{Unknown headerborder style `\baposter@headerborder'. + Edit your file to choose a valid option.} +% \renewcommand{\baposterHeaderBorder}{\baposter@box@headerborder@rectangle} +} + + +\define@cmdkey[ba]{posterbox}[baposter@box@]{borderColor} [yellow] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{headerColorOne} [red] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{headerColorTwo} [brown] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{headerFontColor} [black] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{boxColorOne} [magenta] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{boxColorTwo} [cyan] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{headerfont} [\scshape\Large] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{textfont} [{}] {} + +\define@cmdkey[ba]{posterbox}[baposter@box@]{linewidth} [2pt] {} + +\define@cmdkey[ba]{posterbox}[baposter@box@]{below} [notset]{} +\define@cmdkey[ba]{posterbox}[baposter@box@]{above} [notset]{} +\define@cmdkey[ba]{posterbox}[baposter@box@]{aligned}[notset]{} +\define@cmdkey[ba]{posterbox}[baposter@box@]{bottomaligned}[notset]{} +\define@cmdkey[ba]{posterbox}[baposter@box@]{column} [0] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{row} [0] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{span} [1] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{height} [auto] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{name} [noname]{} + +% Set some default values, the poster and posterbox environments can override: +\setkeys[ba]{poster}{ + % Debug grid + grid=false, + % Is there an eyecatcher image + eyecatcher=true, + columns={}, + % Colours + bgColorOne=baposter@silver, + bgColorTwo=green, + % + colspacing=1em, + headerheight=0.1\textheight, + background=shadeLR, +}{} +\setkeys[ba]{posterbox}{ + % Position + column=0,row=0,span=1, + below=notset,above=notset, + bottomaligned=notset, + aligned=notset, + height=auto, + % Name + name=noname, + % Box design: border: + linewidth=2pt, + borderColor=yellow, + cornerradius=1em, + % text box: + textfont={}, + boxshade=plain, + boxColorOne=magenta, + boxColorTwo=cyan, + textborder=faded, + boxpadding=0.5em, + % header + headerfont=\scshape\Large,% or headerfont=\color{white}\textsf\textbf + headerFontColor=black, + headerColorOne=red, + headerColorTwo=brown, + headershape=rectangle, + headershade=shadeLR, + headerborder=none, + boxheaderheight=2em, +}{} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Background options and functions (one function for each possible value) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newcommand{\baposter@background@shadelr}[2]{ + \debug{BAPOSTER: Using shade left right background.} + \begin{tikzpicture}[remember picture,overlay]% + \shade [shading=axis,left color=#1,right color=#2] (current page.north west) + rectangle(current page.south east); + \end{tikzpicture}% +} +\newcommand{\baposter@background@shadetb}[2]{ + \debug{BAPOSTER: Using shade top to bottom background.} + \begin{tikzpicture}[remember picture,overlay]% + \shade [shading=axis,top color=#1,bottom color=#2] (current page.north west) + rectangle(current page.south east); + \end{tikzpicture}% +} +\newcommand{\baposter@background@plain}[2]{ + \debug{BAPOSTER: Using plain background.} + \begin{tikzpicture}[remember picture,overlay]% + \fill [fill=#1] (current page.north west) rectangle(current page.south east); + \end{tikzpicture}% +} +\newcommand{\baposter@background@user}[2]{ + \debug{BAPOSTER: Using user background.} + \baposter@backgroundCmd% +} +\newcommand{\baposter@background@none}[2]{ + \debug{BAPOSTER: Using no background.} +} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Return shape path of text box (depending on the box shape) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newcommand{\baposter@box@boxshape@none}{\baposter@box@boxshape@rectangle} +\newcommand{\baposter@box@boxshape@bars}{ + (\baposter@box@name tnw) -- (\baposter@box@name sw) % + (\baposter@box@name se) -- (\baposter@box@name tne) +} +\newcommand{\baposter@box@boxshape@coils}{\baposter@box@boxshape@bars} +\newcommand{\baposter@box@boxshape@triangles}{\baposter@box@boxshape@bars} +\newcommand{\baposter@box@boxshape@rectangle}{ + (\baposter@box@name tnw) -- (\baposter@box@name sw) -- % + (\baposter@box@name se) -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@boxshape@faded}{ + (\baposter@box@name tnw) -- (\baposter@box@name sw) % + (\baposter@box@name tne) -- (\baposter@box@name se) + } +\newcommand{\baposter@box@boxshape@rounded}{ + [rc] \baposter@box@boxshape@rectangle% +} +\newcommand{\baposter@box@boxshape@roundedsmall}{ + [src] \baposter@box@boxshape@rectangle +} +\newcommand{\baposter@box@boxshape@roundedleft}{ + (\baposter@box@name tnw) {[rc]-- (\baposter@box@name sw)} -- % + (\baposter@box@name se) -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@boxshape@roundedright}{ + (\baposter@box@name tnw) -- (\baposter@box@name sw) {[rc]-- % + (\baposter@box@name se)} -- (\baposter@box@name tne)% +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Draw box background (one function for each possible value) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% These functions take no arguments +\newcommand{\baposter@box@drawbackground@none}[2]{ + \tikzstyle{box colors}=[] +} +\newcommand{\baposter@box@drawbackground@plain}[2]{ + \tikzstyle{box colors}=[fill=#1] + \fill[box colors] \baposterBoxGetShape; +} +\newcommand{\baposter@box@drawbackground@shadelr}[2]{ + \tikzstyle{box colors}=[shading=axis, left color=#1, right color=#2]% + \fill[box colors] \baposterBoxGetShape; +} +\newcommand{\baposter@box@drawbackground@shadetb}[2]{ + \tikzstyle{box colors}=[shading=axis, top color=#1, bottom color=#2]% + \fill[box colors] \baposterBoxGetShape; +} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Draw box border +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% These functions take two arguments: borderColor +\newcommand{\baposter@box@drawborder@none}[1]{} +\newcommand{\baposter@box@drawborder@bars}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@coils}[1]{ + \draw[color=#1,segment amplitude=0.35em,segment length=0.4em,snake=coil] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@triangles}[1]{ + \draw[color=#1,segment amplitude=0.2em,segment length=0.4em,snake=triangles] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@rectangle}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@rounded}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@roundedleft}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@roundedright}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@faded}[1]{ + % This is the right way to do it, but it does not work with evince, and has problems during printing, so instead we do + %\draw[color=#1,path fading=south] \baposterBoxGetShape;% + % this + \foreach \x in {0,1,...,90} \draw[color=#1!\x] ($(\baposter@box@name tnw)!{(100-\x)/100}!(\baposter@box@name sw)$) -- ($(\baposter@box@name tnw)!{(100-(\x+10))/100}!(\baposter@box@name sw)$);% + \foreach \x in {0,1,...,90} \draw[color=#1!\x] ($(\baposter@box@name tne)!{(100-\x)/100}!(\baposter@box@name se)$) -- ($(\baposter@box@name tne)!{(100-(\x+10))/100}!(\baposter@box@name se)$);% +} +\newcommand{\baposter@box@drawborder@roundedsmall}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Return shape path of text box (depending on the box shape) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% These functions take no arguments +% TODO: For headerborder==none, use (\baposter@box@name outer tnw) instead! +\newcommand{\baposter@box@headershape@rectangle}{% + (\baposter@box@name tnw) -- (\baposter@box@name nw) -- % + (\baposter@box@name ne) -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@headershape@smallrounded}{% + (\baposter@box@name tnw) {[src] -- (\baposter@box@name nw) -- % + (\baposter@box@name ne)} -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@headershape@roundedright}{% + (\baposter@box@name tnw) -- (\baposter@box@name nw) {[rc] -- % + (\baposter@box@name ne)} -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@headershape@roundedleft}{% + (\baposter@box@name tnw) {[rc]-- (\baposter@box@name nw)} -- % + (\baposter@box@name ne) -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@headershape@rounded}{% + (\baposter@box@name tnw) {[rc] -- (\baposter@box@name nw) -- % + (\baposter@box@name ne) } -- (\baposter@box@name tne)% +} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Header text drawing (one function for each possible value of headershape) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% These functions take one argument: the header text +\newcommand{\baposter@box@headerdrawtext@rectangle}[1]{ + \path (\baposter@box@name nw) +(0em,-0.5\baposter@box@@boxheaderheight) node[anchor=west,inner sep=0.4em] {#1};% +} +\newcommand{\baposter@box@headerdrawtext@smallrounded}[1]{ + \path (\baposter@box@name nw) +(0.5\boxwidth,-0.5\baposter@box@@boxheaderheight) node[anchor=center] {#1};% +} +\newcommand{\baposter@box@headerdrawtext@roundedright}[1]{ + \path (\baposter@box@name nw) +(0em,-0.5\baposter@box@@boxheaderheight)% + node[anchor=west,inner sep=0.4em,text depth=0.0em] {#1};% +} +\newcommand{\baposter@box@headerdrawtext@roundedleft}[1]{ + \path (\baposter@box@name nw) +(0em,-0.5\baposter@box@@boxheaderheight)% + node[anchor=west,inner sep=0.4em] {#1};% +} +\newcommand{\baposter@box@headerdrawtext@rounded}[1]{ + \path (\baposter@box@name nw) +(0.5\boxwidth,-0.5\baposter@box@@boxheaderheight) node[anchor=center] {#1};% +} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Header shade options and functions (one function for each possible value) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% These functions take two arguments: headerColorOne, headerColorTwo and borderColor +\newcommand{\baposter@box@headershade@shadelr}[3]{ + \debug{Header-Shade: Shade Left - Right} + \tikzstyle{header colors}=[% + color=#3,% + shading=axis,% + left color=#1,% + right color=#2% + ]% +} +\newcommand{\baposter@box@headershade@shadetb}[3]{ + \debug{Header-Shade: Shade Top - Bottom} + \tikzstyle{header colors}=[% + color=#3,% + shading=axis,% + top color=#1,% + bottom color=#2% + ]% +} +\newcommand{\baposter@box@headershade@shadetbinverse}[3]{ + \tikzstyle{header colors}=[% + top color=#1!75!#2,% + bottom color=#2!100!#1,% + shading angle=20% + ]% + \colorlet{baposterHeaderFontColor}{white}% +} +\newcommand{\baposter@box@headershade@plain}[3]{ + \debug{Header-Shade: Plain} + \tikzstyle{header colors}=[% + color=#3,% + fill=#1% + ]% +} +\newcommand{\baposter@box@headershade@none}[3]{ + \debug{Header-Shade: none} + \tikzstyle{header colors}=[] +} + + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% The main poster environment +%%% \begin{baposter}{settings}{Eye Catcher}{Title}{Author}{University Logo} +%%%----------------------------------------------------------------------------- +%%% The settings are +%%% - grid=true,[false]:Show grid to help with alignment +%%% - colspacing=0.7em: Column spacing +%%% - columns=4: number of columns (default 4 in landscape and 3 in portrait format) (maximum number is 6) +%%% - color=[orange]: xcolor color definition used as the main color of the poster +%%% - colortwo=[white]: The other color for gradient based layouts +%%% - textborder=none,bars,coils,triangles,rectangle,rounded,roundedsmall,roundedleft,roundedright,[faded] +%%% The style of the box around the text area +%%% - headerborder=none,closed,open +%%% No extra border around box header, full border around box header or border that is open below. +%%% - headershape=rectangle,rounded,roundedleft,roundedright +%%% Shape of the box-header region +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newenvironment{poster}[5]{% + \thispagestyle{empty}% Suppress Page Number + \debug{Poster Starts}% + % This setkeys call parses all provided options and depending on the option + % value, assigns different handler functions to the \baposter(Box|Header)* + % functions. Once that is done, we don't have to care about particular + % values for border, shading, etc. All we have to do is call the + % handler functions and let them do their job. + % This also allows the user to override the poster-wide defaults on a per-box + % basis. + \setkeys[ba]{posterbox,poster}{#1}% +% + % TODO: Move all those assignments to the key macros! + % Parse Keys% + \colorlet{bgColorOne}{\baposter@bgColorOne} + \colorlet{bgColorTwo}{\baposter@bgColorTwo} +% + %% Boxes% + \setlength{\headerheight}{\baposter@headerheight}% + \setlength{\colheight}{\textheight-\baposter@headerheight}% + \renewcommand{\@@previousbox}{notset} + + \debug{Format}% + % Set default for columns if unset (4 for landscape, 3 for portrait) + \ifthenelse{\equal{\baposter@columns}{}}{% + \renewcommand{\baposter@columns}{\if@landscape4\else3\fi}% + }{} +% + \debug{Columns: \baposter@columns}% + \setlength{\baposter@@colspacing}{\baposter@colspacing}% + \setlength{\colwidth}{\textwidth}% + \addtolength{\colwidth}{\baposter@@colspacing*(1-\baposter@columns)}% + \ifcase\baposter@columns\relax + \error{You need to have at least one column!} + \or % 1 + \setlength{\colwidth}{\colwidth}% + \or % 2 + \setlength{\colwidth}{0.5\colwidth}% + \or % 3 + \setlength{\colwidth}{0.3333333333333\colwidth}% + \or % 4 + \setlength{\colwidth}{0.25\colwidth}% + \or % 5 + \setlength{\colwidth}{0.2\colwidth}% + \or % 6 + \setlength{\colwidth}{0.16666666666\colwidth}% + \else % >6 + \error{You do not want so many columns} + \fi +% + \newcommand{\baposter@reference}{north west}% +% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % A box with a header and some content. The basic unit of the poster% + %---------------------------------------------------------------------------% + % Each box has a name and can be placed absolutely or relatively.% + % The only inconvenience is that you can only specify a relative position % + % towards an already declared box. So if you have a box attached to the % + % bottom, one to the top and a third one which should be inbetween, you % + % have to specify the top and bottom boxes before you specify the middle % + % box.% + %% + % below= name of other node% + % above= name of other node% + % aligned=name of other node% + % bottomaligned=name of other node% + % column= [0] % + % row= [0] % + % span= [1] % + % height= ,[auto]% + % name= [noname]% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % Backward-compatibility definition (\headerbox command uses posterbox env): + \newcommand{\headerbox}[3]{% + \begin{posterbox}[##2]{##1} + ##3 + \end{posterbox} + } + + \newenvironment{posterbox}[2][]{% + \debug{Poster box options: ##1}% + % Override the poster-wide defaults on a per-box basis + \setkeys[ba]{posterbox}{##1}% +% + \def\baposter@box@title{##2} +% + \colorlet{borderColor}{\baposter@box@borderColor} + \colorlet{headerColorOne}{\baposter@box@headerColorOne} + \colorlet{headerColorTwo}{\baposter@box@headerColorTwo} + \colorlet{headerFontColor}{\baposter@box@headerFontColor} + \colorlet{boxColorOne}{\baposter@box@boxColorOne} + \colorlet{boxColorTwo}{\baposter@box@boxColorTwo} +% + \setlength{\baposter@box@@cornerradius}{\baposter@box@cornerradius}% + \setlength{\baposter@box@@boxheaderheight}{\baposter@box@boxheaderheight}% + \setlength{\baposter@box@@boxpadding}{\baposter@box@boxpadding}% + + + %% The columns is always given absolute + % boxstartx = column * colwidth + column * colspacing + \setlength{\boxstartx}{(\colwidth+\baposter@@colspacing)*\baposter@box@column}% +% + %% The width is gvien absolute + % Box Width = span * colwidth + (span-1) * colspacing + \setlength{\boxwidth}{\baposter@box@span\colwidth} % + \addtolength{\boxwidth}{\baposter@@colspacing*(\baposter@box@span-1)}% +% + %% Measure the content of the box% + \setbox\baposter@box@content=\hbox\bgroup% + \begin{pgfinterruptpicture}% + \begin{minipage}[t]{\boxwidth-\baposter@box@@boxpadding*2}% + \baposter@box@textfont\bgroup% + }% End of posterbox preamble + %%% HERE COME THE ACTUAL CONTENTS OF THE HEADERBOX ENVIRONMENT + {% posterbox handling after contents (i.e. drawing everything) + \egroup% + \end{minipage}% + \end{pgfinterruptpicture}% + \egroup% + \setlength{\boxheight}{\ht\baposter@box@content}% + \addtolength{\boxheight}{\dp\baposter@box@content}% + \addtolength{\boxheight}{\baposter@box@@boxheaderheight} % Header% + \addtolength{\boxheight}{2\baposter@box@@boxpadding} % Inner Sep +% + \ifthenelse{\equal{\baposter@box@height}{bottom}}{% + }{\ifthenelse{\equal{\baposter@box@height}{auto}}{% + }{ % Neither auto nor bottom% + \setlength{\boxheight}{\baposter@box@height\colheight}% + }}% +% + %% Determine the box position% + \debug{Setting Coordinates}% + \debug{Upper Right}% + \debug{\baposter@box@name}% +% + %%% Upper Right Corner% + % if below=auto, set it to the previous box + % TODO: We should generalize this to the previous box of the used column, + % currently we use the previous box, which might be in a different column + \ifthenelse{\equal{\baposter@box@below}{auto}}{% + \edef\baposter@box@below{\@@previousbox} + \debug{Box \baposter@box@name has below=auto, placing it below box \baposter@box@below.} + }{} + \xdef\@@previousbox{\baposter@box@name} + + \ifthenelse{\not\equal{\baposter@box@below}{notset} }{% + %% Below% + \debug{Below}% + \path[shape=coordinate] (\boxstartx,0pt |- \baposter@box@below se) ++(0pt,-\baposter@@colspacing) coordinate(\baposter@box@name nw);% + }{% + \ifthenelse{\not\equal{\baposter@box@aligned}{notset} }{% + %% Aligned% + \debug{Aligned: \baposter@box@aligned}% + \path[shape=coordinate] (\boxstartx,0pt |- \baposter@box@aligned nw) coordinate(\baposter@box@name nw);% + }{% + %% Fixed% + \debug{Fixed}% + \setlength{\boxstarty}{\baposter@box@row\colheight}% + \path[shape=coordinate] (\boxstartx,\colheight-\boxstarty) coordinate(\baposter@box@name nw);% + }}% +% + %% Lower Left Corner% + \debug{Lower Left}% + \ifthenelse{\equal{\baposter@box@above}{bottom}}{% + %% Above = Bottom% + \debug{Above bottom}% + \ifthenelse{\equal{\baposter@box@below}{notset} \and \equal{\baposter@box@aligned}{notset}}{% + \path[shape=coordinate] (\boxstartx,\boxheight) coordinate(\baposter@box@name nw);% + }{}% + \path[shape=coordinate] (\boxstartx+\boxwidth,0pt) coordinate(\baposter@box@name se);% + }{\ifthenelse{\not \equal{\baposter@box@bottomaligned}{notset}}{% + \path[shape=coordinate] (\boxstartx+\boxwidth,0pt |- \baposter@box@bottomaligned se) coordinate(\baposter@box@name se);% + }{{\ifthenelse{\not \equal{\baposter@box@above}{notset}}{% + %% Above = Node% + \path[shape=coordinate] (\boxstartx+\boxwidth,0pt |- \baposter@box@above nw) +(0pt,\baposter@@colspacing) coordinate(\baposter@box@name se);% + }{% + %% Above = notset% + \debug{Above=not set}% + \ifthenelse{\equal{\baposter@box@height}{bottom}}{% + %% height=bottom% + \debug{height=bottom}% + \path[shape=coordinate] (\boxstartx+\boxwidth,0pt) coordinate(\baposter@box@name se);% + }{ %% height=auto or fixed% + \debug{height=auto or fixed}% + \path[shape=coordinate] (\baposter@box@name nw) ++(\boxwidth,-\boxheight) coordinate(\baposter@box@name se);% + }}}}}% +% + % + % Set coordinates relative to nw,se% + \debug{Fixing Coordinates}% + \path[shape=coordinate]% + (\baposter@box@name nw) +(0pt,-\baposter@box@@boxheaderheight) coordinate(\baposter@box@name tnw)% + (\baposter@box@name nw |- \baposter@box@name se) coordinate(\baposter@box@name sw)% + (\baposter@box@name se |- \baposter@box@name nw) coordinate(\baposter@box@name ne)% + (\baposter@box@name ne) +(0pt,-\baposter@box@@boxheaderheight) coordinate(\baposter@box@name tne)% +% + (\baposter@box@name nw) +(-0.025em,0pt) coordinate(\baposter@box@name outer nw)% + (\baposter@box@name tnw) +(-0.025em,0pt) coordinate(\baposter@box@name outer tnw)% + (\baposter@box@name sw) +(-0.025em,0pt) coordinate(\baposter@box@name outer sw)% +% + (\baposter@box@name ne) +( 0.025em,0pt) coordinate(\baposter@box@name outer ne)% + (\baposter@box@name tne) +( 0.025em,0pt) coordinate(\baposter@box@name outer tne)% + (\baposter@box@name se) +( 0.025em,0pt) coordinate(\baposter@box@name outer se);% +% + %% Setting the bg colors of the box header + \baposterHeaderSetShade{headerColorOne}{headerColorTwo}{borderColor} +% + \tikzstyle{rc}=[rounded corners=\baposter@box@@cornerradius];% + \tikzstyle{src}=[rounded corners=0.5em];% +% + + %% Now that everything is set up, draw the actual box, with bg and header + \begin{scope}[line width=\baposter@box@linewidth] + %% Header% + \debug{Header}% + \debug{Header-Shape: \baposter@box@headershape, header-border: \baposter@box@headerborder (\baposter@box@headerbordernr)}% + % TODO: Also turn this last ifcase construct into a handler function + % We only need to determine (fill|shade)(draw|)... +% \baposterHeaderDrawBackground{bgColorOne}{bgColorTwo}{borderColor} +% \baposterHeaderDrawBorder{borderColor} + \ifcase\baposter@box@headerbordernr\relax% + % open + \ifthenelse{\equal{\baposter@box@headershade}{plain}}{ + \filldraw [style=header colors] \baposterHeaderGetShape;% + }{ + \shadedraw [style=header colors] \baposterHeaderGetShape;% + } + \or + % closed + \ifthenelse{\equal{\baposter@box@headershade}{plain}}{ + \filldraw [style=header colors] \baposterHeaderGetShape -- cycle;% + }{ + \shadedraw [style=header colors] \baposterHeaderGetShape -- cycle;% + } + \or + % none + \ifthenelse{\equal{\baposter@box@headershade}{plain}}{ + \fill [style=header colors] \baposterHeaderGetShape;% + }{ + \shade [style=header colors] \baposterHeaderGetShape;% + } + \fi + % + %% Draw the text inside the box header: + \baposterHeaderDrawText{\color{headerFontColor}\baposter@box@headerfont{\baposter@box@title}};% + % + %% Text borders (border around boxes) + \debug{Poster boxes}% + % First set box shade + \baposterBoxDrawBackground{boxColorOne}{boxColorTwo} + \baposterBoxDrawBorder{borderColor} + %% + %% Text Box% + \debug{Drawing Text}% + \path (\baposter@box@name tnw) node(text) [anchor=north west, + outer sep=-0.000em,text width=\boxwidth-2\baposter@box@@boxpadding,inner sep=\baposter@box@@boxpadding, + text justified] {\usebox{\baposter@box@content}};% + \end{scope} + % + % Finally store the box name as the previous box for the next call +% \xdef\@@previousbox{\baposter@box@name}% + }% END of posterbox definition +% + %% Poster Background% + \baposterPosterDrawBackground{bgColorOne}{bgColorTwo}% + %% Poster header/title + \hspace{-1.5em}% + \begin{tikzpicture}[inner sep=0pt,outer sep=0pt,line width=0.05em]% + \useasboundingbox (0em,0em) rectangle(\textwidth,\textheight);% + \path[shape=coordinate]% + (0pt,\colheight) coordinate(north west) (\textwidth,\colheight) coordinate(north east)% + (0pt,0pt) coordinate(south west) (\textwidth,0pt) coordinate(south east);% +% + \ifbaposter@eyecatcher% Has eye catcher + \debug{Eyecatcher found!} + \setbox\baposter@titleimage@left=\hbox{#2}% + \else% Has no eye catcher% + \setbox\baposter@titleimage@left=\hbox{}% + \fi% + \setlength{\baposter@titleimage@left@width}{\wd\baposter@titleimage@left}% + \setbox\baposter@titleimage@right=\hbox{#5}% + \setlength{\baposter@titleimage@right@width}{\wd\baposter@titleimage@right}% + \setlength{\baposter@titleimage@textwidth}{\textwidth}% + \addtolength{\baposter@titleimage@textwidth}{-\baposter@titleimage@left@width}% + \addtolength{\baposter@titleimage@textwidth}{-\baposter@titleimage@right@width}% + + \debug{#3} + % + % + % % Draw Header% + \draw (north west) +(0em,1em+0.5\headerheight) node(image)[anchor=west] { {\usebox{\baposter@titleimage@left }} };% + \draw (north east) +(0em,1em+0.5\headerheight) node(logo) [anchor=east] { {\usebox{\baposter@titleimage@right}} };% + % + \ifbaposter@eyecatcher% Has eye catcher% + \draw (image.east) node(title)[anchor=west,text width=\baposter@titleimage@textwidth]{% + \begin{minipage}{\baposter@titleimage@textwidth}% + \begin{center}% + \textbf{\Huge #3}\\% + {\Large #4}% + \end{center}% + \end{minipage} + };% + \else% Has no eye catcher + \draw (image.east) node(title)[anchor=west] { {\begin{minipage}{\baposter@titleimage@textwidth}{\bfseries\Huge #3}\\{\Large #4}\end{minipage}} };% + \fi + }% END poster begin +% The body + {% BEGIN poster end + % The end, draw gridlines if neccesary + \ifbaposter@grid + \newdimen{\gridpos} + \pgfmathsetmacro{\z}{\baposter@columns-1} + \foreach \y in {0,...,\z} + { + \setlength{\gridpos}{\y\colwidth+\y\baposter@@colspacing} + \draw[draw=green,draw opacity=0.7] (\gridpos,0pt) -- (\gridpos,\colheight) + (\gridpos+\colwidth,0pt) -- (\gridpos+\colwidth,\colheight);% + } + % Horizontal lines, every 0.1: + %% Explicitly list all percentages, because with {0.0, 0.1, ..., 1.0} we + %% get rounding errors in the displayed numbers! + \foreach \y in {0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0} + \draw[draw=green,draw opacity=0.7] (0pt,\colheight-\y\colheight) -- + (\textwidth,\colheight-\y\colheight) node[anchor=west] {\y};% + \fi% + \end{tikzpicture}% + % \xkvview{} + \par + }% END poster end diff --git a/poster scientifique/biblio.bib b/poster scientifique/biblio.bib new file mode 100644 index 0000000..cf53765 --- /dev/null +++ b/poster scientifique/biblio.bib @@ -0,0 +1,39 @@ +@article{madan1990variance, + title={The variance gamma (VG) model for share market returns}, + author={Madan, Dilip B and Seneta, Eugene}, + journal={Journal of business}, + pages={511--524}, + year={1990}, + publisher={JSTOR} +} + +@book{kotz2001laplace, + title={The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance}, + author={Kotz, S. and Kozubowski, T.J. and Podg{\'o}rski, K.}, + isbn={9780817641665}, + lccn={00068900}, + url={http://books.google.ca/books?id=CQvvAAAAMAAJ}, + year={2001}, + publisher={Birkh{\"a}user} +} + +@book{hall2005generalized, + title={Generalized Method of Moments}, + author={Hall, A.R.}, + isbn={9780198775201}, + lccn={2005297434}, + series={Advanced Texts in Econometrics}, + url={http://books.google.ca/books?id=jN-T-fU139wC}, + year={2005}, + publisher={OUP Oxford} +} + +@book{epps2007pricing, + title={Pricing Derivative Securities}, + author={Epps, T.W.}, + isbn={9789812700339}, + lccn={2007006660}, + url={http://books.google.ca/books?id=Wk9Ih3uWM10C}, + year={2007}, + publisher={World Scientific Publishing Company Pte. Limited} +} \ No newline at end of file diff --git a/poster scientifique/images/ul.png b/poster scientifique/images/ul.png new file mode 100644 index 0000000..e1de8c8 Binary files /dev/null and b/poster scientifique/images/ul.png differ diff --git a/poster scientifique/poster_landscape.aux b/poster scientifique/poster_landscape.aux new file mode 100644 index 0000000..b59fc94 --- /dev/null +++ b/poster scientifique/poster_landscape.aux @@ -0,0 +1,22 @@ +\relax +\catcode`:\active +\catcode`;\active +\catcode`!\active +\catcode`?\active +\citation{kotz2001laplace} +\citation{hall2005generalized} +\bibstyle{plain} +\bibdata{biblio} +\bibcite{epps2007pricing}{1} +\bibcite{hall2005generalized}{2} +\bibcite{kotz2001laplace}{3} +\bibcite{madan1990variance}{4} +\citation{epps2007pricing} +\citation{madan1990variance} +\select@language{french} +\@writefile{toc}{\select@language{french}} +\@writefile{lof}{\select@language{french}} +\@writefile{lot}{\select@language{french}} +\pgfsyspdfmark {pgfid1}{1780200}{797160} +\newlabel{eq:defvarY-GAL}{{}{2}} +\newlabel{eq:inversionfncaract}{{}{2}} diff --git a/poster scientifique/poster_landscape.bbl b/poster scientifique/poster_landscape.bbl new file mode 100644 index 0000000..b33db21 --- /dev/null +++ b/poster scientifique/poster_landscape.bbl @@ -0,0 +1,24 @@ +\begin{thebibliography}{1} + +\bibitem{epps2007pricing} +T.W. Epps. +\newblock {\em Pricing Derivative Securities}. +\newblock World Scientific Publishing Company Pte. Limited, 2007. + +\bibitem{hall2005generalized} +A.R. Hall. +\newblock {\em Generalized Method of Moments}. +\newblock Advanced Texts in Econometrics. OUP Oxford, 2005. + +\bibitem{kotz2001laplace} +S.~Kotz, T.J. Kozubowski, and K.~Podg{\'o}rski. +\newblock {\em The Laplace Distribution and Generalizations: A Revisit with + Applications to Communications, Economics, Engineering, and Finance}. +\newblock Birkh{\"a}user, 2001. + +\bibitem{madan1990variance} +Dilip~B Madan and Eugene Seneta. +\newblock The variance gamma (vg) model for share market returns. +\newblock {\em Journal of business}, pages 511--524, 1990. + +\end{thebibliography} diff --git a/poster scientifique/poster_landscape.blg b/poster scientifique/poster_landscape.blg new file mode 100644 index 0000000..aff0d72 --- /dev/null +++ b/poster scientifique/poster_landscape.blg @@ -0,0 +1,46 @@ +This is BibTeX, Version 0.99d (TeX Live 2012/Debian) +Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 +The top-level auxiliary file: poster_landscape.aux +The style file: plain.bst +Database file #1: biblio.bib +You've used 4 entries, + 2118 wiz_defined-function locations, + 520 strings with 4571 characters, +and the built_in function-call counts, 1172 in all, are: += -- 113 +> -- 45 +< -- 1 ++ -- 20 +- -- 14 +* -- 59 +:= -- 199 +add.period$ -- 13 +call.type$ -- 4 +change.case$ -- 16 +chr.to.int$ -- 0 +cite$ -- 4 +duplicate$ -- 56 +empty$ -- 99 +format.name$ -- 14 +if$ -- 252 +int.to.chr$ -- 0 +int.to.str$ -- 4 +missing$ -- 7 +newline$ -- 23 +num.names$ -- 8 +pop$ -- 29 +preamble$ -- 1 +purify$ -- 15 +quote$ -- 0 +skip$ -- 38 +stack$ -- 0 +substring$ -- 54 +swap$ -- 13 +text.length$ -- 1 +text.prefix$ -- 0 +top$ -- 0 +type$ -- 10 +warning$ -- 0 +while$ -- 11 +width$ -- 5 +write$ -- 44 diff --git a/poster scientifique/poster_landscape.log b/poster scientifique/poster_landscape.log new file mode 100644 index 0000000..0f6229a --- /dev/null +++ b/poster scientifique/poster_landscape.log @@ -0,0 +1,1084 @@ +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.2.17) 13 MAR 2013 23:36 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**\input poster_landscape.tex +(./poster_landscape.tex (./baposter.cls +Document Class: baposter 2011/11/26 v2.0 baposter class +(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls +Document Class: article 2007/10/19 v1.4h Standard LaTeX document class +(/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo +File: size10.clo 2007/10/19 v1.4h Standard LaTeX file (size option) +) +\c@part=\count79 +\c@section=\count80 +\c@subsection=\count81 +\c@subsubsection=\count82 +\c@paragraph=\count83 +\c@subparagraph=\count84 +\c@figure=\count85 +\c@table=\count86 +\abovecaptionskip=\skip41 +\belowcaptionskip=\skip42 +\bibindent=\dimen102 +) +baposter: Brian Amberg, 2007, 2008, 2009, 2010, 2011 | http://www.brian-amberg. +de/uni/poster/ +baposter: Reinhold Kainhofer, 2011 | http://reinhold.kainhofer.com/ +\baposter@basepaperwidth=\skip43 +\baposter@basepaperheight=\skip44 +\baposter@basemargin=\skip45 +\baposter@finalpaperwidth=\skip46 +\baposter@finalpaperheight=\skip47 +\baposter@finalmargin=\skip48 +\headerheight=\skip49 +\colwidth=\skip50 +\colheight=\skip51 +\baposter@@colspacing=\skip52 +\baposter@box@@cornerradius=\skip53 +\baposter@box@@boxheaderheight=\skip54 +\baposter@box@@boxpadding=\skip55 +\boxstartx=\skip56 +\boxstarty=\skip57 +\boxwidth=\skip58 +\boxheight=\skip59 +\baposter@titleimage@left@width=\skip60 +\baposter@titleimage@right@width=\skip61 +\baposter@titleimage@textwidth=\skip62 +\baposter@box@content=\box26 +\baposter@titleimage@left=\box27 +\baposter@titleimage@title=\box28 +\baposter@titleimage@right=\box29 +Use Packages +(/usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty +Package: xkeyval 2008/08/13 v2.6a package option processing (HA) + +(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex +\XKV@toks=\toks14 +\XKV@tempa@toks=\toks15 +\XKV@depth=\count87 +File: xkeyval.tex 2008/08/13 v2.6a key=value parser (HA) + +(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/keyval.tex))) +(/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty +Package: calc 2007/08/22 v4.3 Infix arithmetic (KKT,FJ) +\calc@Acount=\count88 +\calc@Bcount=\count89 +\calc@Adimen=\dimen103 +\calc@Bdimen=\dimen104 +\calc@Askip=\skip63 +\calc@Bskip=\skip64 +LaTeX Info: Redefining \setlength on input line 76. +LaTeX Info: Redefining \addtolength on input line 77. +\calc@Ccount=\count90 +\calc@Cskip=\skip65 +) +(/usr/share/texmf/tex/latex/xcolor/xcolor.sty +Package: xcolor 2007/01/21 v2.11 LaTeX color extensions (UK) + +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg +File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive +) +Package xcolor Info: Driver file: pdftex.def on input line 225. + +(/usr/share/texlive/texmf-dist/tex/latex/pdftex-def/pdftex.def +File: pdftex.def 2011/05/27 v0.06d Graphics/color for pdfTeX + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty +Package: infwarerr 2010/04/08 v1.3 Providing info/warning/error messages (HO) +) +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty +Package: ltxcmds 2011/11/09 v1.22 LaTeX kernel commands for general use (HO) +) +\Gread@gobject=\count91 +) +Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1337. +Package xcolor Info: Model `hsb' substituted by `rgb' on input line 1341. +Package xcolor Info: Model `RGB' extended on input line 1353. +Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1355. +Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1356. +Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1357. +Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1358. +Package xcolor Info: Model `Gray' substituted by `gray' on input line 1359. +Package xcolor Info: Model `wave' substituted by `hsb' on input line 1360. +) +(/usr/share/texmf/tex/latex/pgf/frontendlayer/tikz.sty +(/usr/share/texmf/tex/latex/pgf/basiclayer/pgf.sty +(/usr/share/texmf/tex/latex/pgf/utilities/pgfrcs.sty +(/usr/share/texmf/tex/generic/pgf/utilities/pgfutil-common.tex +\pgfutil@everybye=\toks16 +) +(/usr/share/texmf/tex/generic/pgf/utilities/pgfutil-latex.def +\pgfutil@abb=\box30 + +(/usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty +Package: everyshi 2001/05/15 v3.00 EveryShipout Package (MS) +)) +(/usr/share/texmf/tex/generic/pgf/utilities/pgfrcs.code.tex +Package: pgfrcs 2010/10/25 v2.10 (rcs-revision 1.24) +)) +Package: pgf 2008/01/15 v2.10 (rcs-revision 1.12) + +(/usr/share/texmf/tex/latex/pgf/basiclayer/pgfcore.sty +(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty +Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR) + +(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty +Package: graphics 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR) + +(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty +Package: trig 1999/03/16 v1.09 sin cos tan (DPC) +) +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/graphics.cfg +File: graphics.cfg 2010/04/23 v1.9 graphics configuration of TeX Live +) +Package graphics Info: Driver file: pdftex.def on input line 91. +) +\Gin@req@height=\dimen105 +\Gin@req@width=\dimen106 +) +(/usr/share/texmf/tex/latex/pgf/systemlayer/pgfsys.sty +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys.code.tex +Package: pgfsys 2010/06/30 v2.10 (rcs-revision 1.37) + +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeys.code.tex +\pgfkeys@pathtoks=\toks17 +\pgfkeys@temptoks=\toks18 + +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex +\pgfkeys@tmptoks=\toks19 +)) +\pgf@x=\dimen107 +\pgf@y=\dimen108 +\pgf@xa=\dimen109 +\pgf@ya=\dimen110 +\pgf@xb=\dimen111 +\pgf@yb=\dimen112 +\pgf@xc=\dimen113 +\pgf@yc=\dimen114 +\w@pgf@writea=\write3 +\r@pgf@reada=\read1 +\c@pgf@counta=\count92 +\c@pgf@countb=\count93 +\c@pgf@countc=\count94 +\c@pgf@countd=\count95 + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgf.cfg +File: pgf.cfg 2008/05/14 (rcs-revision 1.7) +) +Package pgfsys Info: Driver file for pgf: pgfsys-pdftex.def on input line 900. + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys-pdftex.def +File: pgfsys-pdftex.def 2009/05/22 (rcs-revision 1.26) + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def +File: pgfsys-common-pdf.def 2008/05/19 (rcs-revision 1.10) +))) +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +File: pgfsyssoftpath.code.tex 2008/07/18 (rcs-revision 1.7) +\pgfsyssoftpath@smallbuffer@items=\count96 +\pgfsyssoftpath@bigbuffer@items=\count97 +) +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +File: pgfsysprotocol.code.tex 2006/10/16 (rcs-revision 1.4) +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcore.code.tex +Package: pgfcore 2010/04/11 v2.10 (rcs-revision 1.7) + +(/usr/share/texmf/tex/generic/pgf/math/pgfmath.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathcalc.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathutil.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathparser.code.tex +\pgfmath@dimen=\dimen115 +\pgfmath@count=\count98 +\pgfmath@box=\box31 +\pgfmath@toks=\toks20 +\pgfmath@stack@operand=\toks21 +\pgfmath@stack@operation=\toks22 +) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.random.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.base.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.round.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex))) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfloat.code.tex +\c@pgfmathroundto@lastzeros=\count99 +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex +File: pgfcorepoints.code.tex 2010/04/09 (rcs-revision 1.20) +\pgf@picminx=\dimen116 +\pgf@picmaxx=\dimen117 +\pgf@picminy=\dimen118 +\pgf@picmaxy=\dimen119 +\pgf@pathminx=\dimen120 +\pgf@pathmaxx=\dimen121 +\pgf@pathminy=\dimen122 +\pgf@pathmaxy=\dimen123 +\pgf@xx=\dimen124 +\pgf@xy=\dimen125 +\pgf@yx=\dimen126 +\pgf@yy=\dimen127 +\pgf@zx=\dimen128 +\pgf@zy=\dimen129 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex +File: pgfcorepathconstruct.code.tex 2010/08/03 (rcs-revision 1.24) +\pgf@path@lastx=\dimen130 +\pgf@path@lasty=\dimen131 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex +File: pgfcorepathusage.code.tex 2008/04/22 (rcs-revision 1.12) +\pgf@shorten@end@additional=\dimen132 +\pgf@shorten@start@additional=\dimen133 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex +File: pgfcorescopes.code.tex 2010/09/08 (rcs-revision 1.34) +\pgfpic=\box32 +\pgf@hbox=\box33 +\pgf@layerbox@main=\box34 +\pgf@picture@serial@count=\count100 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex +File: pgfcoregraphicstate.code.tex 2008/04/22 (rcs-revision 1.9) +\pgflinewidth=\dimen134 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex +File: pgfcoretransformations.code.tex 2009/06/10 (rcs-revision 1.11) +\pgf@pt@x=\dimen135 +\pgf@pt@y=\dimen136 +\pgf@pt@temp=\dimen137 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorequick.code.tex +File: pgfcorequick.code.tex 2008/10/09 (rcs-revision 1.3) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex +File: pgfcoreobjects.code.tex 2006/10/11 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex +File: pgfcorepathprocessing.code.tex 2008/10/09 (rcs-revision 1.8) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex +File: pgfcorearrows.code.tex 2008/04/23 (rcs-revision 1.11) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex +File: pgfcoreshade.code.tex 2008/11/23 (rcs-revision 1.13) +\pgf@max=\dimen138 +\pgf@sys@shading@range@num=\count101 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex +File: pgfcoreimage.code.tex 2010/03/25 (rcs-revision 1.16) + +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex +File: pgfcoreexternal.code.tex 2010/09/01 (rcs-revision 1.17) +\pgfexternal@startupbox=\box35 +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex +File: pgfcorelayers.code.tex 2010/08/27 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex +File: pgfcoretransparency.code.tex 2008/01/17 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex +File: pgfcorepatterns.code.tex 2009/07/02 (rcs-revision 1.3) +))) +(/usr/share/texmf/tex/generic/pgf/modules/pgfmoduleshapes.code.tex +File: pgfmoduleshapes.code.tex 2010/09/09 (rcs-revision 1.13) +\pgfnodeparttextbox=\box36 +) +(/usr/share/texmf/tex/generic/pgf/modules/pgfmoduleplot.code.tex +File: pgfmoduleplot.code.tex 2010/10/22 (rcs-revision 1.8) +) +(/usr/share/texmf/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +Package: pgfcomp-version-0-65 2007/07/03 v2.10 (rcs-revision 1.7) +\pgf@nodesepstart=\dimen139 +\pgf@nodesepend=\dimen140 +) +(/usr/share/texmf/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +Package: pgfcomp-version-1-18 2007/07/23 v2.10 (rcs-revision 1.1) +)) +(/usr/share/texmf/tex/latex/pgf/utilities/pgffor.sty +(/usr/share/texmf/tex/latex/pgf/utilities/pgfkeys.sty +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeys.code.tex)) +(/usr/share/texmf/tex/generic/pgf/utilities/pgffor.code.tex +Package: pgffor 2010/03/23 v2.10 (rcs-revision 1.18) +\pgffor@iter=\dimen141 +\pgffor@skip=\dimen142 +\pgffor@stack=\toks23 +\pgffor@toks=\toks24 +)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +Package: tikz 2010/10/13 v2.10 (rcs-revision 1.76) + +(/usr/share/texmf/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex +File: pgflibraryplothandlers.code.tex 2010/05/31 v2.10 (rcs-revision 1.15) +\pgf@plot@mark@count=\count102 +\pgfplotmarksize=\dimen143 +) +\tikz@lastx=\dimen144 +\tikz@lasty=\dimen145 +\tikz@lastxsaved=\dimen146 +\tikz@lastysaved=\dimen147 +\tikzleveldistance=\dimen148 +\tikzsiblingdistance=\dimen149 +\tikz@figbox=\box37 +\tikz@tempbox=\box38 +\tikztreelevel=\count103 +\tikznumberofchildren=\count104 +\tikznumberofcurrentchild=\count105 +\tikz@fig@count=\count106 + +(/usr/share/texmf/tex/generic/pgf/modules/pgfmodulematrix.code.tex +File: pgfmodulematrix.code.tex 2010/08/24 (rcs-revision 1.4) +\pgfmatrixcurrentrow=\count107 +\pgfmatrixcurrentcolumn=\count108 +\pgf@matrix@numberofcolumns=\count109 +) +\tikz@expandcount=\count110 + +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopat +hs.code.tex +File: tikzlibrarytopaths.code.tex 2008/06/17 v2.10 (rcs-revision 1.2) +))) (/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty +Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC) +) +(/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +Package: fontenc 2005/09/27 v1.99g Standard LaTeX package + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.def +File: t1enc.def 2005/09/27 v1.99g Standard LaTeX file +LaTeX Font Info: Redeclaring font encoding T1 on input line 43. +)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecor +ations.code.tex +(/usr/share/texmf/tex/generic/pgf/modules/pgfmoduledecorations.code.tex +\pgfdecoratedcompleteddistance=\dimen150 +\pgfdecoratedremainingdistance=\dimen151 +\pgfdecoratedinputsegmentcompleteddistance=\dimen152 +\pgfdecoratedinputsegmentremainingdistance=\dimen153 +\pgf@decorate@distancetomove=\dimen154 +\pgf@decorate@repeatstate=\count111 +\pgfdecorationsegmentamplitude=\dimen155 +\pgfdecorationsegmentlength=\dimen156 +) +\tikz@lib@dec@box=\box39 +) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryfadin +gs.code.tex +File: tikzlibraryfadings.code.tex 2009/11/15 v2.10 (rcs-revision 1.2) + +(/usr/share/texmf/tex/generic/pgf/libraries/pgflibraryfadings.code.tex +File: pgflibraryfadings.code.tex 2008/02/07 v2.10 (rcs-revision 1.3) +)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarysnake +s.code.tex +File: tikzlibrarysnakes.code.tex 2008/02/05 v2.10 (rcs-revision 1.6) + + +Package tikz Warning: Snakes have been superseded by decorations. Please use th +e decoration libraries instead of the snakes library on input line 14. + + +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecor +ations.pathmorphing.code.tex +(/usr/share/texmf/tex/generic/pgf/libraries/decorations/pgflibrarydecorations.p +athmorphing.code.tex)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecor +ations.pathreplacing.code.tex +(/usr/share/texmf/tex/generic/pgf/libraries/decorations/pgflibrarydecorations.p +athreplacing.code.tex)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecor +ations.shapes.code.tex +(/usr/share/texmf/tex/generic/pgf/libraries/decorations/pgflibrarydecorations.s +hapes.code.tex))) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarycalc. +code.tex +File: tikzlibrarycalc.code.tex 2009/09/04 v2.10 (rcs-revision 1.6) +) +\baposter@movebody=\skip66 +\baposter@basemarginright=\skip67 +\baposter@basemarginleft=\skip68 + +Paperwidth=2392.87677pt +Paperheight=3383.03267pt +BasePaperwidth=681.97864pt +BasePaperheight=964.1767pt +(/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +Package: geometry 2010/09/12 v5.6 Page Geometry + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifpdf.sty +Package: ifpdf 2011/01/30 v2.3 Provides the ifpdf switch (HO) +Package ifpdf Info: pdfTeX in PDF mode is detected. +) +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifvtex.sty +Package: ifvtex 2010/03/01 v1.5 Detect VTeX and its facilities (HO) +Package ifvtex Info: VTeX not detected. +) +(/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional +) +\Gm@cnth=\count112 +\Gm@cntv=\count113 +\c@Gm@tempcnt=\count114 +\Gm@bindingoffset=\dimen157 +\Gm@wd@mp=\dimen158 +\Gm@odd@mp=\dimen159 +\Gm@even@mp=\dimen160 +\Gm@layoutwidth=\dimen161 +\Gm@layoutheight=\dimen162 +\Gm@layouthoffset=\dimen163 +\Gm@layoutvoffset=\dimen164 +\Gm@dimlist=\toks25 +) +(/usr/share/texmf/tex/latex/pgf/utilities/pgfpages.sty +Package: pgfpages 2004/11/05 ver 0.01 +\pgf@logicalpages=\count115 +\pgf@firstshipout=\count116 +\pgf@lastshipout=\count117 +\pgf@currentshipout=\count118 +\pgf@cpn=\count119 +\pgf@shipoutnextto=\count120 +\pgfphysicalheight=\dimen165 +\pgfphysicalwidth=\dimen166 +\pgfpages@shipoutbox=\box40 +) +\pgfpages@box@1=\box41 +) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2000/07/18 v2.13 AMS math features +\@mathmargin=\skip69 + +For additional information on amsmath, use the `?' option. +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2000/06/29 v2.01 + +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 +\@emptytoks=\toks26 +\ex@=\dimen167 +)) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d +\pmbraise@=\dimen168 +) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +Package: amsopn 1999/12/14 v2.01 operator names +) +\inf@bad=\count121 +LaTeX Info: Redefining \frac on input line 211. +\uproot@=\count122 +\leftroot@=\count123 +LaTeX Info: Redefining \overline on input line 307. +\classnum@=\count124 +\DOTSCASE@=\count125 +LaTeX Info: Redefining \ldots on input line 379. +LaTeX Info: Redefining \dots on input line 382. +LaTeX Info: Redefining \cdots on input line 467. +\Mathstrutbox@=\box42 +\strutbox@=\box43 +\big@size=\dimen169 +LaTeX Font Info: Redeclaring font encoding OML on input line 567. +LaTeX Font Info: Redeclaring font encoding OMS on input line 568. +\macc@depth=\count126 +\c@MaxMatrixCols=\count127 +\dotsspace@=\muskip10 +\c@parentequation=\count128 +\dspbrk@lvl=\count129 +\tag@help=\toks27 +\row@=\count130 +\column@=\count131 +\maxfields@=\count132 +\andhelp@=\toks28 +\eqnshift@=\dimen170 +\alignsep@=\dimen171 +\tagshift@=\dimen172 +\tagwidth@=\dimen173 +\totwidth@=\dimen174 +\lineht@=\dimen175 +\@envbody=\toks29 +\multlinegap=\skip70 +\multlinetaggap=\skip71 +\mathdisplay@stack=\toks30 +LaTeX Info: Redefining \[ on input line 2666. +LaTeX Info: Redefining \] on input line 2667. +) +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +Package: amssymb 2009/06/22 v3.00 + +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +Package: amsfonts 2009/06/22 v3.00 Basic AMSFonts support +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 96. +)) +(/usr/share/texlive/texmf-dist/tex/latex/relsize/relsize.sty +Package: relsize 2011/09/21 ver 4.0 + +Examine \normalsize starts \@setfontsize size may be \@xpt. +Examine \small starts \@setfontsize size may be \@ixpt. +Examine \footnotesize starts \@setfontsize size may be \@viiipt. +Examine \large starts \@setfontsize size may be \@xiipt. +Examine \Large starts \@setfontsize size may be \@xivpt. +Examine \LARGE starts \@setfontsize size may be \@xviipt. +Examine \scriptsize starts \@setfontsize size may be \@viipt. +Examine \tiny starts \@setfontsize size may be \@vpt. +Examine \huge starts \@setfontsize size may be \@xxpt. +Examine \Huge starts \@setfontsize size may be \@xxvpt. ) +(/usr/share/texlive/texmf-dist/tex/latex/multirow/multirow.sty +\bigstrutjot=\dimen176 +) +(/usr/share/texlive/texmf-dist/tex/latex/rotating/rotating.sty +Package: rotating 2009/03/28 v2.16a rotated objects in LaTeX +\c@r@tfl@t=\count133 +\rotFPtop=\skip72 +\rotFPbot=\skip73 +\rot@float@box=\box44 +\rot@mess@toks=\toks31 +) +(/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty +Package: bm 2004/02/26 v1.1c Bold Symbol Support (DPC/FMi) +\symboldoperators=\mathgroup6 +\symboldletters=\mathgroup7 +\symboldsymbols=\mathgroup8 +LaTeX Font Info: Redeclaring math alphabet \mathbf on input line 138. +LaTeX Info: Redefining \bm on input line 204. +) +(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty +\Urlmuskip=\muskip11 +Package: url 2006/04/12 ver 3.3 Verb mode for urls, etc. +) +(/var/lib/texmf/tex/generic/babel/babel.sty +Package: babel 2008/07/08 v3.8m The Babel package + +(/usr/share/texlive/texmf-dist/tex/generic/babel/frenchb.ldf +Language: frenchb 2010/08/21 v2.5a French support from the babel system + +(/usr/share/texlive/texmf-dist/tex/generic/babel/babel.def +File: babel.def 2008/07/08 v3.8m Babel common definitions +\babel@savecnt=\count134 +\U@D=\dimen177 +) +Package babel Info: Making : an active character on input line 234. +Package babel Info: Making ; an active character on input line 235. +Package babel Info: Making ! an active character on input line 236. +Package babel Info: Making ? an active character on input line 237. +\FB@Mht=\dimen178 +\std@mcc=\count135 +\dec@mcc=\count136 +\parindentFFN=\dimen179 + +************************************* +* Local config file frenchb.cfg used +* +(/usr/share/texlive/texmf-dist/tex/generic/babel/frenchb.cfg))) +(/usr/share/texlive/texmf-dist/tex/latex/carlisle/scalefnt.sty) +(/usr/share/texlive/texmf-dist/tex/latex/tools/multicol.sty +Package: multicol 2011/06/27 v1.7a multicolumn formatting (FMi) +\c@tracingmulticols=\count137 +\mult@box=\box45 +\multicol@leftmargin=\dimen180 +\c@unbalance=\count138 +\c@collectmore=\count139 +\doublecol@number=\count140 +\multicoltolerance=\count141 +\multicolpretolerance=\count142 +\full@width=\dimen181 +\page@free=\dimen182 +\premulticols=\dimen183 +\postmulticols=\dimen184 +\multicolsep=\skip74 +\multicolbaselineskip=\skip75 +\partial@page=\box46 +\last@line=\box47 +\mult@rightbox=\box48 +\mult@grightbox=\box49 +\mult@gfirstbox=\box50 +\mult@firstbox=\box51 +\@tempa=\box52 +\@tempa=\box53 +\@tempa=\box54 +\@tempa=\box55 +\@tempa=\box56 +\@tempa=\box57 +\@tempa=\box58 +\@tempa=\box59 +\@tempa=\box60 +\@tempa=\box61 +\@tempa=\box62 +\@tempa=\box63 +\@tempa=\box64 +\@tempa=\box65 +\@tempa=\box66 +\@tempa=\box67 +\@tempa=\box68 +\c@columnbadness=\count143 +\c@finalcolumnbadness=\count144 +\last@try=\dimen185 +\multicolovershoot=\dimen186 +\multicolundershoot=\dimen187 +\mult@nat@firstbox=\box69 +\colbreak@box=\box70 +\multicol@sort@counter=\count145 +) +(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +Package: inputenc 2008/03/30 v1.1d Input encoding file +\inpenc@prehook=\toks32 +\inpenc@posthook=\toks33 + +(/usr/share/texlive/texmf-dist/tex/latex/base/utf8.def +File: utf8.def 2008/04/05 v1.1m UTF-8 support for inputenc +Now handling font encoding OML ... +... no UTF-8 mapping file for font encoding OML +Now handling font encoding T1 ... +... processing UTF-8 mapping file for font encoding T1 + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu +File: t1enc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A1 (decimal 161) + defining Unicode char U+00A3 (decimal 163) + defining Unicode char U+00AB (decimal 171) + defining Unicode char U+00BB (decimal 187) + defining Unicode char U+00BF (decimal 191) + defining Unicode char U+00C0 (decimal 192) + defining Unicode char U+00C1 (decimal 193) + defining Unicode char U+00C2 (decimal 194) + defining Unicode char U+00C3 (decimal 195) + defining Unicode char U+00C4 (decimal 196) + defining Unicode char U+00C5 (decimal 197) + defining Unicode char U+00C6 (decimal 198) + defining Unicode char U+00C7 (decimal 199) + defining Unicode char U+00C8 (decimal 200) + defining Unicode char U+00C9 (decimal 201) + defining Unicode char U+00CA (decimal 202) + defining Unicode char U+00CB (decimal 203) + defining Unicode char U+00CC (decimal 204) + defining Unicode char U+00CD (decimal 205) + defining Unicode char U+00CE (decimal 206) + defining Unicode char U+00CF (decimal 207) + defining Unicode char U+00D0 (decimal 208) + defining Unicode char U+00D1 (decimal 209) + defining Unicode char U+00D2 (decimal 210) + defining Unicode char U+00D3 (decimal 211) + defining Unicode char U+00D4 (decimal 212) + defining Unicode char U+00D5 (decimal 213) + defining Unicode char U+00D6 (decimal 214) + defining Unicode char U+00D8 (decimal 216) + defining Unicode char U+00D9 (decimal 217) + defining Unicode char U+00DA (decimal 218) + defining Unicode char U+00DB (decimal 219) + defining Unicode char U+00DC (decimal 220) + defining Unicode char U+00DD (decimal 221) + defining Unicode char U+00DE (decimal 222) + defining Unicode char U+00DF (decimal 223) + defining Unicode char U+00E0 (decimal 224) + defining Unicode char U+00E1 (decimal 225) + defining Unicode char U+00E2 (decimal 226) + defining Unicode char U+00E3 (decimal 227) + defining Unicode char U+00E4 (decimal 228) + defining Unicode char U+00E5 (decimal 229) + defining Unicode char U+00E6 (decimal 230) + defining Unicode char U+00E7 (decimal 231) + defining Unicode char U+00E8 (decimal 232) + defining Unicode char U+00E9 (decimal 233) + defining Unicode char U+00EA (decimal 234) + defining Unicode char U+00EB (decimal 235) + defining Unicode char U+00EC (decimal 236) + defining Unicode char U+00ED (decimal 237) + defining Unicode char U+00EE (decimal 238) + defining Unicode char U+00EF (decimal 239) + defining Unicode char U+00F0 (decimal 240) + defining Unicode char U+00F1 (decimal 241) + defining Unicode char U+00F2 (decimal 242) + defining Unicode char U+00F3 (decimal 243) + defining Unicode char U+00F4 (decimal 244) + defining Unicode char U+00F5 (decimal 245) + defining Unicode char U+00F6 (decimal 246) + defining Unicode char U+00F8 (decimal 248) + defining Unicode char U+00F9 (decimal 249) + defining Unicode char U+00FA (decimal 250) + defining Unicode char U+00FB (decimal 251) + defining Unicode char U+00FC (decimal 252) + defining Unicode char U+00FD (decimal 253) + defining Unicode char U+00FE (decimal 254) + defining Unicode char U+00FF (decimal 255) + defining Unicode char U+0102 (decimal 258) + defining Unicode char U+0103 (decimal 259) + defining Unicode char U+0104 (decimal 260) + defining Unicode char U+0105 (decimal 261) + defining Unicode char U+0106 (decimal 262) + defining Unicode char U+0107 (decimal 263) + defining Unicode char U+010C (decimal 268) + defining Unicode char U+010D (decimal 269) + defining Unicode char U+010E (decimal 270) + defining Unicode char U+010F (decimal 271) + defining Unicode char U+0110 (decimal 272) + defining Unicode char U+0111 (decimal 273) + defining Unicode char U+0118 (decimal 280) + defining Unicode char U+0119 (decimal 281) + defining Unicode char U+011A (decimal 282) + defining Unicode char U+011B (decimal 283) + defining Unicode char U+011E (decimal 286) + defining Unicode char U+011F (decimal 287) + defining Unicode char U+0130 (decimal 304) + defining Unicode char U+0131 (decimal 305) + defining Unicode char U+0132 (decimal 306) + defining Unicode char U+0133 (decimal 307) + defining Unicode char U+0139 (decimal 313) + defining Unicode char U+013A (decimal 314) + defining Unicode char U+013D (decimal 317) + defining Unicode char U+013E (decimal 318) + defining Unicode char U+0141 (decimal 321) + defining Unicode char U+0142 (decimal 322) + defining Unicode char U+0143 (decimal 323) + defining Unicode char U+0144 (decimal 324) + defining Unicode char U+0147 (decimal 327) + defining Unicode char U+0148 (decimal 328) + defining Unicode char U+014A (decimal 330) + defining Unicode char U+014B (decimal 331) + defining Unicode char U+0150 (decimal 336) + defining Unicode char U+0151 (decimal 337) + defining Unicode char U+0152 (decimal 338) + defining Unicode char U+0153 (decimal 339) + defining Unicode char U+0154 (decimal 340) + defining Unicode char U+0155 (decimal 341) + defining Unicode char U+0158 (decimal 344) + defining Unicode char U+0159 (decimal 345) + defining Unicode char U+015A (decimal 346) + defining Unicode char U+015B (decimal 347) + defining Unicode char U+015E (decimal 350) + defining Unicode char U+015F (decimal 351) + defining Unicode char U+0160 (decimal 352) + defining Unicode char U+0161 (decimal 353) + defining Unicode char U+0162 (decimal 354) + defining Unicode char U+0163 (decimal 355) + defining Unicode char U+0164 (decimal 356) + defining Unicode char U+0165 (decimal 357) + defining Unicode char U+016E (decimal 366) + defining Unicode char U+016F (decimal 367) + defining Unicode char U+0170 (decimal 368) + defining Unicode char U+0171 (decimal 369) + defining Unicode char U+0178 (decimal 376) + defining Unicode char U+0179 (decimal 377) + defining Unicode char U+017A (decimal 378) + defining Unicode char U+017B (decimal 379) + defining Unicode char U+017C (decimal 380) + defining Unicode char U+017D (decimal 381) + defining Unicode char U+017E (decimal 382) + defining Unicode char U+200C (decimal 8204) + defining Unicode char U+2013 (decimal 8211) + defining Unicode char U+2014 (decimal 8212) + defining Unicode char U+2018 (decimal 8216) + defining Unicode char U+2019 (decimal 8217) + defining Unicode char U+201A (decimal 8218) + defining Unicode char U+201C (decimal 8220) + defining Unicode char U+201D (decimal 8221) + defining Unicode char U+201E (decimal 8222) + defining Unicode char U+2030 (decimal 8240) + defining Unicode char U+2031 (decimal 8241) + defining Unicode char U+2039 (decimal 8249) + defining Unicode char U+203A (decimal 8250) + defining Unicode char U+2423 (decimal 9251) +) +Now handling font encoding OT1 ... +... processing UTF-8 mapping file for font encoding OT1 + +(/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu +File: ot1enc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A1 (decimal 161) + defining Unicode char U+00A3 (decimal 163) + defining Unicode char U+00B8 (decimal 184) + defining Unicode char U+00BF (decimal 191) + defining Unicode char U+00C5 (decimal 197) + defining Unicode char U+00C6 (decimal 198) + defining Unicode char U+00D8 (decimal 216) + defining Unicode char U+00DF (decimal 223) + defining Unicode char U+00E6 (decimal 230) + defining Unicode char U+00EC (decimal 236) + defining Unicode char U+00ED (decimal 237) + defining Unicode char U+00EE (decimal 238) + defining Unicode char U+00EF (decimal 239) + defining Unicode char U+00F8 (decimal 248) + defining Unicode char U+0131 (decimal 305) + defining Unicode char U+0141 (decimal 321) + defining Unicode char U+0142 (decimal 322) + defining Unicode char U+0152 (decimal 338) + defining Unicode char U+0153 (decimal 339) + defining Unicode char U+2013 (decimal 8211) + defining Unicode char U+2014 (decimal 8212) + defining Unicode char U+2018 (decimal 8216) + defining Unicode char U+2019 (decimal 8217) + defining Unicode char U+201C (decimal 8220) + defining Unicode char U+201D (decimal 8221) +) +Now handling font encoding OMS ... +... processing UTF-8 mapping file for font encoding OMS + +(/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu +File: omsenc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A7 (decimal 167) + defining Unicode char U+00B6 (decimal 182) + defining Unicode char U+00B7 (decimal 183) + defining Unicode char U+2020 (decimal 8224) + defining Unicode char U+2021 (decimal 8225) + defining Unicode char U+2022 (decimal 8226) +) +Now handling font encoding OMX ... +... no UTF-8 mapping file for font encoding OMX +Now handling font encoding U ... +... no UTF-8 mapping file for font encoding U + defining Unicode char U+00A9 (decimal 169) + defining Unicode char U+00AA (decimal 170) + defining Unicode char U+00AE (decimal 174) + defining Unicode char U+00BA (decimal 186) + defining Unicode char U+02C6 (decimal 710) + defining Unicode char U+02DC (decimal 732) + defining Unicode char U+200C (decimal 8204) + defining Unicode char U+2026 (decimal 8230) + defining Unicode char U+2122 (decimal 8482) + defining Unicode char U+2423 (decimal 9251) +)) +(/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +Package: fontenc 2005/09/27 v1.99g Standard LaTeX package + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.def +File: t1enc.def 2005/09/27 v1.99g Standard LaTeX file +LaTeX Font Info: Redeclaring font encoding T1 on input line 43. +)) +(/usr/share/texlive/texmf-dist/tex/latex/psnfss/palatino.sty +Package: palatino 2005/04/12 PSNFSS-v9.2a (SPQR) +) + +LaTeX Warning: Unused global option(s): + [final]. + +(./poster_landscape.aux) +\openout1 = `poster_landscape.aux'. + +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 52. +LaTeX Font Info: ... okay on input line 52. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 52. +LaTeX Font Info: ... okay on input line 52. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 52. +LaTeX Font Info: ... okay on input line 52. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 52. +LaTeX Font Info: ... okay on input line 52. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 52. +LaTeX Font Info: ... okay on input line 52. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 52. +LaTeX Font Info: ... okay on input line 52. +LaTeX Font Info: Try loading font information for T1+ppl on input line 52. + +(/usr/share/texlive/texmf-dist/tex/latex/psnfss/t1ppl.fd +File: t1ppl.fd 2001/06/04 font definitions for T1/ppl. +) +(/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii +[Loading MPS to PDF converter (version 2006.09.02).] +\scratchcounter=\count146 +\scratchdimen=\dimen188 +\scratchbox=\box71 +\nofMPsegments=\count147 +\nofMParguments=\count148 +\everyMPshowfont=\toks34 +\MPscratchCnt=\count149 +\MPscratchDim=\dimen189 +\MPnumerator=\count150 +\makeMPintoPDFobject=\count151 +\everyMPtoPDFconversion=\toks35 +) ABD: EveryShipout initializing macros +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/pdftexcmds.sty +Package: pdftexcmds 2011/11/29 v0.20 Utility functions of pdfTeX for LuaTeX (HO +) + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifluatex.sty +Package: ifluatex 2010/03/01 v1.3 Provides the ifluatex switch (HO) +Package ifluatex Info: LuaTeX not detected. +) +Package pdftexcmds Info: LuaTeX not detected. +Package pdftexcmds Info: \pdf@primitive is available. +Package pdftexcmds Info: \pdf@ifprimitive is available. +Package pdftexcmds Info: \pdfdraftmode found. +) +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/epstopdf-base.sty +Package: epstopdf-base 2010/02/09 v2.5 Base part for package epstopdf + +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/grfext.sty +Package: grfext 2010/08/19 v1.1 Manage graphics extensions (HO) + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvdefinekeys.sty +Package: kvdefinekeys 2011/04/07 v1.3 Define keys (HO) +)) +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty +Package: kvoptions 2011/06/30 v3.11 Key value format for package options (HO) + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvsetkeys.sty +Package: kvsetkeys 2012/04/25 v1.16 Key value parser (HO) + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/etexcmds.sty +Package: etexcmds 2011/02/16 v1.5 Avoid name clashes with e-TeX commands (HO) +Package etexcmds Info: Could not find \expanded. +(etexcmds) That can mean that you are not using pdfTeX 1.50 or +(etexcmds) that some package has redefined \expanded. +(etexcmds) In the latter case, load this package earlier. +))) +Package grfext Info: Graphics extension search list: +(grfext) [.png,.pdf,.jpg,.mps,.jpeg,.jbig2,.jb2,.PNG,.PDF,.JPG,.JPE +G,.JBIG2,.JB2,.eps] +(grfext) \AppendGraphicsExtensions on input line 452. + +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg +File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv +e +)) +*geometry* driver: auto-detecting +*geometry* detected driver: pdftex +*geometry* verbose mode - [ preamble ] result: +* driver: pdftex +* paper: custom +* layout: +* layoutoffset:(h,v)=(0.0pt,0.0pt) +* modes: landscape +* h-part:(L,W,R)=(12.1637pt, 939.8493pt, 12.1637pt) +* v-part:(T,H,B)=(12.1637pt, 657.65125pt, 12.1637pt) +* \paperwidth=964.1767pt +* \paperheight=681.97864pt +* \textwidth=939.8493pt +* \textheight=657.65125pt +* \oddsidemargin=-60.1063pt +* \evensidemargin=-60.1063pt +* \topmargin=-97.1063pt +* \headheight=12.0pt +* \headsep=25.0pt +* \topskip=10.0pt +* \footskip=30.0pt +* \marginparwidth=65.0pt +* \marginparsep=11.0pt +* \columnsep=14.99634pt +* \skip\footins=9.0pt plus 4.0pt minus 2.0pt +* \hoffset=0.0pt +* \voffset=0.0pt +* \mag=1000 +* \@twocolumnfalse +* \@twosidefalse +* \@mparswitchfalse +* \@reversemarginfalse +* (1in=72.27pt=25.4mm, 1cm=28.453pt) + +LaTeX Info: Redefining \degres on input line 52. +LaTeX Info: Redefining \dots on input line 52. +LaTeX Info: Redefining \up on input line 52. + +File: images/ullong.pdf Graphic file (type pdf) + +Package pdftex.def Info: images/ullong.pdf used on input line 107. +(pdftex.def) Requested size: 120.67516pt x 50.0pt. +LaTeX Font Info: Font shape `T1/ppl/bx/n' in size <10> not available +(Font) Font shape `T1/ppl/b/n' tried instead on input line 107. +LaTeX Font Info: Font shape `T1/ppl/bx/n' in size <24.88> not available +(Font) Font shape `T1/ppl/b/n' tried instead on input line 107. +LaTeX Font Info: Font shape `T1/ppl/bx/sc' in size <24.88> not available +(Font) Font shape `T1/ppl/b/sc' tried instead on input line 107. +LaTeX Font Info: Try loading font information for U+msa on input line 107. + +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2009/06/22 v3.00 AMS symbols A +) +LaTeX Font Info: Try loading font information for U+msb on input line 107. + +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2009/06/22 v3.00 AMS symbols B +) +Underfull \hbox (badness 1910) in paragraph at lines 131--131 +\T1/ppl/m/n/9 On veut mo-dé-li-ser $\OML/cmm/m/it/9 R[]$ \T1/ppl/m/n/9 par la d +is-tri-bu-tion de La- + [] + +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +LaTeX Font Info: Font shape `T1/ppl/bx/n' in size <14.4> not available +(Font) Font shape `T1/ppl/b/n' tried instead on input line 131. +LaTeX Font Info: Font shape `T1/ppl/bx/sc' in size <14.4> not available +(Font) Font shape `T1/ppl/b/sc' tried instead on input line 131. +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! + +File: images/GraphiqueAffiche1.pdf Graphic file (type pdf) + + +Package pdftex.def Info: images/GraphiqueAffiche1.pdf used on input line 165. +(pdftex.def) Requested size: 182.0949pt x 159.33302pt. +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! + (./poster_landscape.bbl) +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! + +Underfull \hbox (badness 2103) in paragraph at lines 232--232 +[]\T1/ppl/m/n/9 Inspirés par les mo-dèles Man-del-brot, Press et + [] + +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +[2 + +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} <./images/ullong.pdf + +pdfTeX warning: pdflatex (file ./images/ullong.pdf): PDF inclusion: multiple pd +fs with page group included in a single page +> <./images/GraphiqueAffiche1.pdf>] (./poster_landscape.aux) ) +Here is how much of TeX's memory you used: + 14904 strings out of 493485 + 280344 string characters out of 3143526 + 387885 words of memory out of 3000000 + 17817 multiletter control sequences out of 15000+200000 + 32759 words of font info for 72 fonts, out of 3000000 for 9000 + 959 hyphenation exceptions out of 8191 + 63i,17n,70p,434b,1000s stack positions out of 5000i,500n,10000p,200000b,50000s +{/usr/share/texli +ve/texmf-dist/fonts/enc/dvips/base/8r.enc} + +Output written on poster_landscape.pdf (1 page, 232832 bytes). +PDF statistics: + 124 PDF objects out of 1000 (max. 8388607) + 77 compressed objects within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 125 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/poster scientifique/poster_landscape.pdf b/poster scientifique/poster_landscape.pdf new file mode 100644 index 0000000..97d9f5b Binary files /dev/null and b/poster scientifique/poster_landscape.pdf differ diff --git a/poster scientifique/poster_landscape.tex b/poster scientifique/poster_landscape.tex new file mode 100644 index 0000000..3ea222f --- /dev/null +++ b/poster scientifique/poster_landscape.tex @@ -0,0 +1,257 @@ +\documentclass[landscape,final,a0paper,fontscale=0.285]{baposter} + +\usepackage{calc} +\usepackage{graphicx} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{relsize} +\usepackage{multirow} +\usepackage{rotating} +\usepackage{bm} +\usepackage{url} +\usepackage[francais]{babel} +\usepackage{graphicx} +\usepackage{multicol} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +%\usepackage{times} +%\usepackage{helvet} +%\usepackage{bookman} +\usepackage{palatino} + +\newcommand{\captionfont}{\footnotesize} + +\graphicspath{{images/}{../images/}} +\usetikzlibrary{calc} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Some math symbols used in the text +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Multicol Settings +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\setlength{\columnsep}{1.5em} +\setlength{\columnseprule}{0mm} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Save space in lists. Use this after the opening of the list +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newcommand{\compresslist}{% +\setlength{\itemsep}{1pt}% +\setlength{\parskip}{0pt}% +\setlength{\parsep}{0pt}% +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Begin of Document +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Here starts the poster +%%%--------------------------------------------------------------------------- +%%% Format it to your taste with the options +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Define some colors + +%\definecolor{lightblue}{cmyk}{0.83,0.24,0,0.12} +\definecolor{lightblue}{rgb}{0.145,0.6666,1} +\hyphenation{resolution occlusions} +%% +\begin{poster}% + % Poster Options + { + % Show grid to help with alignment + grid=false, + % Column spacing + colspacing=1em, + % Color style + bgColorOne=white, + bgColorTwo=white, + borderColor=lightblue, + headerColorOne=black, + headerColorTwo=lightblue, + headerFontColor=white, + boxColorOne=white, + boxColorTwo=lightblue, + % Format of textbox + textborder=roundedleft, + % Format of text header + eyecatcher=true, + headerborder=closed, + headerheight=0.1\textheight, +% textfont=\sc, An example of changing the text font + headershape=roundedright, + headershade=shadelr, + headerfont=\Large\bf\textsc, %Sans Serif + textfont={\setlength{\parindent}{1.5em}}, + boxshade=plain, +% background=shade-tb, + background=plain, + linewidth=2pt + } + % Eye Catcher + {\includegraphics[height=5em]{images/ullong.pdf}} + % Title + {\bf\textsc{Distribution de Laplace asymétrique généralisée}\vspace{0.5em}} + % Authors + {\textsc{François Pelletier, Andrew Luong} \\ \small{École d'actuariat, Université Laval}} + % University logo + {% The makebox allows the title to flow into the logo, this is a hack because of the L shaped logo. + + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Now define the boxes that make up the poster +%%%--------------------------------------------------------------------------- +%%% Each box has a name and can be placed absolutely or relatively. +%%% The only inconvenience is that you can only specify a relative position +%%% towards an already declared box. So if you have a box attached to the +%%% bottom, one to the top and a third one which should be in between, you +%%% have to specify the top and bottom boxes before you specify the middle +%%% box. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % + % A coloured circle useful as a bullet with an adjustably strong filling + \newcommand{\colouredcircle}{% + \tikz{\useasboundingbox (-0.2em,-0.32em) rectangle(0.2em,0.32em); \draw[draw=black,fill=lightblue,line width=0.03em] (0,0) circle(0.18em);}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Introduction}{name=problem,column=0,row=0}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \small{On définit le prix au temps $t$ d'un titre financier $S(t)$ et le rendement cumulé sur ce titre +\begin{equation*}L_t = R_1 + \ldots + R_{t} = \log S(t) - \log S(0). \end{equation*} +On veut modéliser $R_t$ par la distribution de Laplace asymétrique généralisée, aussi appelée variance-gamma. L'estimation paramétrique se fera à partir de la méthode GMM. +On pourra ainsi obtenir la distribution de $L_t$ par convolution, afin d'évaluer le prix d'options européennes.} + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Simulation}{name=contribution,column=1,row=0,bottomaligned=problem}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \small{Le processus de Laplace $Y(t)$ peut être représenté comme la différence de deux processus gamma $G(t)$ \cite{kotz2001laplace}. + \begin{equation*} Y \stackrel{d}{=} \theta + \frac{\sigma}{\sqrt{2}} \left( \frac{1}{\kappa} G_1 - \kappa G_2 \right) + \end{equation*} où $ G_1,G_2 \sim \Gamma\left(\alpha=\tau,\beta=1 \right)$. +Il suffit donc de simuler deux réalisations de cette variable +aléatoire pour obtenir une réalisation de la distribution Laplace +asymétrique généralisée. +} +} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\headerbox{Estimation du modèle}{name=results,column=2,span=1,row=0}{ + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \small{On utilise les conditions de moment basées sur l'espérance et la variance + \begin{equation*} + g(Y_t,\theta) = + \left[\begin{array}[]{c} + \bar{Y} - \left(\theta+\mu\tau\right) \\ + \frac{1}{T}\sum_{t=1}^T \left(Y_t - \bar{Y} \right)^2 - \tau\left(\sigma^2+\mu^2\right) + \end{array}\right] +\end{equation*} +La matrice optimale $W$ est estimée en utilisant la méthode du GMM itératif \cite{hall2005generalized}: + \begin{align*} + \hat{W}_{(1)} &= \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\hat\theta_{(0)})g(Y_t,\hat\theta_{(0)})^{\prime}\bigg)^{-1} \\ + \hat{W}_{(i)} &= \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\hat\theta_{(i-1)})g(Y_t,\hat\theta_{(i-1)})^{\prime}\bigg)^{-1} + \end{align*}} +} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Graphique}{name=references,column=3,span=1,bottomaligned=results}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \includegraphics[width=64mm,height=56mm]{GraphiqueAffiche1} +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Processus de Laplace}{name=references,column=0,above=bottom}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \small{Le processus de Laplace $Y(t)$ est un processus de Lévy subordonné, c'est-à-dire qu'il est formé de deux processus de Lévy, l'un qui détermine l'amplitude des évènements (sauts) et l'autre qui détermine le temps entre les évènements. + Le premier est un processus de Wiener $W(t+\tau)-W(t) \sim N(0,\tau\sigma^2)$ et le second un processus gamma $G(t+\tau)-G(t) \sim \Gamma(\tau,\nu\tau)$. + \begin{equation*}Y(t) = W(G(t))\end{equation*}} + } +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Méthode GMM}{name=questions,column=1,span=1,aligned=references,above=bottom}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + La méthode GMM a pour objectif d'estimer les paramètres $\theta$ d'une distribution en minimisant une norme quadratique +\begin{equation*} + \|\hat{m}\left(\theta\right) \|_W^2 = \hat{m}\left(\theta\right)^{\prime} W \hat{m}\left(\theta\right) +\end{equation*} +pondérée par une matrice définie positive $W$ et où $\hat{m}\left(\theta\right) = \left(\frac{1}{T}\sum_{t=1}^T g(Y_t,\theta)\right)$ est la moyenne empirique des conditions de moment $g(Y_t,\theta)$. + } +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Références}{name=references,column=2,span=2,aligned=references,above=bottom}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \bibliographystyle{plain} % (uses file "plain.bst") + \bibliography{biblio} % expects file "myrefs.bib" + } +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\headerbox{Prix d'options}{name=speed,column=2,span=2,row=0,below=results,above=references}{ + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \small{L'évaluation du prix d'un option de vente européenne équivaut à calculer la valeur espérée de la réclamation contingente définie par le contrat + \begin{equation} + P(Y(t),T-t) = B(t,T) \int_{0}^{K} (K-Y(T)) d\hat{F}_t(Y(T)) + \end{equation} + où $B(t,T)$ est la valeur d'une obligation zéro-coupon au taux sans + risque $r$ d'échéance $T$ et $K$ le prix d'exercice. Cette évaluation doit de faire sous une mesure neutre au risque, selon laquelle les investisseurs n'exigent pas une prime de risque. Cette mesure doit répondre à la propriété martingale. Pour la distribution étudiée, ceci se fait par une modification du paramètre de dérive $\theta$ en $\theta^{\star}$ + \begin{align*} + e^{rt} = E\left[exp(L_t) \right] = M_{Y_1+\ldots+Y_t}(1) = (\phi_{Y}(-i))^t = \left(\frac{e^{\theta^{\star}}}{(1-\mu-\sigma^2/2)^{\tau}} \right)^t \Rightarrow \theta^{\star} = r + \tau\log(1-\mu-\frac{\sigma^2}{2}) + \end{align*} + On pourra ensuite évaluer la valeur du prix de l'option de vente avec la formule de Heston (1993) décrite dans \cite{epps2007pricing} + \begin{align*} + P(Y(t),T-t) &= B(t,T)K \int_{0}^{K} d{F}_t(Y(T)) - B(t,T)\int_{0}^{K} Y(T)\cdot d{F}_t(Y(T)) = B(t,T)K \hat{F}_t(K) - Y(t)\hat{G}_t(K) + \end{align*} +où $\hat{G}_t(K)$ est la fonction de répartition de la transformée d'Esscher (h=1) de la mesure neutre au risque. + }} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Choix du modèle}{name=method,column=0,below=problem,bottomaligned=speed}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \small{Inspirés par les modèles Mandelbrot, Press et Praetz , Madan et Seneta \cite{madan1990variance} présentent un ensemble de caractéristiques essentielles pour un modèle de rendements financiers:} + \small{\begin{enumerate} + \item Distribution de $R_t$ ayant une queue longue + \item Distribution de $R_t$ ayant des moments finis + \item Processus en temps continu ayant des accroissements stationnaires et indépendants. Distribution des accroissement de même famille peu importe la longueur. + \item Extension multivariée afin de conserver la validité du modèle CAPM. + \end{enumerate}} + \small{Ce tableau décrit le respect des conditions pour les différents modèles étudiés par les auteurs} +\begin{center} + \begin{tabular}{|l|cccc|} + \hline + & \multicolumn{4}{c|}{\scriptsize{Conditions}} \\ + \hline + \scriptsize{Modèle} & \scriptsize{1} & \scriptsize{2} & \scriptsize{3} & \scriptsize{4} \\ + \hline + \scriptsize{Mandelbrot} & \scriptsize{X} & & & \scriptsize{X} \\ + \scriptsize{Press} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} \\ + \scriptsize{Praetz} & \scriptsize{X} & \scriptsize{X} & & \scriptsize{X} \\ + \scriptsize{Madan et Seneta} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} \\ + \hline + \end{tabular} +\end{center} + } +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Fonction caractéristique}{name=background model,column=1,below=problem,bottomaligned=speed}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \small{On définit la variable aléatoire $Y$ + \begin{equation*} + \label{eq:defvarY-GAL} + Y = \theta + \mu W + \sigma \sqrt{W} Z + \end{equation*} + La fonction caractéristique de la distribution de $Y$ peut être + obtenue en utilisant la formule de l'espérance conditionnelle. + \begin{align*} + \phi_Y&(t;\theta,\sigma,\mu,\tau) = E\left[e^{ity}\right] \\ + &= E\left[E\left[e^{ity} | W \right] \right] \\ + &= \int_0^{\infty} E \left[ e^{it(\theta + \mu w+\sigma\sqrt{w}Z)} \right] g(w) dw \\ + &= e^{i\theta t}\int_0^{\infty} e^{ i\mu w t-\frac{\sigma^2t^2w}{2}} \times \frac{1}{\Gamma (\tau)} w^{\tau-1}e^{-w} \\ + &= \frac{e^{i\theta t}}{\left(1+\frac{1}{2} \sigma^2 t^2 - i\mu t \right)^{\tau}} + \end{align*} + On retrouve la fonction de répartition par le théorème de Gil-Pelaez + \begin{equation*} + \label{eq:inversionfncaract} + F_Y(y) = \frac{1}{2} - \frac{1}{\pi}\int_{0}^{\infty} \frac{Im\left[e^{-ity}\phi_Y(t)\right]}{t} dt + \end{equation*}} + } +\end{poster} + +\end{document} diff --git a/poster scientifique/poster_portrait.aux b/poster scientifique/poster_portrait.aux new file mode 100644 index 0000000..4c5bfef --- /dev/null +++ b/poster scientifique/poster_portrait.aux @@ -0,0 +1,22 @@ +\relax +\catcode`:\active +\catcode`;\active +\catcode`!\active +\catcode`?\active +\citation{kotz2001laplace} +\citation{madan1990variance} +\bibstyle{plain} +\bibdata{biblio} +\bibcite{epps2007pricing}{1} +\bibcite{hall2005generalized}{2} +\bibcite{kotz2001laplace}{3} +\bibcite{madan1990variance}{4} +\citation{epps2007pricing} +\citation{hall2005generalized} +\select@language{french} +\@writefile{toc}{\select@language{french}} +\@writefile{lof}{\select@language{french}} +\@writefile{lot}{\select@language{french}} +\pgfsyspdfmark {pgfid1}{1757794}{774754} +\newlabel{eq:defvarY-GAL}{{}{2}} +\newlabel{eq:inversionfncaract}{{}{2}} diff --git a/poster scientifique/poster_portrait.bbl b/poster scientifique/poster_portrait.bbl new file mode 100644 index 0000000..b33db21 --- /dev/null +++ b/poster scientifique/poster_portrait.bbl @@ -0,0 +1,24 @@ +\begin{thebibliography}{1} + +\bibitem{epps2007pricing} +T.W. Epps. +\newblock {\em Pricing Derivative Securities}. +\newblock World Scientific Publishing Company Pte. Limited, 2007. + +\bibitem{hall2005generalized} +A.R. Hall. +\newblock {\em Generalized Method of Moments}. +\newblock Advanced Texts in Econometrics. OUP Oxford, 2005. + +\bibitem{kotz2001laplace} +S.~Kotz, T.J. Kozubowski, and K.~Podg{\'o}rski. +\newblock {\em The Laplace Distribution and Generalizations: A Revisit with + Applications to Communications, Economics, Engineering, and Finance}. +\newblock Birkh{\"a}user, 2001. + +\bibitem{madan1990variance} +Dilip~B Madan and Eugene Seneta. +\newblock The variance gamma (vg) model for share market returns. +\newblock {\em Journal of business}, pages 511--524, 1990. + +\end{thebibliography} diff --git a/poster scientifique/poster_portrait.blg b/poster scientifique/poster_portrait.blg new file mode 100644 index 0000000..5157a36 --- /dev/null +++ b/poster scientifique/poster_portrait.blg @@ -0,0 +1,46 @@ +This is BibTeX, Version 0.99d (TeX Live 2012/Debian) +Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 +The top-level auxiliary file: poster_portrait.aux +The style file: plain.bst +Database file #1: biblio.bib +You've used 4 entries, + 2118 wiz_defined-function locations, + 520 strings with 4569 characters, +and the built_in function-call counts, 1172 in all, are: += -- 113 +> -- 45 +< -- 1 ++ -- 20 +- -- 14 +* -- 59 +:= -- 199 +add.period$ -- 13 +call.type$ -- 4 +change.case$ -- 16 +chr.to.int$ -- 0 +cite$ -- 4 +duplicate$ -- 56 +empty$ -- 99 +format.name$ -- 14 +if$ -- 252 +int.to.chr$ -- 0 +int.to.str$ -- 4 +missing$ -- 7 +newline$ -- 23 +num.names$ -- 8 +pop$ -- 29 +preamble$ -- 1 +purify$ -- 15 +quote$ -- 0 +skip$ -- 38 +stack$ -- 0 +substring$ -- 54 +swap$ -- 13 +text.length$ -- 1 +text.prefix$ -- 0 +top$ -- 0 +type$ -- 10 +warning$ -- 0 +while$ -- 11 +width$ -- 5 +write$ -- 44 diff --git a/poster scientifique/poster_portrait.log b/poster scientifique/poster_portrait.log new file mode 100644 index 0000000..f36c147 --- /dev/null +++ b/poster scientifique/poster_portrait.log @@ -0,0 +1,1089 @@ +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.2.17) 14 MAR 2013 21:45 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**\input poster_portrait.tex +(./poster_portrait.tex (./baposter.cls +Document Class: baposter 2011/11/26 v2.0 baposter class +(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls +Document Class: article 2007/10/19 v1.4h Standard LaTeX document class +(/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo +File: size10.clo 2007/10/19 v1.4h Standard LaTeX file (size option) +) +\c@part=\count79 +\c@section=\count80 +\c@subsection=\count81 +\c@subsubsection=\count82 +\c@paragraph=\count83 +\c@subparagraph=\count84 +\c@figure=\count85 +\c@table=\count86 +\abovecaptionskip=\skip41 +\belowcaptionskip=\skip42 +\bibindent=\dimen102 +) +baposter: Brian Amberg, 2007, 2008, 2009, 2010, 2011 | http://www.brian-amberg. +de/uni/poster/ +baposter: Reinhold Kainhofer, 2011 | http://reinhold.kainhofer.com/ +\baposter@basepaperwidth=\skip43 +\baposter@basepaperheight=\skip44 +\baposter@basemargin=\skip45 +\baposter@finalpaperwidth=\skip46 +\baposter@finalpaperheight=\skip47 +\baposter@finalmargin=\skip48 +\headerheight=\skip49 +\colwidth=\skip50 +\colheight=\skip51 +\baposter@@colspacing=\skip52 +\baposter@box@@cornerradius=\skip53 +\baposter@box@@boxheaderheight=\skip54 +\baposter@box@@boxpadding=\skip55 +\boxstartx=\skip56 +\boxstarty=\skip57 +\boxwidth=\skip58 +\boxheight=\skip59 +\baposter@titleimage@left@width=\skip60 +\baposter@titleimage@right@width=\skip61 +\baposter@titleimage@textwidth=\skip62 +\baposter@box@content=\box26 +\baposter@titleimage@left=\box27 +\baposter@titleimage@title=\box28 +\baposter@titleimage@right=\box29 +Use Packages +(/usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty +Package: xkeyval 2008/08/13 v2.6a package option processing (HA) + +(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex +\XKV@toks=\toks14 +\XKV@tempa@toks=\toks15 +\XKV@depth=\count87 +File: xkeyval.tex 2008/08/13 v2.6a key=value parser (HA) + +(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/keyval.tex))) +(/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty +Package: calc 2007/08/22 v4.3 Infix arithmetic (KKT,FJ) +\calc@Acount=\count88 +\calc@Bcount=\count89 +\calc@Adimen=\dimen103 +\calc@Bdimen=\dimen104 +\calc@Askip=\skip63 +\calc@Bskip=\skip64 +LaTeX Info: Redefining \setlength on input line 76. +LaTeX Info: Redefining \addtolength on input line 77. +\calc@Ccount=\count90 +\calc@Cskip=\skip65 +) +(/usr/share/texmf/tex/latex/xcolor/xcolor.sty +Package: xcolor 2007/01/21 v2.11 LaTeX color extensions (UK) + +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg +File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive +) +Package xcolor Info: Driver file: pdftex.def on input line 225. + +(/usr/share/texlive/texmf-dist/tex/latex/pdftex-def/pdftex.def +File: pdftex.def 2011/05/27 v0.06d Graphics/color for pdfTeX + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty +Package: infwarerr 2010/04/08 v1.3 Providing info/warning/error messages (HO) +) +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty +Package: ltxcmds 2011/11/09 v1.22 LaTeX kernel commands for general use (HO) +) +\Gread@gobject=\count91 +) +Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1337. +Package xcolor Info: Model `hsb' substituted by `rgb' on input line 1341. +Package xcolor Info: Model `RGB' extended on input line 1353. +Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1355. +Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1356. +Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1357. +Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1358. +Package xcolor Info: Model `Gray' substituted by `gray' on input line 1359. +Package xcolor Info: Model `wave' substituted by `hsb' on input line 1360. +) +(/usr/share/texmf/tex/latex/pgf/frontendlayer/tikz.sty +(/usr/share/texmf/tex/latex/pgf/basiclayer/pgf.sty +(/usr/share/texmf/tex/latex/pgf/utilities/pgfrcs.sty +(/usr/share/texmf/tex/generic/pgf/utilities/pgfutil-common.tex +\pgfutil@everybye=\toks16 +) +(/usr/share/texmf/tex/generic/pgf/utilities/pgfutil-latex.def +\pgfutil@abb=\box30 + +(/usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty +Package: everyshi 2001/05/15 v3.00 EveryShipout Package (MS) +)) +(/usr/share/texmf/tex/generic/pgf/utilities/pgfrcs.code.tex +Package: pgfrcs 2010/10/25 v2.10 (rcs-revision 1.24) +)) +Package: pgf 2008/01/15 v2.10 (rcs-revision 1.12) + +(/usr/share/texmf/tex/latex/pgf/basiclayer/pgfcore.sty +(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty +Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR) + +(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty +Package: graphics 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR) + +(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty +Package: trig 1999/03/16 v1.09 sin cos tan (DPC) +) +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/graphics.cfg +File: graphics.cfg 2010/04/23 v1.9 graphics configuration of TeX Live +) +Package graphics Info: Driver file: pdftex.def on input line 91. +) +\Gin@req@height=\dimen105 +\Gin@req@width=\dimen106 +) +(/usr/share/texmf/tex/latex/pgf/systemlayer/pgfsys.sty +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys.code.tex +Package: pgfsys 2010/06/30 v2.10 (rcs-revision 1.37) + +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeys.code.tex +\pgfkeys@pathtoks=\toks17 +\pgfkeys@temptoks=\toks18 + +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex +\pgfkeys@tmptoks=\toks19 +)) +\pgf@x=\dimen107 +\pgf@y=\dimen108 +\pgf@xa=\dimen109 +\pgf@ya=\dimen110 +\pgf@xb=\dimen111 +\pgf@yb=\dimen112 +\pgf@xc=\dimen113 +\pgf@yc=\dimen114 +\w@pgf@writea=\write3 +\r@pgf@reada=\read1 +\c@pgf@counta=\count92 +\c@pgf@countb=\count93 +\c@pgf@countc=\count94 +\c@pgf@countd=\count95 + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgf.cfg +File: pgf.cfg 2008/05/14 (rcs-revision 1.7) +) +Package pgfsys Info: Driver file for pgf: pgfsys-pdftex.def on input line 900. + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys-pdftex.def +File: pgfsys-pdftex.def 2009/05/22 (rcs-revision 1.26) + +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def +File: pgfsys-common-pdf.def 2008/05/19 (rcs-revision 1.10) +))) +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +File: pgfsyssoftpath.code.tex 2008/07/18 (rcs-revision 1.7) +\pgfsyssoftpath@smallbuffer@items=\count96 +\pgfsyssoftpath@bigbuffer@items=\count97 +) +(/usr/share/texmf/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +File: pgfsysprotocol.code.tex 2006/10/16 (rcs-revision 1.4) +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcore.code.tex +Package: pgfcore 2010/04/11 v2.10 (rcs-revision 1.7) + +(/usr/share/texmf/tex/generic/pgf/math/pgfmath.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathcalc.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathutil.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathparser.code.tex +\pgfmath@dimen=\dimen115 +\pgfmath@count=\count98 +\pgfmath@box=\box31 +\pgfmath@toks=\toks20 +\pgfmath@stack@operand=\toks21 +\pgfmath@stack@operation=\toks22 +) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.code.tex +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.random.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.base.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.round.code.tex) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex))) +(/usr/share/texmf/tex/generic/pgf/math/pgfmathfloat.code.tex +\c@pgfmathroundto@lastzeros=\count99 +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex +File: pgfcorepoints.code.tex 2010/04/09 (rcs-revision 1.20) +\pgf@picminx=\dimen116 +\pgf@picmaxx=\dimen117 +\pgf@picminy=\dimen118 +\pgf@picmaxy=\dimen119 +\pgf@pathminx=\dimen120 +\pgf@pathmaxx=\dimen121 +\pgf@pathminy=\dimen122 +\pgf@pathmaxy=\dimen123 +\pgf@xx=\dimen124 +\pgf@xy=\dimen125 +\pgf@yx=\dimen126 +\pgf@yy=\dimen127 +\pgf@zx=\dimen128 +\pgf@zy=\dimen129 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex +File: pgfcorepathconstruct.code.tex 2010/08/03 (rcs-revision 1.24) +\pgf@path@lastx=\dimen130 +\pgf@path@lasty=\dimen131 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex +File: pgfcorepathusage.code.tex 2008/04/22 (rcs-revision 1.12) +\pgf@shorten@end@additional=\dimen132 +\pgf@shorten@start@additional=\dimen133 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex +File: pgfcorescopes.code.tex 2010/09/08 (rcs-revision 1.34) +\pgfpic=\box32 +\pgf@hbox=\box33 +\pgf@layerbox@main=\box34 +\pgf@picture@serial@count=\count100 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex +File: pgfcoregraphicstate.code.tex 2008/04/22 (rcs-revision 1.9) +\pgflinewidth=\dimen134 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex +File: pgfcoretransformations.code.tex 2009/06/10 (rcs-revision 1.11) +\pgf@pt@x=\dimen135 +\pgf@pt@y=\dimen136 +\pgf@pt@temp=\dimen137 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorequick.code.tex +File: pgfcorequick.code.tex 2008/10/09 (rcs-revision 1.3) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex +File: pgfcoreobjects.code.tex 2006/10/11 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex +File: pgfcorepathprocessing.code.tex 2008/10/09 (rcs-revision 1.8) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex +File: pgfcorearrows.code.tex 2008/04/23 (rcs-revision 1.11) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex +File: pgfcoreshade.code.tex 2008/11/23 (rcs-revision 1.13) +\pgf@max=\dimen138 +\pgf@sys@shading@range@num=\count101 +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex +File: pgfcoreimage.code.tex 2010/03/25 (rcs-revision 1.16) + +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex +File: pgfcoreexternal.code.tex 2010/09/01 (rcs-revision 1.17) +\pgfexternal@startupbox=\box35 +)) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex +File: pgfcorelayers.code.tex 2010/08/27 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex +File: pgfcoretransparency.code.tex 2008/01/17 (rcs-revision 1.2) +) +(/usr/share/texmf/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex +File: pgfcorepatterns.code.tex 2009/07/02 (rcs-revision 1.3) +))) +(/usr/share/texmf/tex/generic/pgf/modules/pgfmoduleshapes.code.tex +File: pgfmoduleshapes.code.tex 2010/09/09 (rcs-revision 1.13) +\pgfnodeparttextbox=\box36 +) +(/usr/share/texmf/tex/generic/pgf/modules/pgfmoduleplot.code.tex +File: pgfmoduleplot.code.tex 2010/10/22 (rcs-revision 1.8) +) +(/usr/share/texmf/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +Package: pgfcomp-version-0-65 2007/07/03 v2.10 (rcs-revision 1.7) +\pgf@nodesepstart=\dimen139 +\pgf@nodesepend=\dimen140 +) +(/usr/share/texmf/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +Package: pgfcomp-version-1-18 2007/07/23 v2.10 (rcs-revision 1.1) +)) +(/usr/share/texmf/tex/latex/pgf/utilities/pgffor.sty +(/usr/share/texmf/tex/latex/pgf/utilities/pgfkeys.sty +(/usr/share/texmf/tex/generic/pgf/utilities/pgfkeys.code.tex)) +(/usr/share/texmf/tex/generic/pgf/utilities/pgffor.code.tex +Package: pgffor 2010/03/23 v2.10 (rcs-revision 1.18) +\pgffor@iter=\dimen141 +\pgffor@skip=\dimen142 +\pgffor@stack=\toks23 +\pgffor@toks=\toks24 +)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +Package: tikz 2010/10/13 v2.10 (rcs-revision 1.76) + +(/usr/share/texmf/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex +File: pgflibraryplothandlers.code.tex 2010/05/31 v2.10 (rcs-revision 1.15) +\pgf@plot@mark@count=\count102 +\pgfplotmarksize=\dimen143 +) +\tikz@lastx=\dimen144 +\tikz@lasty=\dimen145 +\tikz@lastxsaved=\dimen146 +\tikz@lastysaved=\dimen147 +\tikzleveldistance=\dimen148 +\tikzsiblingdistance=\dimen149 +\tikz@figbox=\box37 +\tikz@tempbox=\box38 +\tikztreelevel=\count103 +\tikznumberofchildren=\count104 +\tikznumberofcurrentchild=\count105 +\tikz@fig@count=\count106 + +(/usr/share/texmf/tex/generic/pgf/modules/pgfmodulematrix.code.tex +File: pgfmodulematrix.code.tex 2010/08/24 (rcs-revision 1.4) +\pgfmatrixcurrentrow=\count107 +\pgfmatrixcurrentcolumn=\count108 +\pgf@matrix@numberofcolumns=\count109 +) +\tikz@expandcount=\count110 + +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopat +hs.code.tex +File: tikzlibrarytopaths.code.tex 2008/06/17 v2.10 (rcs-revision 1.2) +))) (/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty +Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC) +) +(/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +Package: fontenc 2005/09/27 v1.99g Standard LaTeX package + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.def +File: t1enc.def 2005/09/27 v1.99g Standard LaTeX file +LaTeX Font Info: Redeclaring font encoding T1 on input line 43. +)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecor +ations.code.tex +(/usr/share/texmf/tex/generic/pgf/modules/pgfmoduledecorations.code.tex +\pgfdecoratedcompleteddistance=\dimen150 +\pgfdecoratedremainingdistance=\dimen151 +\pgfdecoratedinputsegmentcompleteddistance=\dimen152 +\pgfdecoratedinputsegmentremainingdistance=\dimen153 +\pgf@decorate@distancetomove=\dimen154 +\pgf@decorate@repeatstate=\count111 +\pgfdecorationsegmentamplitude=\dimen155 +\pgfdecorationsegmentlength=\dimen156 +) +\tikz@lib@dec@box=\box39 +) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryfadin +gs.code.tex +File: tikzlibraryfadings.code.tex 2009/11/15 v2.10 (rcs-revision 1.2) + +(/usr/share/texmf/tex/generic/pgf/libraries/pgflibraryfadings.code.tex +File: pgflibraryfadings.code.tex 2008/02/07 v2.10 (rcs-revision 1.3) +)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarysnake +s.code.tex +File: tikzlibrarysnakes.code.tex 2008/02/05 v2.10 (rcs-revision 1.6) + + +Package tikz Warning: Snakes have been superseded by decorations. Please use th +e decoration libraries instead of the snakes library on input line 14. + + +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecor +ations.pathmorphing.code.tex +(/usr/share/texmf/tex/generic/pgf/libraries/decorations/pgflibrarydecorations.p +athmorphing.code.tex)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecor +ations.pathreplacing.code.tex +(/usr/share/texmf/tex/generic/pgf/libraries/decorations/pgflibrarydecorations.p +athreplacing.code.tex)) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecor +ations.shapes.code.tex +(/usr/share/texmf/tex/generic/pgf/libraries/decorations/pgflibrarydecorations.s +hapes.code.tex))) +(/usr/share/texmf/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarycalc. +code.tex +File: tikzlibrarycalc.code.tex 2009/09/04 v2.10 (rcs-revision 1.6) +) +\baposter@movebody=\skip66 +\baposter@basemarginright=\skip67 +\baposter@basemarginleft=\skip68 + +Paperwidth=2392.87677pt +Paperheight=3383.03267pt +BasePaperwidth=662.80963pt +BasePaperheight=937.07568pt +(/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +Package: geometry 2010/09/12 v5.6 Page Geometry + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifpdf.sty +Package: ifpdf 2011/01/30 v2.3 Provides the ifpdf switch (HO) +Package ifpdf Info: pdfTeX in PDF mode is detected. +) +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifvtex.sty +Package: ifvtex 2010/03/01 v1.5 Detect VTeX and its facilities (HO) +Package ifvtex Info: VTeX not detected. +) +(/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional +) +\Gm@cnth=\count112 +\Gm@cntv=\count113 +\c@Gm@tempcnt=\count114 +\Gm@bindingoffset=\dimen157 +\Gm@wd@mp=\dimen158 +\Gm@odd@mp=\dimen159 +\Gm@even@mp=\dimen160 +\Gm@layoutwidth=\dimen161 +\Gm@layoutheight=\dimen162 +\Gm@layouthoffset=\dimen163 +\Gm@layoutvoffset=\dimen164 +\Gm@dimlist=\toks25 +) +(/usr/share/texmf/tex/latex/pgf/utilities/pgfpages.sty +Package: pgfpages 2004/11/05 ver 0.01 +\pgf@logicalpages=\count115 +\pgf@firstshipout=\count116 +\pgf@lastshipout=\count117 +\pgf@currentshipout=\count118 +\pgf@cpn=\count119 +\pgf@shipoutnextto=\count120 +\pgfphysicalheight=\dimen165 +\pgfphysicalwidth=\dimen166 +\pgfpages@shipoutbox=\box40 +) +\pgfpages@box@1=\box41 +) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2000/07/18 v2.13 AMS math features +\@mathmargin=\skip69 + +For additional information on amsmath, use the `?' option. +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2000/06/29 v2.01 + +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 +\@emptytoks=\toks26 +\ex@=\dimen167 +)) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d +\pmbraise@=\dimen168 +) +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +Package: amsopn 1999/12/14 v2.01 operator names +) +\inf@bad=\count121 +LaTeX Info: Redefining \frac on input line 211. +\uproot@=\count122 +\leftroot@=\count123 +LaTeX Info: Redefining \overline on input line 307. +\classnum@=\count124 +\DOTSCASE@=\count125 +LaTeX Info: Redefining \ldots on input line 379. +LaTeX Info: Redefining \dots on input line 382. +LaTeX Info: Redefining \cdots on input line 467. +\Mathstrutbox@=\box42 +\strutbox@=\box43 +\big@size=\dimen169 +LaTeX Font Info: Redeclaring font encoding OML on input line 567. +LaTeX Font Info: Redeclaring font encoding OMS on input line 568. +\macc@depth=\count126 +\c@MaxMatrixCols=\count127 +\dotsspace@=\muskip10 +\c@parentequation=\count128 +\dspbrk@lvl=\count129 +\tag@help=\toks27 +\row@=\count130 +\column@=\count131 +\maxfields@=\count132 +\andhelp@=\toks28 +\eqnshift@=\dimen170 +\alignsep@=\dimen171 +\tagshift@=\dimen172 +\tagwidth@=\dimen173 +\totwidth@=\dimen174 +\lineht@=\dimen175 +\@envbody=\toks29 +\multlinegap=\skip70 +\multlinetaggap=\skip71 +\mathdisplay@stack=\toks30 +LaTeX Info: Redefining \[ on input line 2666. +LaTeX Info: Redefining \] on input line 2667. +) +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +Package: amssymb 2009/06/22 v3.00 + +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +Package: amsfonts 2009/06/22 v3.00 Basic AMSFonts support +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 96. +)) +(/usr/share/texlive/texmf-dist/tex/latex/relsize/relsize.sty +Package: relsize 2011/09/21 ver 4.0 + +Examine \normalsize starts \@setfontsize size may be \@xpt. +Examine \small starts \@setfontsize size may be \@ixpt. +Examine \footnotesize starts \@setfontsize size may be \@viiipt. +Examine \large starts \@setfontsize size may be \@xiipt. +Examine \Large starts \@setfontsize size may be \@xivpt. +Examine \LARGE starts \@setfontsize size may be \@xviipt. +Examine \scriptsize starts \@setfontsize size may be \@viipt. +Examine \tiny starts \@setfontsize size may be \@vpt. +Examine \huge starts \@setfontsize size may be \@xxpt. +Examine \Huge starts \@setfontsize size may be \@xxvpt. ) +(/usr/share/texlive/texmf-dist/tex/latex/multirow/multirow.sty +\bigstrutjot=\dimen176 +) +(/usr/share/texlive/texmf-dist/tex/latex/rotating/rotating.sty +Package: rotating 2009/03/28 v2.16a rotated objects in LaTeX +\c@r@tfl@t=\count133 +\rotFPtop=\skip72 +\rotFPbot=\skip73 +\rot@float@box=\box44 +\rot@mess@toks=\toks31 +) +(/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty +Package: bm 2004/02/26 v1.1c Bold Symbol Support (DPC/FMi) +\symboldoperators=\mathgroup6 +\symboldletters=\mathgroup7 +\symboldsymbols=\mathgroup8 +LaTeX Font Info: Redeclaring math alphabet \mathbf on input line 138. +LaTeX Info: Redefining \bm on input line 204. +) +(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty +\Urlmuskip=\muskip11 +Package: url 2006/04/12 ver 3.3 Verb mode for urls, etc. +) +(/var/lib/texmf/tex/generic/babel/babel.sty +Package: babel 2008/07/08 v3.8m The Babel package + +(/usr/share/texlive/texmf-dist/tex/generic/babel/frenchb.ldf +Language: frenchb 2010/08/21 v2.5a French support from the babel system + +(/usr/share/texlive/texmf-dist/tex/generic/babel/babel.def +File: babel.def 2008/07/08 v3.8m Babel common definitions +\babel@savecnt=\count134 +\U@D=\dimen177 +) +Package babel Info: Making : an active character on input line 234. +Package babel Info: Making ; an active character on input line 235. +Package babel Info: Making ! an active character on input line 236. +Package babel Info: Making ? an active character on input line 237. +\FB@Mht=\dimen178 +\std@mcc=\count135 +\dec@mcc=\count136 +\parindentFFN=\dimen179 + +************************************* +* Local config file frenchb.cfg used +* +(/usr/share/texlive/texmf-dist/tex/generic/babel/frenchb.cfg))) +(/usr/share/texlive/texmf-dist/tex/latex/carlisle/scalefnt.sty) +(/usr/share/texlive/texmf-dist/tex/latex/tools/multicol.sty +Package: multicol 2011/06/27 v1.7a multicolumn formatting (FMi) +\c@tracingmulticols=\count137 +\mult@box=\box45 +\multicol@leftmargin=\dimen180 +\c@unbalance=\count138 +\c@collectmore=\count139 +\doublecol@number=\count140 +\multicoltolerance=\count141 +\multicolpretolerance=\count142 +\full@width=\dimen181 +\page@free=\dimen182 +\premulticols=\dimen183 +\postmulticols=\dimen184 +\multicolsep=\skip74 +\multicolbaselineskip=\skip75 +\partial@page=\box46 +\last@line=\box47 +\mult@rightbox=\box48 +\mult@grightbox=\box49 +\mult@gfirstbox=\box50 +\mult@firstbox=\box51 +\@tempa=\box52 +\@tempa=\box53 +\@tempa=\box54 +\@tempa=\box55 +\@tempa=\box56 +\@tempa=\box57 +\@tempa=\box58 +\@tempa=\box59 +\@tempa=\box60 +\@tempa=\box61 +\@tempa=\box62 +\@tempa=\box63 +\@tempa=\box64 +\@tempa=\box65 +\@tempa=\box66 +\@tempa=\box67 +\@tempa=\box68 +\c@columnbadness=\count143 +\c@finalcolumnbadness=\count144 +\last@try=\dimen185 +\multicolovershoot=\dimen186 +\multicolundershoot=\dimen187 +\mult@nat@firstbox=\box69 +\colbreak@box=\box70 +\multicol@sort@counter=\count145 +) +(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +Package: inputenc 2008/03/30 v1.1d Input encoding file +\inpenc@prehook=\toks32 +\inpenc@posthook=\toks33 + +(/usr/share/texlive/texmf-dist/tex/latex/base/utf8.def +File: utf8.def 2008/04/05 v1.1m UTF-8 support for inputenc +Now handling font encoding OML ... +... no UTF-8 mapping file for font encoding OML +Now handling font encoding T1 ... +... processing UTF-8 mapping file for font encoding T1 + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu +File: t1enc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A1 (decimal 161) + defining Unicode char U+00A3 (decimal 163) + defining Unicode char U+00AB (decimal 171) + defining Unicode char U+00BB (decimal 187) + defining Unicode char U+00BF (decimal 191) + defining Unicode char U+00C0 (decimal 192) + defining Unicode char U+00C1 (decimal 193) + defining Unicode char U+00C2 (decimal 194) + defining Unicode char U+00C3 (decimal 195) + defining Unicode char U+00C4 (decimal 196) + defining Unicode char U+00C5 (decimal 197) + defining Unicode char U+00C6 (decimal 198) + defining Unicode char U+00C7 (decimal 199) + defining Unicode char U+00C8 (decimal 200) + defining Unicode char U+00C9 (decimal 201) + defining Unicode char U+00CA (decimal 202) + defining Unicode char U+00CB (decimal 203) + defining Unicode char U+00CC (decimal 204) + defining Unicode char U+00CD (decimal 205) + defining Unicode char U+00CE (decimal 206) + defining Unicode char U+00CF (decimal 207) + defining Unicode char U+00D0 (decimal 208) + defining Unicode char U+00D1 (decimal 209) + defining Unicode char U+00D2 (decimal 210) + defining Unicode char U+00D3 (decimal 211) + defining Unicode char U+00D4 (decimal 212) + defining Unicode char U+00D5 (decimal 213) + defining Unicode char U+00D6 (decimal 214) + defining Unicode char U+00D8 (decimal 216) + defining Unicode char U+00D9 (decimal 217) + defining Unicode char U+00DA (decimal 218) + defining Unicode char U+00DB (decimal 219) + defining Unicode char U+00DC (decimal 220) + defining Unicode char U+00DD (decimal 221) + defining Unicode char U+00DE (decimal 222) + defining Unicode char U+00DF (decimal 223) + defining Unicode char U+00E0 (decimal 224) + defining Unicode char U+00E1 (decimal 225) + defining Unicode char U+00E2 (decimal 226) + defining Unicode char U+00E3 (decimal 227) + defining Unicode char U+00E4 (decimal 228) + defining Unicode char U+00E5 (decimal 229) + defining Unicode char U+00E6 (decimal 230) + defining Unicode char U+00E7 (decimal 231) + defining Unicode char U+00E8 (decimal 232) + defining Unicode char U+00E9 (decimal 233) + defining Unicode char U+00EA (decimal 234) + defining Unicode char U+00EB (decimal 235) + defining Unicode char U+00EC (decimal 236) + defining Unicode char U+00ED (decimal 237) + defining Unicode char U+00EE (decimal 238) + defining Unicode char U+00EF (decimal 239) + defining Unicode char U+00F0 (decimal 240) + defining Unicode char U+00F1 (decimal 241) + defining Unicode char U+00F2 (decimal 242) + defining Unicode char U+00F3 (decimal 243) + defining Unicode char U+00F4 (decimal 244) + defining Unicode char U+00F5 (decimal 245) + defining Unicode char U+00F6 (decimal 246) + defining Unicode char U+00F8 (decimal 248) + defining Unicode char U+00F9 (decimal 249) + defining Unicode char U+00FA (decimal 250) + defining Unicode char U+00FB (decimal 251) + defining Unicode char U+00FC (decimal 252) + defining Unicode char U+00FD (decimal 253) + defining Unicode char U+00FE (decimal 254) + defining Unicode char U+00FF (decimal 255) + defining Unicode char U+0102 (decimal 258) + defining Unicode char U+0103 (decimal 259) + defining Unicode char U+0104 (decimal 260) + defining Unicode char U+0105 (decimal 261) + defining Unicode char U+0106 (decimal 262) + defining Unicode char U+0107 (decimal 263) + defining Unicode char U+010C (decimal 268) + defining Unicode char U+010D (decimal 269) + defining Unicode char U+010E (decimal 270) + defining Unicode char U+010F (decimal 271) + defining Unicode char U+0110 (decimal 272) + defining Unicode char U+0111 (decimal 273) + defining Unicode char U+0118 (decimal 280) + defining Unicode char U+0119 (decimal 281) + defining Unicode char U+011A (decimal 282) + defining Unicode char U+011B (decimal 283) + defining Unicode char U+011E (decimal 286) + defining Unicode char U+011F (decimal 287) + defining Unicode char U+0130 (decimal 304) + defining Unicode char U+0131 (decimal 305) + defining Unicode char U+0132 (decimal 306) + defining Unicode char U+0133 (decimal 307) + defining Unicode char U+0139 (decimal 313) + defining Unicode char U+013A (decimal 314) + defining Unicode char U+013D (decimal 317) + defining Unicode char U+013E (decimal 318) + defining Unicode char U+0141 (decimal 321) + defining Unicode char U+0142 (decimal 322) + defining Unicode char U+0143 (decimal 323) + defining Unicode char U+0144 (decimal 324) + defining Unicode char U+0147 (decimal 327) + defining Unicode char U+0148 (decimal 328) + defining Unicode char U+014A (decimal 330) + defining Unicode char U+014B (decimal 331) + defining Unicode char U+0150 (decimal 336) + defining Unicode char U+0151 (decimal 337) + defining Unicode char U+0152 (decimal 338) + defining Unicode char U+0153 (decimal 339) + defining Unicode char U+0154 (decimal 340) + defining Unicode char U+0155 (decimal 341) + defining Unicode char U+0158 (decimal 344) + defining Unicode char U+0159 (decimal 345) + defining Unicode char U+015A (decimal 346) + defining Unicode char U+015B (decimal 347) + defining Unicode char U+015E (decimal 350) + defining Unicode char U+015F (decimal 351) + defining Unicode char U+0160 (decimal 352) + defining Unicode char U+0161 (decimal 353) + defining Unicode char U+0162 (decimal 354) + defining Unicode char U+0163 (decimal 355) + defining Unicode char U+0164 (decimal 356) + defining Unicode char U+0165 (decimal 357) + defining Unicode char U+016E (decimal 366) + defining Unicode char U+016F (decimal 367) + defining Unicode char U+0170 (decimal 368) + defining Unicode char U+0171 (decimal 369) + defining Unicode char U+0178 (decimal 376) + defining Unicode char U+0179 (decimal 377) + defining Unicode char U+017A (decimal 378) + defining Unicode char U+017B (decimal 379) + defining Unicode char U+017C (decimal 380) + defining Unicode char U+017D (decimal 381) + defining Unicode char U+017E (decimal 382) + defining Unicode char U+200C (decimal 8204) + defining Unicode char U+2013 (decimal 8211) + defining Unicode char U+2014 (decimal 8212) + defining Unicode char U+2018 (decimal 8216) + defining Unicode char U+2019 (decimal 8217) + defining Unicode char U+201A (decimal 8218) + defining Unicode char U+201C (decimal 8220) + defining Unicode char U+201D (decimal 8221) + defining Unicode char U+201E (decimal 8222) + defining Unicode char U+2030 (decimal 8240) + defining Unicode char U+2031 (decimal 8241) + defining Unicode char U+2039 (decimal 8249) + defining Unicode char U+203A (decimal 8250) + defining Unicode char U+2423 (decimal 9251) +) +Now handling font encoding OT1 ... +... processing UTF-8 mapping file for font encoding OT1 + +(/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu +File: ot1enc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A1 (decimal 161) + defining Unicode char U+00A3 (decimal 163) + defining Unicode char U+00B8 (decimal 184) + defining Unicode char U+00BF (decimal 191) + defining Unicode char U+00C5 (decimal 197) + defining Unicode char U+00C6 (decimal 198) + defining Unicode char U+00D8 (decimal 216) + defining Unicode char U+00DF (decimal 223) + defining Unicode char U+00E6 (decimal 230) + defining Unicode char U+00EC (decimal 236) + defining Unicode char U+00ED (decimal 237) + defining Unicode char U+00EE (decimal 238) + defining Unicode char U+00EF (decimal 239) + defining Unicode char U+00F8 (decimal 248) + defining Unicode char U+0131 (decimal 305) + defining Unicode char U+0141 (decimal 321) + defining Unicode char U+0142 (decimal 322) + defining Unicode char U+0152 (decimal 338) + defining Unicode char U+0153 (decimal 339) + defining Unicode char U+2013 (decimal 8211) + defining Unicode char U+2014 (decimal 8212) + defining Unicode char U+2018 (decimal 8216) + defining Unicode char U+2019 (decimal 8217) + defining Unicode char U+201C (decimal 8220) + defining Unicode char U+201D (decimal 8221) +) +Now handling font encoding OMS ... +... processing UTF-8 mapping file for font encoding OMS + +(/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu +File: omsenc.dfu 2008/04/05 v1.1m UTF-8 support for inputenc + defining Unicode char U+00A7 (decimal 167) + defining Unicode char U+00B6 (decimal 182) + defining Unicode char U+00B7 (decimal 183) + defining Unicode char U+2020 (decimal 8224) + defining Unicode char U+2021 (decimal 8225) + defining Unicode char U+2022 (decimal 8226) +) +Now handling font encoding OMX ... +... no UTF-8 mapping file for font encoding OMX +Now handling font encoding U ... +... no UTF-8 mapping file for font encoding U + defining Unicode char U+00A9 (decimal 169) + defining Unicode char U+00AA (decimal 170) + defining Unicode char U+00AE (decimal 174) + defining Unicode char U+00BA (decimal 186) + defining Unicode char U+02C6 (decimal 710) + defining Unicode char U+02DC (decimal 732) + defining Unicode char U+200C (decimal 8204) + defining Unicode char U+2026 (decimal 8230) + defining Unicode char U+2122 (decimal 8482) + defining Unicode char U+2423 (decimal 9251) +)) +(/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +Package: fontenc 2005/09/27 v1.99g Standard LaTeX package + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.def +File: t1enc.def 2005/09/27 v1.99g Standard LaTeX file +LaTeX Font Info: Redeclaring font encoding T1 on input line 43. +)) +(/usr/share/texlive/texmf-dist/tex/latex/psnfss/palatino.sty +Package: palatino 2005/04/12 PSNFSS-v9.2a (SPQR) +) + +LaTeX Warning: Unused global option(s): + [final]. + +(./poster_portrait.aux) +\openout1 = `poster_portrait.aux'. + +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 53. +LaTeX Font Info: ... okay on input line 53. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 53. +LaTeX Font Info: ... okay on input line 53. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 53. +LaTeX Font Info: ... okay on input line 53. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 53. +LaTeX Font Info: ... okay on input line 53. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 53. +LaTeX Font Info: ... okay on input line 53. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 53. +LaTeX Font Info: ... okay on input line 53. +LaTeX Font Info: Try loading font information for T1+ppl on input line 53. + +(/usr/share/texlive/texmf-dist/tex/latex/psnfss/t1ppl.fd +File: t1ppl.fd 2001/06/04 font definitions for T1/ppl. +) +(/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii +[Loading MPS to PDF converter (version 2006.09.02).] +\scratchcounter=\count146 +\scratchdimen=\dimen188 +\scratchbox=\box71 +\nofMPsegments=\count147 +\nofMParguments=\count148 +\everyMPshowfont=\toks34 +\MPscratchCnt=\count149 +\MPscratchDim=\dimen189 +\MPnumerator=\count150 +\makeMPintoPDFobject=\count151 +\everyMPtoPDFconversion=\toks35 +) ABD: EveryShipout initializing macros +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/pdftexcmds.sty +Package: pdftexcmds 2011/11/29 v0.20 Utility functions of pdfTeX for LuaTeX (HO +) + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifluatex.sty +Package: ifluatex 2010/03/01 v1.3 Provides the ifluatex switch (HO) +Package ifluatex Info: LuaTeX not detected. +) +Package pdftexcmds Info: LuaTeX not detected. +Package pdftexcmds Info: \pdf@primitive is available. +Package pdftexcmds Info: \pdf@ifprimitive is available. +Package pdftexcmds Info: \pdfdraftmode found. +) +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/epstopdf-base.sty +Package: epstopdf-base 2010/02/09 v2.5 Base part for package epstopdf + +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/grfext.sty +Package: grfext 2010/08/19 v1.1 Manage graphics extensions (HO) + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvdefinekeys.sty +Package: kvdefinekeys 2011/04/07 v1.3 Define keys (HO) +)) +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty +Package: kvoptions 2011/06/30 v3.11 Key value format for package options (HO) + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvsetkeys.sty +Package: kvsetkeys 2012/04/25 v1.16 Key value parser (HO) + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/etexcmds.sty +Package: etexcmds 2011/02/16 v1.5 Avoid name clashes with e-TeX commands (HO) +Package etexcmds Info: Could not find \expanded. +(etexcmds) That can mean that you are not using pdfTeX 1.50 or +(etexcmds) that some package has redefined \expanded. +(etexcmds) In the latter case, load this package earlier. +))) +Package grfext Info: Graphics extension search list: +(grfext) [.png,.pdf,.jpg,.mps,.jpeg,.jbig2,.jb2,.PNG,.PDF,.JPG,.JPE +G,.JBIG2,.JB2,.eps] +(grfext) \AppendGraphicsExtensions on input line 452. + +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg +File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv +e +)) +*geometry* driver: auto-detecting +*geometry* detected driver: pdftex +*geometry* verbose mode - [ preamble ] result: +* driver: pdftex +* paper: custom +* layout: +* layoutoffset:(h,v)=(0.0pt,0.0pt) +* modes: +* h-part:(L,W,R)=(11.82181pt, 639.16602pt, 11.82181pt) +* v-part:(T,H,B)=(11.82181pt, 913.43207pt, 11.82181pt) +* \paperwidth=662.80963pt +* \paperheight=937.07568pt +* \textwidth=639.16602pt +* \textheight=913.43207pt +* \oddsidemargin=-60.44818pt +* \evensidemargin=-60.44818pt +* \topmargin=-97.44818pt +* \headheight=12.0pt +* \headsep=25.0pt +* \topskip=10.0pt +* \footskip=30.0pt +* \marginparwidth=65.0pt +* \marginparsep=11.0pt +* \columnsep=14.99634pt +* \skip\footins=9.0pt plus 4.0pt minus 2.0pt +* \hoffset=0.0pt +* \voffset=0.0pt +* \mag=1000 +* \@twocolumnfalse +* \@twosidefalse +* \@mparswitchfalse +* \@reversemarginfalse +* (1in=72.27pt=25.4mm, 1cm=28.453pt) + +LaTeX Info: Redefining \degres on input line 53. +LaTeX Info: Redefining \dots on input line 53. +LaTeX Info: Redefining \up on input line 53. + +File: images/ullong.pdf Graphic file (type pdf) + +Package pdftex.def Info: images/ullong.pdf used on input line 107. +(pdftex.def) Requested size: 120.67516pt x 50.0pt. +LaTeX Font Info: Font shape `T1/ppl/bx/n' in size <10> not available +(Font) Font shape `T1/ppl/b/n' tried instead on input line 107. +LaTeX Font Info: Font shape `T1/ppl/bx/n' in size <24.88> not available +(Font) Font shape `T1/ppl/b/n' tried instead on input line 107. +LaTeX Font Info: Font shape `T1/ppl/bx/sc' in size <24.88> not available +(Font) Font shape `T1/ppl/b/sc' tried instead on input line 107. +LaTeX Font Info: Try loading font information for U+msa on input line 107. + +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2009/06/22 v3.00 AMS symbols A +) +LaTeX Font Info: Try loading font information for U+msb on input line 107. + +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2009/06/22 v3.00 AMS symbols B +) +Underfull \hbox (badness 1454) in paragraph at lines 131--131 +\T1/ppl/m/n/9 On veut mo-dé-li-ser $\OML/cmm/m/it/9 R[]$ \T1/ppl/m/n/9 par la d +is-tri-bu-tion de + [] + + +Underfull \hbox (badness 1194) in paragraph at lines 131--131 +\T1/ppl/m/n/9 variance-gamma. L'es-ti-ma-tion pa-ra-mé-trique se + [] + +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +LaTeX Font Info: Font shape `T1/ppl/bx/n' in size <14.4> not available +(Font) Font shape `T1/ppl/b/n' tried instead on input line 131. +LaTeX Font Info: Font shape `T1/ppl/bx/sc' in size <14.4> not available +(Font) Font shape `T1/ppl/b/sc' tried instead on input line 131. +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +(./poster_portrait.bbl) +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! + + +File: images/GraphiqueAffiche1.pdf Graphic file (type pdf) + + +Package pdftex.def Info: images/GraphiqueAffiche1.pdf used on input line 207. +(pdftex.def) Requested size: 182.0949pt x 204.86116pt. + +Overfull \hbox (0.70938pt too wide) in paragraph at lines 207--207 +[][] + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 207--207 + + [] + +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +[2 + +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} <./images/ullong.pdf + +pdfTeX warning: pdflatex (file ./images/ullong.pdf): PDF inclusion: multiple pd +fs with page group included in a single page +> <./images/GraphiqueAffiche1.pdf>] (./poster_portrait.aux) ) +Here is how much of TeX's memory you used: + 14904 strings out of 493485 + 280405 string characters out of 3143526 + 380647 words of memory out of 3000000 + 17817 multiletter control sequences out of 15000+200000 + 32759 words of font info for 72 fonts, out of 3000000 for 9000 + 959 hyphenation exceptions out of 8191 + 63i,17n,70p,433b,1000s stack positions out of 5000i,500n,10000p,200000b,50000s +{/usr/share/texliv +e/texmf-dist/fonts/enc/dvips/base/8r.enc}< +/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb> + +Output written on poster_portrait.pdf (1 page, 231701 bytes). +PDF statistics: + 124 PDF objects out of 1000 (max. 8388607) + 77 compressed objects within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 125 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/poster scientifique/poster_portrait.pdf b/poster scientifique/poster_portrait.pdf new file mode 100644 index 0000000..a857e92 Binary files /dev/null and b/poster scientifique/poster_portrait.pdf differ diff --git a/poster scientifique/poster_portrait.tex b/poster scientifique/poster_portrait.tex new file mode 100644 index 0000000..7a12269 --- /dev/null +++ b/poster scientifique/poster_portrait.tex @@ -0,0 +1,256 @@ +\documentclass[portrait,final,a0paper,fontscale=0.277]{baposter} + +\usepackage{calc} +\usepackage{graphicx} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{relsize} +\usepackage{multirow} +\usepackage{rotating} +\usepackage{bm} +\usepackage{url} +\usepackage[francais]{babel} +\usepackage{graphicx} +\usepackage{multicol} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +%\usepackage{times} +%\usepackage{helvet} +%\usepackage{bookman} +\usepackage{palatino} + +\newcommand{\captionfont}{\footnotesize} + +\graphicspath{{images/}{../images/}} +\usetikzlibrary{calc} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Some math symbols used in the text +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Multicol Settings +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\setlength{\columnsep}{1.5em} +\setlength{\columnseprule}{0mm} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Save space in lists. Use this after the opening of the list +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newcommand{\compresslist}{% +\setlength{\itemsep}{1pt}% +\setlength{\parskip}{0pt}% +\setlength{\parsep}{0pt}% +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Begin of Document +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Here starts the poster +%%%--------------------------------------------------------------------------- +%%% Format it to your taste with the options +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Define some colors + +%\definecolor{lightblue}{cmyk}{0.83,0.24,0,0.12} +\definecolor{lightblue}{rgb}{0.145,0.6666,1} +\hyphenation{resolution occlusions} +%% +\begin{poster}% + % Poster Options + { + % Show grid to help with alignment + grid=false, + % Column spacing + colspacing=1em, + % Color style + bgColorOne=white, + bgColorTwo=white, + borderColor=lightblue, + headerColorOne=black, + headerColorTwo=lightblue, + headerFontColor=white, + boxColorOne=white, + boxColorTwo=lightblue, + % Format of textbox + textborder=roundedleft, + % Format of text header + eyecatcher=true, + headerborder=closed, + headerheight=0.1\textheight, +% textfont=\sc, An example of changing the text font + headershape=roundedright, + headershade=shadelr, + headerfont=\Large\bf\textsc, %Sans Serif + textfont={\setlength{\parindent}{1.5em}}, + boxshade=plain, +% background=shade-tb, + background=plain, + linewidth=2pt + } + % Eye Catcher + {\includegraphics[height=5em]{images/ullong.pdf}} + % Title + {\bf\textsc{Distribution de Laplace asymétrique généralisée}\vspace{0.5em}} + % Authors + {\textsc{François Pelletier, Andrew Luong} \\ \small{École d'actuariat, Université Laval}} + % University logo + {% The makebox allows the title to flow into the logo, this is a hack because of the L shaped logo. + + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Now define the boxes that make up the poster +%%%--------------------------------------------------------------------------- +%%% Each box has a name and can be placed absolutely or relatively. +%%% The only inconvenience is that you can only specify a relative position +%%% towards an already declared box. So if you have a box attached to the +%%% bottom, one to the top and a third one which should be in between, you +%%% have to specify the top and bottom boxes before you specify the middle +%%% box. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % + % A coloured circle useful as a bullet with an adjustably strong filling + \newcommand{\colouredcircle}{% + \tikz{\useasboundingbox (-0.2em,-0.32em) rectangle(0.2em,0.32em); \draw[draw=black,fill=lightblue,line width=0.03em] (0,0) circle(0.18em);}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Introduction}{name=problem,column=0,row=0}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \small{On définit le prix au temps $t$ d'un titre financier $S(t)$ et le rendement cumulé sur ce titre +\begin{equation*}L_t = R_1 + \ldots + R_{t} = \log S(t) - \log S(0). \end{equation*} +On veut modéliser $R_t$ par la distribution de Laplace asymétrique généralisée, aussi appelée variance-gamma. L'estimation paramétrique se fera à partir de la méthode GMM. +On pourra ainsi obtenir la distribution de $L_t$ par convolution, afin d'évaluer le prix d'options européennes.} + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\headerbox{Fonction caractéristique}{name=results,column=1,span=1,row=0}{ + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \small{On définit la variable aléatoire $Y$ + \begin{equation*} + \label{eq:defvarY-GAL} + Y = \theta + \mu W + \sigma \sqrt{W} Z + \end{equation*} + La fonction caractéristique de la distribution de $Y$ peut être + obtenue en utilisant la formule de l'espérance conditionnelle. + \begin{align*} + \phi_Y&(t;\theta,\sigma,\mu,\tau) = E\left[e^{ity}\right] \\ + &= E\left[E\left[e^{ity} | W \right] \right] \\ + &= \int_0^{\infty} E \left[ e^{it(\theta + \mu w+\sigma\sqrt{w}Z)} \right] g(w) dw \\ + &= e^{i\theta t}\int_0^{\infty} e^{ i\mu w t-\frac{\sigma^2t^2w}{2}} \times \frac{1}{\Gamma (\tau)} w^{\tau-1}e^{-w} \\ + &= \frac{e^{i\theta t}}{\left(1+\frac{1}{2} \sigma^2 t^2 - i\mu t \right)^{\tau}} + \end{align*} + On retrouve la fonction de répartition par le théorème de Gil-Pelaez + \begin{equation*} + \label{eq:inversionfncaract} + F_Y(y) = \frac{1}{2} - \frac{1}{\pi}\int_{0}^{\infty} \frac{Im\left[e^{-ity}\phi_Y(t)\right]}{t} dt + \end{equation*}} +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Simulation}{name=contribution,column=1,span=2,below=results}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \small{Le processus de Laplace $Y(t)$ peut être représenté comme la différence de deux processus gamma $G(t)$ \cite{kotz2001laplace}. + \begin{equation*} Y \stackrel{d}{=} \theta + \frac{\sigma}{\sqrt{2}} \left( \frac{1}{\kappa} G_1 - \kappa G_2 \right) + \end{equation*} où $ G_1,G_2 \sim \Gamma\left(\alpha=\tau,\beta=1 \right)$. +Il suffit donc de simuler deux réalisations de cette variable +aléatoire pour obtenir une réalisation de la distribution Laplace +asymétrique généralisée. +} + } + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Choix du modèle}{name=method,column=0,below=problem}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \small{Inspirés par les modèles Mandelbrot, Press et Praetz , Madan et Seneta \cite{madan1990variance} présentent un ensemble de caractéristiques essentielles pour un modèle de rendements financiers:} + \small{\begin{enumerate} + \item Distribution de $R_t$ ayant une queue longue + \item Distribution de $R_t$ ayant des moments finis + \item Processus en temps continu ayant des accroissements stationnaires et indépendants. Distribution des accroissements de même famille peu importe la longueur. + \item Extension multivariée afin de conserver la validité du modèle CAPM. + \end{enumerate}} + \small{Ce tableau décrit le respect des conditions pour les différents modèles étudiés par les auteurs} +\begin{center} + \begin{tabular}{|l|cccc|} + \hline + & \multicolumn{4}{c|}{\scriptsize{Conditions}} \\ + \hline + \scriptsize{Modèle} & \scriptsize{1} & \scriptsize{2} & \scriptsize{3} & \scriptsize{4} \\ + \hline + \scriptsize{Mandelbrot} & \scriptsize{X} & & & \scriptsize{X} \\ + \scriptsize{Press} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} \\ + \scriptsize{Praetz} & \scriptsize{X} & \scriptsize{X} & & \scriptsize{X} \\ + \scriptsize{Madan et Seneta} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} & \scriptsize{X} \\ + \hline + \end{tabular} +\end{center} + } +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Références}{name=source,column=0,span=3,above=bottom}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \bibliographystyle{plain} % (uses file "plain.bst") + \bibliography{biblio} % expects file "myrefs.bib" + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Graphique}{name=references,column=0,below=method,above=source}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \includegraphics[width=64mm,height=72mm]{GraphiqueAffiche1} + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Prix d'options}{name=background model,column=1,span=2,below=contribution,above=source}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\small{L'évaluation du prix d'un option de vente européenne équivaut à calculer la valeur espérée de la réclamation contingente définie par le contrat + \begin{equation*} + P(Y(t),T-t) = B(t,T) \int_{0}^{K} (K-Y(T)) d\hat{F}_t(Y(T)) + \end{equation*} + où $B(t,T)$ est la valeur d'une obligation zéro-coupon au taux sans + risque $r$ d'échéance $T$ et $K$ le prix d'exercice. Cette évaluation doit de faire sous une mesure neutre au risque, selon laquelle les investisseurs n'exigent pas une prime de risque. Cette mesure doit répondre à la propriété martingale. Pour la distribution étudiée, ceci se fait par une modification du paramètre de dérive $\theta$ en $\theta^{\star}$ + \begin{align*} + e^{rt} = E\left[exp(L_t) \right] = M_{Y_1+\ldots+Y_t}(1) = (\phi_{Y}(-i))^t = \left(\frac{e^{\theta^{\star}}}{(1-\mu-\sigma^2/2)^{\tau}} \right)^t \Rightarrow \theta^{\star} = r + \tau\log(1-\mu-\frac{\sigma^2}{2}) + \end{align*} + On pourra ensuite évaluer la valeur du prix de l'option de vente avec la formule de Heston (1993) décrite dans \cite{epps2007pricing} + \begin{align*} + P(Y(t),T-t) &= B(t,T)K \int_{0}^{K} d{F}_t(Y(T)) - B(t,T)\int_{0}^{K} Y(T)\cdot d{F}_t(Y(T)) \\ + &= B(t,T)K \hat{F}_t(K) - Y(t)\hat{G}_t(K) + \end{align*} +où $\hat{G}_t(K)$ est la fonction de répartition de la transformée d'Esscher (h=1) de la mesure neutre au risque. + } + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \headerbox{Méthode GMM}{name=questions,column=2,span=1,above=background model,bottomaligned=results}{ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + La méthode GMM a pour objectif d'estimer les paramètres $\theta$ d'une distribution en minimisant une norme quadratique +\begin{equation*} + \|\hat{m}\left(\theta\right) \|_W^2 = \hat{m}\left(\theta\right)^{\prime} W \hat{m}\left(\theta\right) +\end{equation*} +pondérée par une matrice définie positive $W$ et où $\hat{m}\left(\theta\right) = \left(\frac{1}{T}\sum_{t=1}^T g(Y_t,\theta)\right)$ est la moyenne empirique des conditions de moment $g(Y_t,\theta)$. +\small{On utilise les conditions de moment basées sur l'espérance et la variance + \begin{equation*} + g(Y_t,\theta) = + \left[\begin{array}[]{c} + \bar{Y} - \left(\theta+\mu\tau\right) \\ + \frac{1}{T}\sum_{t=1}^T \left(Y_t - \bar{Y} \right)^2 - \tau\left(\sigma^2+\mu^2\right) + \end{array}\right] +\end{equation*} +La matrice optimale $W$ est estimée en utilisant la méthode du GMM itératif \cite{hall2005generalized}: + \begin{align*} + \hat{W}_{(1)} &= \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\hat\theta_{(0)})g(Y_t,\hat\theta_{(0)})^{\prime}\bigg)^{-1} \\ + \hat{W}_{(i)} &= \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\hat\theta_{(i-1)})g(Y_t,\hat\theta_{(i-1)})^{\prime}\bigg)^{-1} + \end{align*}} + } + +\end{poster} + +\end{document} +