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Normalisation de champs de vecteurs holomorphes et équations différentielles implicites / Normalization of holomorphic vector fields and implicit differential equationsAurouet, Julien 06 December 2013 (has links)
La théorie classique des formes normales a pour but de simplifier des problèmes compliqués grâce à des changements de coordonnées réguliers pour ne conserver que les caractéristiques dynamiques du système. Plus précisément, on considère un système dynamique que l'on dit "élémentaire", comme par exemple la partie linéaire d'un champ de vecteurs au voisinage d'un point singulier, et on se donne une perturbation de ce système élémentaire. Les formes normales sont alors l'ensemble des représentants de ces perturbations à la conjugaison près d'une transformation régulière. Elles ne sont constituées que des termes qui caractérisent la dynamique du système perturbé et que l'on appelle "résonances". Dans la première partie de la thèse on cherche à comprendre la dynamique locale d'équations différentielles implicites de la forme F(x,y,y')=0, où F est un germe de fonction holomorphe au voisinage d'un point singulier. Pour cela on utilise la relation intime entre les systèmes implicites et les champs liouvilliens. La classification par transformation de contact des équations implicites provient de la classification symplectique des champs liouvilliens. On utilise alors toute la théorie des formes normales pour les champs de vecteurs, dans le cas holomorphe (Brjuno, Siegel, Stolovitch) et dans le cas réel (Sternberg), que l'on adapte pour les champs liouviliens avec des transformations symplectiques. On établit alors des résultats de classification des équations implicites en fonction des invariants dynamiques, ainsi que des conditions d'existence de solutions locales via les formes normales. / The aim of the classical theory of normal forms is to simplify complicated problems by using regular changes of coordinates, in order to keep the dynamical characteristics of the system. More precisely, we consider a dynamic system said to be "elementary", like a linear part of a vector field in the neighborhood of a singular point, and we focus on a perturbation of this elementary system. Normal forms are the set of all representatives of those perturbations under the action of the group of regular transformation. They are composed of terms which caracterise the dynamics of the perturbed system, and which are called "resonances". In the first part, we try to understand the local dynamic of implicit equations of the form $F(x,y,y')=0$, where $F$ is a germ of holomorphic function in a neighborhood of a singular point. To this end we use the relation between implicit systems and liouvillian vector fields. The classification by contact transformations of implicit equations come from the symplectic classification of liouvillian vector fields. We use all normal forms theory for vector fields, in complex case (Bjruno, Siegel, Stolovitch), and in real case (Sternberg), adapted to liouvillian fields with symplectic transformations. We establish classification results for implicit equations according to the dynamical invariants, and existence conditions of local solutions using normal forms. In the second part, we undertake the normalization of an analytic vector field in a neighborhood of the torus. Brjuno enunciates a theorem of normalization, under conditions of control of small divisors and integrability of the normal forms ; however he doesn't give any proof of that theorem.
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Teorema de Riemann-Roch e aplicaçõesArruda, Rafael Lucas de [UNESP] 25 February 2011 (has links) (PDF)
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arruda_rl_me_sjrp.pdf: 624072 bytes, checksum: 23ddd00e27d1ad781e2d1cec2cb65dee (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / O objetivo principal deste trabalho é estudar o Teorema de Riemann-Roch, um dos resultados fundamentais na teoria de curvas algébricas, e apresentar algumas de suas aplicações. Este teorema é uma importante ferramenta para a classificação das curvas algébricas, pois relaciona propriedades algébricas e topológicas. Daremos uma descrição das curvas algébricas de gênero g, 1≤ g ≤ 5, e faremos um breve estudo dos pontos de inflexão de um sistema linear sobre uma curva algébrica / The main purpose of this work is to discuss The Riemann-Roch Theorem, wich is one of the most important results of the theory algebraic curves, and to present some applications. This theorem is an important tool of the classification of algebraic curves, sinces relates algebraic and topological properties. We will describle the algebraic curves of genus g, 1≤ g ≤ 5, and also study inflection points of a linear system on an algebraic curve
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Divisors on graphs, binomial and monomial ideals, and cellular resolutionsShokrieh, Farbod 27 August 2014 (has links)
We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs.
We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their Z-graded Betti tables coincide.
As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.
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Sobre divisores livres homogêneosSilva, Mauri Pereira da 16 July 2015 (has links)
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Previous issue date: 2015-07-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The maingoalofthisdissertationisthepresentationofconcepts,examplesand
characterizations{bothclassicalandrecent{concerningtheimportantandin
uential
theory oftheso-called freedivisors in thestandardhomogeneouscase.Tothisend,we
beginwithabasicstudyonderivationsandwefocusonthemoduledubbed tangential
idealizer of agivenhomogeneouspolynomial,whichgeometricallycorrespondstothe
moduleoflogarithmicvector eldsalongthegivenprojectivehypersurface(thedivisor
is saidtobe free if suchmoduleisfreeoverthegradedpolynomialring).Wewillalso
discuss, inparticular,resultsaboutfreedivisorsintheprojectiveplane. / O principalobjetivodestadisserta c~ao eaapresenta c~aodeconceitos,exemplose
caracteriza c~oes{tantocl assicasquantorecentes{arespeitodaimportanteein
uente
teoria doschamados divisoreslivres no casohomog^eneopadr~ao.Paraesta nalidade,
iniciamos comumestudob asicosobrederiva c~oesefocalizamosnom odulodenomi-
nado idealizadortangencial de umdadopolin^omiohomog^eneo,oquegeometricamente
correspondeaom odulodoscamposvetoriaislogar tmicosaolongodahipersuperf cie
projetivadada(odivisor edito livre quando talm odulo elivresobreoanelgraduado
de polin^omios).Tamb emdiscutiremos,emparticular,resultadossobredivisoreslivres
no planoprojetivo.
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Elementary proof of the Riemann—Roch TheoremSundgren, Hampus January 2023 (has links)
This thesis will cover an elementary proof of the Riemann–Roch Theorem for planecurves. We will introduce the notions of divisors, which is a convenient way of com-puting multiplicities of rational function, then continuing by introducing differentials.Furthermore we will introduce the K-vector space L(D), consisting of rational func-tions which are controlled by a divisor D. This is followed by presenting some moreresults before we arrive at an elementary proof of the Riemann–Roch Theorem.
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Distribution asymptotique du nombre de diviseurs premiers distincts inférieurs ou égaux à mPersechino, Roberto 05 1900 (has links)
Le sujet principal de ce mémoire est l'étude de la distribution asymptotique de la fonction f_m qui compte le nombre de diviseurs premiers distincts parmi les nombres premiers $p_1,...,p_m$.
Au premier chapitre, nous présentons les sept résultats qui seront démontrés au chapitre 4.
Parmi ceux-ci figurent l'analogue du théorème d'Erdos-Kac et un résultat sur les grandes déviations.
Au second chapitre, nous définissons les espaces de probabilités qui serviront à calculer les probabilités asymptotiques des événements considérés, et éventuellement à calculer les densités qui leur correspondent.
Le troisième chapitre est la partie centrale du mémoire. On y définit la promenade aléatoire qui,
une fois normalisée, convergera vers le mouvement brownien. De là, découleront les résultats qui
formeront la base des démonstrations de ceux chapitre 1. / The main topic of this masters thesis is the study of the asymptotic distribution of the fonction
f_m which counts the number of distinct prime divisors among the first $m$ prime numbers, i.e. $p_1,...,p_m$.
The first chapter provides the seven main results which will later on be proved in chapter 4.
Among these we find the analogue of the Erdos-Kac central limit theorem and a result on large deviations.
In the following chapter, we define several probability spaces on which we will calculate asymptotic probabilities of specific events. These will become necessary for calculating their corresponding densities.
The third chapter is the main part of this masters thesis. In it, we introduce a random walk which, when suitably normalized, will converge to the Brownian motion. We will then obtain results which will form the basis of the proofs of those of
chapiter 1.
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b-divisors on toric and toroidal embeddingsBotero, Ana María 11 August 2017 (has links)
In dieser Dissertation entwickeln wir eine Schnittheorie von torischen bzw. toroidalen
b-Divisoren auf torischen bzw. toroidalen Einbettungen. Motiviert wird
dies durch das Ziel, eine arithmetische Schnittheorie auf gemischten Shimura-
Varietäten von nicht-kompaktem Typ zu begründen. Die bisher zur Verfügung
stehenden Werkzeuge definieren keine numerischen Invarianten, die birational
invariant sind.
Zuerst definieren wir torische b-Divisoren auf torischen Varietäten und einen
Integrabilitätsbegriff für solche Divisoren. Wir zeigen, dass torische b-Divisoren
unter geeigneten Annahmen an die Positivität integrierbar sind und dass ihr Grad
als das Volumen einer konvexen Menge gegeben ist. Außerdem zeigen wir, dass die
Dimension des Vektorraums der globalen Schnitte eines torischen b-Divisors, der
nef ist, gleich der Anzahl der Gitterpunkte in besagter konvexer Menge ist und
wir geben eine Hilbert–Samuel-Formel für das asymptotische Wachstum dieser
Dimension. Dies verallgemeinert klassische Resultate für klassische torische
Divisoren auf torischen Varietäten. Als ein zusätzliches Resultat setzen wir
konvexe Mengen, die von torischen b-Divisoren kommen, mit Newton–Okounkov-
Körpern in Beziehung.
Anschließend definieren wir toroidale b-Divisoren auf toroidalen Varietäten
und einen Integrierbarkeitsbegriff für solche Divisoren. Wir zeigen, dass unter
geeigneten Positivitätsannahmen toroidale b-Divisoren integrierbar sind und ihr
Grad als ein Integral bezüglich eines Grenzmaßes aufgefasst werden kann. Dieses
Grenzmaß ist ein schwacher Grenzwert von diskreten Maßen, deren Gewichte
über tropische Schnittheorie auf rationalen konischen polyedrischen Komplexen
definiert sind, welche zu der toroidalen Varietät gehören. Wir setzen dieses
Grenzmaß ebenfalls in Beziehung zum zu einem konvexen Körper assoziierten
Flächeninhaltsmaß. Diese Beziehung erlaubt es uns, Integrale bezüglich des
Grenzmaßes explizit auszurechnen. Zusätzlich erhalten wir eine kanonische Zerlegung
der Differenz zweier konvexer Mengen und eine Beziehung zwischen das
Volumen von den Teilen und tropische Schnittheoretische Mengen.
Schließlich berechnen wir als Anwendung den Grad des b-Divisors von Jacobiformen
vom Gewicht k und Index m bezüglich der Hauptkongruenzuntergruppe
zum Level N >= 3 auf der verallgemeinerten universellen elliptischen Kurve und
wir zeigen, dass der b-divisoriale Ansatz gegenüber lediglich einer kanonischen
Kompaktifizierung Vorteile bietet. / In this thesis we develop an intersection theory of toric and toroidal b-divisors on
toric and toroidal embeddings, respectively. Our motivation comes from wanting
to establish an arithmetic intersection theory on mixed Shimura varieties of non-
compact type. The tools available until now do not define numerical invariants
which are birationally invariant.
First, we define toric b-divisors on toric varieties and an integrability notion
of such divisors. We show that under suitable positivity assumptions toric b-
divisors are integrable and that their degree is given as the volume of a convex
set. Moreover, we show that the dimension of the space of global sections of a nef
toric b-divisor is equal to the number of lattice points in this convex set and we
give a Hilbert-Samuel type formula for its asymptotic growth. This generalizes
classical results for classical toric divisors on toric varieties. As a by-product, we
relate convex sets arising from toric b-divisors with Newton-Okounkov bodies.
Then, we define toroidal b-divisors on toroidal varieties and an integrability
notion of such divisors. We show that under suitable positivity assumptions
toroidal b-divisors are integrable and that their degree is given as an integral
with respect to a limit measure, which is a weak limit of discrete measures
whose weights are defined via tropical intersection theory on the rational con-
ical polyhedral complex attached to the toroidal variety. We also relate this
limit measure with the surface area measure associated to a convex body. This
relation enables us to compute integrals with respect to these limit measures ex-
plicitly. Additionally, we give a canonical decomposition of the difference of two
convex sets and we relate the volume of the pieces to tropical top intersection
numbers.
Finally, as an application, we compute the degree of the b-divisor of Jacobi
forms of weight k and index m with respect to the principal congruence subgroup
of level N >= 3 on the generalized universal elliptic curve and we show that it
is meaningful to consider the b-divisorial approach instead of just fixing one
canonical compactification.
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Distribution asymptotique du nombre de diviseurs premiers distincts inférieurs ou égaux à mPersechino, Roberto 05 1900 (has links)
Le sujet principal de ce mémoire est l'étude de la distribution asymptotique de la fonction f_m qui compte le nombre de diviseurs premiers distincts parmi les nombres premiers $p_1,...,p_m$.
Au premier chapitre, nous présentons les sept résultats qui seront démontrés au chapitre 4.
Parmi ceux-ci figurent l'analogue du théorème d'Erdos-Kac et un résultat sur les grandes déviations.
Au second chapitre, nous définissons les espaces de probabilités qui serviront à calculer les probabilités asymptotiques des événements considérés, et éventuellement à calculer les densités qui leur correspondent.
Le troisième chapitre est la partie centrale du mémoire. On y définit la promenade aléatoire qui,
une fois normalisée, convergera vers le mouvement brownien. De là, découleront les résultats qui
formeront la base des démonstrations de ceux chapitre 1. / The main topic of this masters thesis is the study of the asymptotic distribution of the fonction
f_m which counts the number of distinct prime divisors among the first $m$ prime numbers, i.e. $p_1,...,p_m$.
The first chapter provides the seven main results which will later on be proved in chapter 4.
Among these we find the analogue of the Erdos-Kac central limit theorem and a result on large deviations.
In the following chapter, we define several probability spaces on which we will calculate asymptotic probabilities of specific events. These will become necessary for calculating their corresponding densities.
The third chapter is the main part of this masters thesis. In it, we introduce a random walk which, when suitably normalized, will converge to the Brownian motion. We will then obtain results which will form the basis of the proofs of those of
chapiter 1.
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Teorema de Riemann-Roch e aplicações /Arruda, Rafael Lucas de. January 2011 (has links)
Orientador: Parham Salehyan / Banca: Eduardo de Sequeira Esteves / Banca: Jéfferson Luiz Rocha Bastos / Resumo: O objetivo principal deste trabalho é estudar o Teorema de Riemann-Roch, um dos resultados fundamentais na teoria de curvas algébricas, e apresentar algumas de suas aplicações. Este teorema é uma importante ferramenta para a classificação das curvas algébricas, pois relaciona propriedades algébricas e topológicas. Daremos uma descrição das curvas algébricas de gênero g, 1≤ g ≤ 5, e faremos um breve estudo dos pontos de inflexão de um sistema linear sobre uma curva algébrica / Abstract: The main purpose of this work is to discuss The Riemann-Roch Theorem, wich is one of the most important results of the theory algebraic curves, and to present some applications. This theorem is an important tool of the classification of algebraic curves, sinces relates algebraic and topological properties. We will describle the algebraic curves of genus g, 1≤ g ≤ 5, and also study inflection points of a linear system on an algebraic curve / Mestre
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Bernstein--Sato Ideals and the Logarithmic Data of a DivisorDaniel L Bath (10724076) 05 May 2021 (has links)
We study a multivariate version of the Bernstein–Sato polynomial, the so-called Bernstein–Sato ideal, associated to an arbitrary factorization of an analytic germ <i>f - f</i><sub>1</sub>···<i>f</i><sub>r</sub>. We identify a large class of geometrically characterized germs so that the <i>D</i><sub>X,x</sub>[<i>s</i><sub>1</sub>,...,<i>s</i><sub>r</sub>]-annihilator of <i>f</i><sup>s</sup><sub>1</sub><sup>1</sup>···<i>f</i><sup>s</sup><sub>r</sub><sup>r</sup> admits the simplest possible description and, more-over, has a particularly nice associated graded object. As a consequence we are able to verify Budur’s Topological Multivariable Strong Monodromy Conjecture for arbitrary factorizations of tame hyperplane arrangements by showing the zero locus of the associated Bernstein–Sato ideal contains a special hyperplane. By developing ideas of Maisonobe and Narvaez-Macarro, we are able to find many more hyperplanes contained in the zero locus of this Bernstein–Sato ideal. As an example, for reduced, tame hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial contained in [−1,0) are combinatorially determined; for reduced, free hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial are all combinatorially determined. Finally, outside the hyperplane arrangement setting, we prove many results about a certain <i>D</i><sub>X,x</sub>-map ∇<sub><i>A</i></sub> that is expected to characterize the roots of the Bernstein–Sato ideal.
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