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J(Cr);J(X)isalsoaproductofJacobiansofsmoothcurves,hence codimJ(X)Sing()4 Thisworksonlyindimension3!Theorem3.1.AsmoothcubicthreefoldX3P4isnotrational.Wew Example2.Asasecondexample,weconsiderthe3-Colourabilityproblemforgraphs.Itsobjectiveistocolourtheverticesofagraphwith3coloursinsuchawaythatadjacentverticesgetdierentcolours.EachgraphG,viewedasaninstan tery$tempo$attoo$ato$$opt tery$tempo$attoo$ato$$potTheegdelabelsarefactorsoftheinputstrings.Thustheycanbestoredinconstantspaceusingpointerstothestrings,whicharestoredseparately.Inafullycompacttrie,any Lecture5: Context . Free Languages. Prof. Amos Israeli. On the last lecture we completed our study of regular languages. (There is still a lot to learn but our time is limited…).. Introduction and Motivation. xandthelinesx=0andy=3.Solution.Wewillusedierentialelementsconsistingofrectangularhorizontalslicesofheightxandwidthdy.Thismeansthatvariableywillbethevariableofintegration. x y 3 9 0 dy y x ( x e ; y e Example2.AfundamentallydierentinequalitythatcanalsobeexpressedintheformofEqua-tion1isthefactthattheLpnormisanon{increasingfunctionofp.Forp2[0;1]wehavetheinequalityPjXpjPjXjp1;correspondingtothe Example2.5Considerthetwomappings:N!Ndenedby(n)=2nand:N!Ndenedby(n)=n+1 2ifnisoddn 2ifnisevenShowthatandareone-to-onebutisnot.Solution.Since(n1)=(n2)implies2n1=2n2andthisinturnsimpliest Pre-Assessment. 1. . Either the physicians in this hospital or the chief administrator is /are going to have to make a decision. . 2.. Either the Committee on Course Design or the Committee on College Operations decide/ decides these matters. . Intro Unit and Unit 1 Vocabulary Contest Geography , Clothes and Accesories , Food , Adjectives , Social expressions , Invitations , Idioms and Phrasal Verbs . ribera puente Capital /’ i O Wewi Outline12Examplespart13TheGITconstruction4TheHilbert-Mumfordcriterion5Examplespart26ConclusionAtanasAtanasovHarvardUniversityGeometricInvariantTheoryZhejiangUniversity20112/41IntroductionAtheoremfromd