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Cirilo, Patricia Romano. "Órbitas de Birkhoff e não Birkhoff para aplicações do tipo Twist." Universidade Federal de Minas Gerais, 2007. http://hdl.handle.net/1843/EABA-72VJWU.
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Para estudar a dinâmica de transformações que preservam área é interessante se perguntar sobre a existência de órbitas "ordenadas". A importância desta condição geométrica foi observada por G. D. Birkhoff no início do século XX e desde então as órbitas de Birkhoff vêm sendo estudadas com afinco. Nesta dissertação será estudado um critério para que uma aplicação possua entropia topológica positiva e utilizando este critério serão apresentadas condições para a existência de órbitas de Birkhoff. A aplicação em questão é um homeomorfismo do cilindro nele mesmo e são requeridas as hipóteses de que ela seja twist monótona e que preserve orientação. Tal aplicação é obtida através de uma relação de recorrência. Será apresentado um teorema que permite obter soluções da relação de recorrência com certas propriedades de periodicidade e ordem. Com isto é possível, a partir de órbitas da aplicação inicial com estas propriedades, concluir a existência de órbitas de Birkhoff, donde segue, em particular, um teorema de G. R. Hall. Com algumas hipóteses, mostra-se também a existência de órbitas de Birkhoff com um número de rotação pré-determinado. Para terminar, mostra-se que se a aplicação em questão tem entropia topológica nula então toda órbita tem número de rotação para frente e para trás e ainda, um resultado atribuído originalmente a P. Boyland, que se a entropia topológica é nula e a órbita é do tipo (p,q), com mdc(p,q) =1, então esta é necessariamente uma órbita de Birkhoff. Já que não se supõe nenhuma diferenciabilidade sobre a transformação em questão, não podem ser utilizados argumentos como os de hiperbolicidade, transversalidade e nem procedimentos variacionais para a construção de conjuntos caóticos, portanto os métodos aqui utilizados são puramente topológicos, o que ressalta a beleza do assunto. A referência básica do estudo apresentado é o artigo de S. B. Angenent, Monotone recurrence relations, their Birkhoff orbits and topological entropy publicado na Ergodic Theory & Dynamical Systems.
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Costa, Liliana Manuela Gaspar Cerveira da. "Politopo de Birkhoff acíclico." Doctoral thesis, Universidade de Aveiro, 2011. http://hdl.handle.net/10773/8510.
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Doutoramento em MatemáticaNeste trabalho estabelece-se uma interpreta c~ao geom etrica, em termos da teoria dos grafos, para v ertices, arestas e faces de uma qualquer dimens~ao do politopo de Birkho ac clico, Tn = n(T), onde T e uma arvore com n v ertices. Generaliza-se o resultado obtido por G. Dahl, [18], para o c alculo do di^ametro do grafo G( t n), onde t n e o politopo das matrizes tridiagonais duplamente estoc asticas. Adicionalmente, para q = 0; 1; 2; 3 s~ao obtidas f ormulas expl citas para a contagem do n umero de qfaces do politopo de Birkho tridiagonal, t n, e e feito o estudo da natureza geom etrica dessas mesmas faces. S~ao, tamb em, apresentados algoritmos para efectuar contagens do n umero de faces de dimens~ao inferior a de uma dada face do politopo de Birkho ac clico.
In this work using graph theory, we give a geometrical interpretation of vertices, edges, and faces of any dimension of the acyclic Birkho polytope, Tn = n(T), were T is a tree with n vertices. We generalize a proposition from G. Dahl, [18], that allows the calculation of the diameter of the graph G( t n), where t n denotes the polytope of tridiagonal doubly stochastic matrices. Furthermore, for q = 0; 1; 2; 3 we obtain some explicit formulae for counting the number of qfaces of the tridiagonal Birkho polytope, t n, and the study of its geometrical nature is done. For a given p-face of t n we determine the number of faces of lower dimension that are contained in it and we discuss its nature. Some algorithms allowing an exhaustive account on the number of edges and faces of the acyclic Birkho polytope are presented.
3
Le, Calvez Patrice. "Propriétés des attracteurs de Birkhoff." Grenoble 2 : ANRT, 1987. http://catalogue.bnf.fr/ark:/12148/cb376071668.
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MARSON, Guilherme Porfírio. "Órbitas Birkhoff na Ferradura Rotacional." reponame:Repositório Institucional da UNIFEI, 2017. http://repositorio.unifei.edu.br/xmlui/handle/123456789/887.
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Neste trabalho, estudamos difeomorfismos de classe C¹ do anel com uma órbita homoclínica transversal K-rotacional a um ponto fixo hiperbólico. Primeiramente, recuperamos um resultado clássico de Poincaré, Birkhoff e Smale: Um ponto homoclínico implica a existência de uma ferradura topológica para alguma iterada. Além disso, obtemos informações interessantes sobre o comportamento rotacional das órbitas em um conjunto de Cantor invariante e maximal (chamado ferradura rotacional). Usando conjugação e dinâmica simbólica associada ao conjunto de Cantor não-errante da ferradura, provamos a existência de um intervalo de rotação não trivial I, e de incontáveis conjuntos de Cantor invariantes para cada número de rotação irracional em I. Finalizamos o trabalho caracterizando a codificação das órbitas Birkhoff da aplicação de duplicação em S¹, as quais implicam a existência de órbitas Birkhoff da ferradura rotacional.
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Le, Calvez Patrice. "Proprietes des attracteurs de birkhoff." Paris 7, 1987. http://www.theses.fr/1987PA077014.
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Ce travail a pour base un article de g. D. Birkhoff consacre aux diffeomorphismes de l'anneau deviant la verticale, et se divise en trois parties. Dans le premier chapitre, on donne des demonstrations completes et rigoureuses des resultats de cet article, on definit l'attracteur de birkhoff d'un diffeomorphisme de l'anneau dissipatif et deviant la verticale, ainsi que les nombres de rotation inferieur et superieur d'un tel ensemble. Dans le second chapitre, on montre qu'il existe pour tout reel compris entre les deux nombres de rotation d'un attracteur de birkhoff, un ensemble d'aubry-mather, contenu dans cet ensemble, dont c'est le nombre de rotation. Dans le troisieme chapitre, on repond a diverses questions sur les attracteurs de birkhoff, on montre en particulier que ceux-ci ne dependent pas continument des diffeomorphismes
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Paolantoni, Thibault. "Application de Riemann-Hilbert-Birkhoff." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLS410/document.
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L'application exponentielle duale est une façon d'encoder les matrices de Stokes d'une connexion sur un fibré trivial sur la sphère de Riemann avec deux pôles : un pôle double en 0 et un pôle simple en l'infini.On donne ici une formule pour l'application exponentielle duale comme une série formelle non commutative. D'autres généralisations de cette formule sont donnéesThe exponential dual map is a way to encode Stokes data of a connection on a trivial vector bundle on the Riemann sphere with two poles: one double pole at 0 and one simple pole at infinity.We give here a formula for the exponential dual map expressed as a non commutative serie. Others generalizations of this formula are given
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Palacios, Quiñonero Francesc. "Contribución al problema de interpolación de Birkhoff." Doctoral thesis, Universitat Politècnica de Catalunya, 2004. http://hdl.handle.net/10803/6711.
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El objetivo de esta tesis es desarrollar la interpolación de Birkhoff mediante polinomios lacunarios.En la interpolación algebraica de Birkhoff se determina un polinomio de grado menor que n, para ello se emplean n condiciones que fijan el valor del polinomio o sus derivadas. Los problemas clásicos de interpolación de Lagrange, Taylor, Hermite, Hermite-Sylvester y Abel-Gontcharov son casos particulares de interpolación algebraica de Birkhoff.
Un espacio de polinomios lacunarios de dimensión n es el conjunto de los polinomios que pueden generarse por combinación lineal de n potencias distintas de grados, en general, no consecutivos. En particular, cuando tomamos potencias de grados 0,1,.,n-1, se obtiene el espacio de polinomios de grado menor que n, empleado en la interpolación algebraica clásica.
En la interpolación algebraica clásica, el número de condiciones determina el espacio de interpolación. En contraste, en la interpolación mediante polinomios lacunarios las condiciones de interpolación determinan únicamente la dimensión del espacio de interpolación y pueden existir una infinidad de espacios sobre los que realizar la interpolación. Esto nos permite construir mejores estrategias de interpolación en ciertos casos, como la interpolación de funciones de gran crecimiento (interpolación de exponenciales y de ramas asintóticas).
La aportación de la tesis consiste en la definición de un marco teórico adecuado para la interpolación de Birkhoff mediante polinomios lacunarios y en la extensión al nuevo marco de los principales elementos de la interpolación algebraica de Birkhoff. En concreto, se generaliza la condición de Pólya, se caracteriza la regularidad condicionada, se establecen condiciones suficientes de regularidad ordenada que extienden el teorema de Atkhison-Sharma, se extiende la descomposición normal y se establecen condiciones suficientes de singularidad en los casos indescomponibles.
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Cortiñas, Guillermo. "Cuantización y teorema de Poincaré-Birkhoff-Witt." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/95761.
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Nguyen, Thu Huong. "Strong Stability Preserving Hermite-Birkhoff Time Discretization Methods." Thèse, Université d'Ottawa / University of Ottawa, 2012. http://hdl.handle.net/10393/23491.
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The main goal of the thesis is to construct explicit, s-stage, strong-stability-preserving (SSP) Hermite–Birkhoff (HB) time discretization methods of order p with nonnegative coefficients for the integration of hyperbolic conservation laws. The Shu–Osher form and the canonical Shu–Osher form by means of the vector formulation for SSP Runge–Kutta (RK) methods are extended to SSP HB methods. The SSP coefficients of k-step, s-stage methods of order p, HB(k,s,p), as combinations of k-step methods of order (p − 3) with s-stage explicit RK methods of order 4, and k-step methods of order (p-4) with s-stage explicit RK methods of order 5, respectively, for s = 4, 5,..., 10 and p = 4, 5,..., 12, are constructed and compared
with other methods. The good efficiency gains of the new, optimal, SSP HB methods over other SSP
methods, such as Huang’s hybrid methods and RK methods, are numerically shown by means of their effective SSP coefficients and largest effective CFL numbers. The formulae of these new, optimal methods are presented in their Shu–Osher form.
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Reff, Nathan. "A generalization of the Birkhoff-von Neumann theorem /." Online version of thesis, 2007. http://hdl.handle.net/1850/5967.
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Premi, Lorenzo. "Geometria Proiettiva Sintetica, da Birkhoff a Faigle ed Herrmann." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/14775/.
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Questa tesi presenta una descrizione dei primi aspetti della Teoria dei Reticoli e della Teoria delle Categorie e una loro applicazione alla descrizione di un sistema di corrispondenze fra geometrie proiettive di incidenza in senso ampio
e reticoli modulari nel caso finito dimensionale. Nell'ordine:
dualità fra insiemi e reticoli booleani; dualità fra poset e reticoli distributivi (Birkhoff, 1937); equivalenza fra geometrie proiettive e reticoli modulari complementati (Birkhoff, 1935), equivalenza fra geometrie proiettive su poset e reticoli modulari (Faigle ed Herrmann, 1981).
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Sundararajan, Jay Kumar 1982. "Extending the Birkhoff-von Neumann switching strategy to multicast switching." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/30185.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (p. 63-64).
The Birkhoff-von Neumann (BVN) strategy for offline switching does not support multicast, as it considers only permutation-based switch configurations. This thesis extends the BVN strategy to multicast switching. Using a graph theoretic model, we show that the capacity region for a traffic pattern is precisely the stable set polytope of the pattern's "conflict graph", in the no-fanout-splitting case. We construct examples to show that, if dynamic fanout splitting is excluded, there is no clear winner in terms of rate region among various fanout splitting strategies. The problem of deciding whether a given set of rates is achievable in a multicast switch is also addressed. We show that, in general, the problem is equivalent to the membership problem for the stable set polytope of a graph, and is therefore NP-hard. We also prove that the problem is NP-hard for the case that splitting of the set of destinations, or fanout, is allowed. However, in the no-splitting case, it is polynomial time solvable when the number of multicast flows in the N x N switch is O(logN). The algorithm naturally leads to a schedule to serve the flows in a stable manner, if the rates are achievable. For an arbitrary number of multicasts, we show that, computing the offline schedule is equivalent to fractional weighted graph coloring which takes polynomial time for perfect graphs. We present several types of traffic patterns whose conflict graphs are perfect. [18] proposed a simple online algorithm called i-SLIP based on parallel iterative matching, for online unicast scheduling. We propose an online algorithm for multicast, based on i-SLIP and the conflict graph idea, and compare them with ESLIP([19]) and the copy-and- use-i-SLIP strategy, through simulations.
by Jay Kumar Sundararajan.
S.M.
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Cloutier, John. "A Combinatorial Analog of the Poincaré–Birkhoff Fixed Point Theorem." Scholarship @ Claremont, 2003. https://scholarship.claremont.edu/hmc_theses/145.
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Results from combinatorial topology have shown that certain combinatorial lemmas are equivalent to certain topologocal fixed point theorems. For example, Sperner’s lemma about labelings of triangulated simplices is equivalent to the fixed point theorem of Brouwer. Moreover, since Sperner’s lemma has a constructive proof, its equivalence to the Brouwer fixed point theorem provides a constructive method for actually finding the fixed points rather than just stating their existence. The goal of this research project is to develop a combinatorial analogue for the Poincare ́-Birkhoff fixed point theorem.
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Albishi, Njwd. "Three-and four-derivative Hermite-Birkhoff-Obrechkoff solvers for stiff ODE." Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/34332.
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Three- and four-derivative k-step Hermite-Birkhoff-Obrechkoff (HBO) methods are
constructed for solving stiff systems of first-order differential equations of the form
y'= f(t,y), y(t0) = y0. These methods use higher derivatives of the solution y as
in Obrechkoff methods. We compute their regions of absolute stability and show
the three- and four-derivative HBO are A( 𝜶)-stable with 𝜶 > 71 ° and 𝜶 > 78 °
respectively. We conduct numerical tests and show that our new methods are more
efficient than several existing well-known methods.
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Bozic, Vladan. "Three-stage Hermite-Birkhoff-Taylor ODE solver with a C++ program." Thesis, University of Ottawa (Canada), 2008. http://hdl.handle.net/10393/27751.
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One-step 3-stage Hermite-Birkhoff-Taylor methods, denoted by HBT( p)3, are constructed for solving nonstiff systems of first-order differential equations of the form y' = f( x, y), y(x0) = y0. The method uses derivatives y' to y(p--2) as in Taylor methods and is combined with a 3-stage Runge-Kutta method of order 3. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to Taylor- and Runge-Kutta-type order conditions, which are then reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method yields impressive results with regards to intervals of absolute stability. A detailed formulation of variable step size (VS) fixed order HBT( p)3 is presented, as well as the formulation of variable-step variable-order (VSVO) HBT(p)3. Several problems often used to test high order ODE solvers on the basis the number of steps, CPU time, maximum global error, and maximum global energy error are considered. The results stress that both VS and VSVO HBT(p)3 methods are superior to Dormand-Prince DP (8,7)13M and Taylor method of order p, denoted by T( p). To obtain results at high precision, high order VS and VSVO HBT( p)3 methods have been implemented in multiple precision. These numerical results clearly show the benefit of formulating a method by adding high order derivatives to Runge-Kutta method.
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Ronchetti, Niccolò. "Il Diamond lemma e il teorema di Poincaré-Birkhoff-Witt sugli anelli." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2010. http://amslaurea.unibo.it/1185/.
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Demeneghi, Paulinho. "Aplicações completamente positivas em algebras de matrizes e o teorema de Birkhoff." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2014. http://hdl.handle.net/10183/109992.
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Descrevemos propriedades espectrais de aplicações positivas em C*- álgebras de dimensão finita, seguindo o trabalho clássico de Evans e Hoegh-Krohn [EH-K]. Conjuntamente, estudamos os pontos extremais do conjunto das aplicações duplamente estocásticas completamente positivas sobre Mn(C), seguindo Landau e Streater [LS].We describe spectral properties of positive maps over nite dimensional C* -algebras, following the classical work of Evans and H egh-Krohn [EH-K]. We also study the extremal points of the set of completely positive doubly-stochastic maps over Mn(C), following Landau and Streater [LS].
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Berger, Roland. "Propriété de Poincaré-Birkhoff-Witt dans les espaces et groupes quantiques différentiels." Lyon 1, 1992. http://www.theses.fr/1992LYO10001.
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La propriete de poincare-birkhoff-witt (pbw) classique dit que les monomes ordonnes forment une base de l'algebre enveloppante d'une algebre de lie. Jimbo, rosso, lusztig ont demontre que cette propriete subsiste dans les algebres enveloppantes quantiques de drinfeld-jimbo. Dans cette these, nous abordons le probleme de la propriete de pbw d'un point de vue purement algebrique. Le cadre est suffisamment general pour inclure la quantification des superalgebres enveloppantes et des supergroupes lineaires. La question cruciale des rapports entre la propriete de pbw et l'equation de yang-baxter quantique est etudiee de facon systematique. Les consequences de la propriete de pbw sur le calcul differentiel non commutatif sont examinees. Le dernier chapitre est consacre au groupe lineaire quantique differentiel associe a la matrice r multiparametree de type a
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Habermas, Derek. "Compact Symmetric Spaces, Triangular Factorization, and Cayley Coordinates." Diss., The University of Arizona, 2006. http://hdl.handle.net/10150/195953.
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Let X be a simply connected, compact Riemannian symmetric space. We can represent X as the homogeneous space U/K, where U is a simply connected compact Lie group, and K is the fixed point set of an involution θ of U. Let G be the complexification of U. We consider the intersections of the image of the Cartan embedding Φ : U/K → U ⊂ G : uK → uu⁻ᶿ with the strata of the Birkhoff (or triangular, or LDU) decomposition G = ⫫(w∈W) ∑(G/w), ∑(G/w) = N⁻wHN⁺ relative to a θ-stable decomposition of the Lie algebra, g = n⁻ ⊕h ⊕ n⁺. For a generic element g in this intersection, g ∈ Φ(U/K) ∩ ∑(G/1), this yields a unique triangular factorization g = ldu. Our main contribution is to produce explicit formulas for the diagonal term d in classical cases, using Cayley coordinates (this choice of coordinate is motivated by considerations beyond sheer convenience). These formulas have several applications: 1) we can compute π₀(Φ(U/K) \ ∩ ∑(G/1) ) explicitly; 2) we can compute ʃ(Φ(U/K))ᵃΦ^-iλ (where ᵃΦ is the positive part of d) using elementary techniques in rank 1 cases; 3) they are useful in explicitly calculating Evens-Lu Poisson structures on U=K (see [Caine(2006)]). Our set-up involves choosing specific representations of the various u in su(n;C) that are compatible with θ; that is, θ fixes each of the subspaces n⁻; h; and n⁺ which, in our setup, always consist of strictly lower triangular, diagonal, and strictly upper triangular matrices, respectively. The formulas contain determinants such as det(1 + X), where X is in ip, the -1-eigenspace of θ acting on the Lie algebra u. Due to the relatively sparse nature of these matrices, these determinants are often easily calculable, and we illustrate this with many examples.
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Silva, Giovane Ferreira. "Formalismo termodinâmico do conjunto irregular para médias de Birkhoff e expoentes de Lyapunov." Universidade Federal de Alagoas, 2011. http://repositorio.ufal.br/handle/riufal/1050.
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In this work, we study the set X ̇(φ,f) of points such that the Birkhoff averages do not exist. Following Thompson, our main result here is to show that the topological pressure of X ̇(φ,f) is total. As corollary, we get the some result for the Oseledets Irregular set for Lyapunov exponent in one dimension. For higher dimensions, this question is still open.Conselho Nacional de Desenvolvimento Científico e Tecnológico
Neste trabalho, estudamos o conjunto X ̇(φ,f) de pontos tal que as médias de Birkhoff não existe. Seguindo Thompson, nosso resultado principal aqui é mostrar que a pressão topológica de X ̇(φ,f) é total. Como corolário, damos o mesmo resultado para o conjunto Irregular de Oseledets para os expoentes de Lyapunov em dimensão um. Para dimensões maiores, esta questão está em aberto.
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Karouma, Abdulrahman. "A Class of Contractivity Preserving Hermite-Birkhoff-Taylor High Order Time Discretization Methods." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/32403.
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In this thesis, we study the contractivity preserving, high order, time discretization methods for solving non-stiff ordinary differential equations. We construct a class of one-step, explicit, contractivity preserving, multi-stage, multi-derivative, Hermite-Birkhoff-Taylor methods of order p=5,6, ..., 15, that we denote by CPHBT, with nonnegative coefficients by casting s-stage Runge-Kutta methods of order 4 and 5 with Taylor methods of order p-3 and p-4, respectively.
The constructed CPHBT methods are implemented using an efficient variable step algorithm and are compared to other well-known methods on a variety of initial value problems. The results show that CPHBT methods have larger regions of absolute stability, require less function evaluations and hence they require less CPU time to achieve the same accuracy requirements as other methods in the literature. Also, we show that the contractivity preserving property of CPHBT is very efficient in suppressing the effect of the propagation of discretization errors when a long-term integration of a standard N-body problem is considered.
The formulae of 49 CPHBT methods of various orders are provided in Butcher form.
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Wallez, Thomas. "Invariants iso-spectraux et théorèmes KAM." Thesis, Nantes, 2018. http://www.theses.fr/2018NANT4067/document.
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L’objectif de ce travail est d’établir des résultats de rigidité spectrale pour des familles C1 d’opérateurs (pseudo-)différentiels elliptiques auto-adjoints Pt, t ϵ [0, ẟ] sur une variété lisse compacte M sans bord de dimension n ≥ 2. Dans les deux premiers chapitres, on étudie des hamiltoniens proches d’un hamiltonien intégrable qui est non dégénéré au sens de Kolmogorov (Système KAM). On y construit une forme normale de Birkhoff au voisinage de chaque tore KAM ayant une fréquence diophantienne. Dans les chapitres 3 et 4 on établit une forme normale de Birkfoff quantique afin de construire des familles C1 de quasi-modes. Ces dernières permettent de relier les propriétés spectrales de Pt aux propriétés dynamiques des tores KAM. Les deux derniers chapitres proposent des applications en lien avec la transformée de Radon ainsi qu’une étude sur les surfaces de rotationThe aim of this work is to obtain spectral rigidity results for C1 families of elliptic self-adjoint (pseudo-)differential operators Pt, t ϵ [0, ẟ], on a smooth closed manifold M of dimension n ≥ 2. In the first two chapters, we investigate Hamiltonians close to a given integrable Hamiltonian which is non-degenerate in the sense of Kolmogorov (KAM system). This allows us to obtain a Birkhoff normal form in a neighborhood of any KAM tori with a Diophantine frequency. In the third and fourth chapters, we construct a quantum Birkhoff normal form and obtain C1 families of quasimodes. Using the quasi-modes, we establish a connection between the spectral properties of Pt and the dynamical properties of the KAM tori. The last two chapters provide applications of these results to the Radon transform and the surfaces of revolution
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Yagoub, Hemza. "Variable-step variable-order 3-stage Hermite-Birkhoff ODEDDE solver of order 5 to 15." Thesis, University of Ottawa (Canada), 2009. http://hdl.handle.net/10393/28075.
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This thesis presents the variable-step variable-order 3-stage Hermite-Birkhoff numerical solver HB515DDE of order 5 to 15. This method can solve ordinary and delay differential equations (ODEs/DDEs) with state-dependent, non-vanishing, small, vanishing and asymptotically vanishing delays. Delayed values are computed using Hermite interpolation and small delays are dealt with using extrapolation. Discontinuities in DDEs are located by a bisection method. HB515DDE was tested and compared with other solvers. The results are given along with the convergence theory which supports the experimentation.
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Silva, Romenique da Rocha. "Toros incompressíveis para ações Anosov de \'R POT. k\' sobre uma variedade de dimensão K+2." Universidade de São Paulo, 2011. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-14102011-160937/.
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Dentre todos os sistemas dinâmicos os sistemas Anosov têm atraído a atenção de muitos matemáticos. No caso de fluxo Anosov em uma variedade fechada M de dimensão três, Sérgio Fenley definiu o conceito de losangos no recobrimento universal de M e obteve resultados importantes envolvendo losangos e automorfismos do recobrimento universal. Seguindo o que foi feito por Fenley, e utilizando o conceito de losangos no espaço das órbitas do fluxo levantado (no recobrimento universal), Thierry Barbot obteve condições suficientes para que um toro incompressível numa 3-variedade fechada suportando um fluxo Anosov seja isotópico a um outro que é transverso ao fluxo. Neste trabalho consideramos ações Anosov de \'R POT. k\' sobre uma variedade fechada M de dimensão k + 2. Primeiramente, conseguimos resultados análogos aos de Fenley (sobre existência de losangos) para estas ações, e usando isso, finalmente obtemos condições suficientes para que um toro incompressível seja isotópico a um toro transverso à ação. Este último resultado é uma generalização de Barbot mencionado acimaAmong all dynamical systems the Anosov systems has attracted the attention of many mathematicians. In the case of an Anosov flow in a closed manifold M of dimension three, Sérgio Fenley defined the concept of lozenges in the universal covering of M and obtained important results involving lozenges and covering automorphism. Following what was made by Fenley, and using the concept of lozenge on the orbit space of the lifted flow (in the universal covering). Thierry Barbot obtains sufficient conditions for an incompressible torus in a closed 3-manifold supporting an Anosov flow to be isotopic to another which is transverse to flow. If this work we considered Anosov of \'R POT. k\' on a closed manifold M of dimension k + 2. First, we obtain analogous results those of Fenley (about existence of lozenges) for this actions, and using this, finally we obtain sufficient conditions for an incompressible torus to be isotopic to another torus which is transverse to action. This last result is a generalization of Barbot\'s result mentioned above
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Niles, David G. "Problème inverse de Riemann-Hilbert-Birkhoff et formules de connexion pour les transendentes [sic] de Painlevé III." Dijon, 2009. http://www.theses.fr/2009DIJOS069.
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Un système linéaire d'équations différentielles ordinaires est considéré qui est lié à la troisième équation de Painlevé par la méthode des déformations isomonodromiques. Ce système linéaire génère une application de monodromie pour laquelle la surjectivité est démontrée en utilisant la methode de la décomposition Riemann-Hilbert. Cette surjectivité permet de déterminer complètement les comportements asymptotiques des fonctions de Painlevé comme x tend vers le zéro dans un secteur qui contient la droite réelle positive. Les positions des singularités dans ce secteur sont aussi établiesA linear system of ordinary differential equations corresponding by the isomonodromy deformation method to the third Painlevé equation is considered. The surjectivity of the monodromy map generated by this system is proven using the Riemann-Hilbert factorization method. This allows a complete determination of the small x asymptotic behavior of the Painlevé III functions in a sector containing the positive real line. The locations of the singularities within this sector are also given
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Thomine, Damien. "Théorèmes limites pour les sommes de Birkhoff de fonctions d'intégrale nulle en théorie ergodique en mesure infinie." Thesis, Rennes 1, 2013. http://www.theses.fr/2013REN1S194/document.
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Ce travail est consacré à certaines classes de systèmes dynamiques ergodiques, munis d'une mesure invariante infinie, telles que des applications de l'intervalle avec un point fixe neutre ou des marches aléatoires. Le comportement asymptotique des sommes de Birkhoff d'observables d'intégrale non nulle est assez bien connu, pour peu que le système ait une certaine forme d'hyperbolicité. Une situation particulièrement intéressante est celle des tours au-dessus d'une application Gibbs-Markov. Nous cherchons dans ce contexte à étudier le cas d'observables d'intégrale nulle. Nous obtenons ainsi une forme de théorème central limite pour des systèmes dynamiques munis d'une mesure infinie. Après avoir introduit l'ensemble des notions nécessaires, nous adaptons des résultats de E. Csáki et A. Földes sur les marches aléatoires au cas des applications Gibbs-Markov. Les théorèmes d'indépendance asymptotique qui en découlent forment le cœur de cette thèse, et permettent de démontrer un théorème central limite généralisé. Quelques variations sur l'énoncé de ce théorème sont obtenues. Ensuite, nous abordons les processus en temps continu, tels que des semi-flots et des flots. Un premier travail consiste à étudier les propriété en temps grand du temps de premier retour et du temps local pour des extensions de systèmes dynamiques, ce qui se fait par des méthodes spectrales. Enfin, par réductions successives, nous pouvons obtenir une version du théorème central limite pour des flots périodiques, et en particulier le flot géodésique sur le fibré tangent unitaire de certaines variétés périodiques hyperboliquesThis work is focused on some classes of ergodic dynamical systems endowed with an infinite invariant measure, such as transformations of the interval with a neutral fixed point or random walks. The asymptotic behavior of the Birkhoff sums of observables with a non-zero integral is well known, as long as the system shows some kind of hyperbolicity. The towers over a Gibbs-Markov map are especially interesting. In this context, we aim to study the case of observables whose integral is zero. We get the equivalent of a central limit theorem for some dynamical systems endowed with an infinite measure. After we introduce the necessary definitions, we adapt some results by E. Csáki and A. Földes on random walks to the case of Gibbs-Markov maps. We derive a theorem on the asymptotic independence of Birhoff sums, which is the core of this thesis, and from this point we work out a generalised central limit theorem. We also prove a few variations on this generalised central limit theorem. Then, we study dynamical systems in continuous time, such as semi-flows and flows. We first work on the asymptotic properties of the first return time and the local time for extensions of dynamical systems; this is done by spectral methods. Finally, step by step, we extend our generalised central limit theorem to cover some periodic flows, and in particular the geodesic flow on the unitary tangent bundle of some hyperbolic periodic manifolds
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Alzahrani, Abdulrahman. "Contractivity-Preserving Explicit 2-Step, 6-Stage, 6-Derivative Hermite-Birkhoff–Obrechkoff Ode Solver of Order 13." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/32564.
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In this thesis, we construct a new optimal contractivity-preserving (CP) explicit, 2-step, 6-stage, 6-derivative, Hermite--Birkhoff--Obrechkoff method of order 13, denoted by HBO(13) with nonnegative coefficients, for solving nonstiff first-order initial value problems y'=f(t,y), y(t_0)=y_0. This new method is the combination of a CP 2-step, 6-derivative, Hermite--Obrechkoff of order 9, denoted by HO(9), and a 6-stage Runge-Kutta method of order 5, denoted by RK(6,5). The new HBO(13) method has order 13.
We compare this new method, programmed in Matlab, to Adams-Bashforth-Moulton method of order 13 in PECE mode, denoted by ABM(13), by testing them on several frequently used test problems, and show that HBO(13) is more efficient with respect to the CPU time, the global error at the endpoint of integration and the relative energy error. We show that the new HBO(13) method has a larger scaled interval of absolute stability than ABM(13) in PECE mode.
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Gill, Jonna. "The k-assignment Polytope and the Space of Evolutionary Trees." Licentiate thesis, Linköping : Univ, 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-5677.
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Matacchione, Roberto. "Una rassegna di teoremi ergodici. Dal teorema ergodico di Birkhoff Khinchin al teorema ergodico quantistico di von Neumann." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2014. http://amslaurea.unibo.it/6958/.
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Choirat, Christine. "Contributions à l'étude du théorème ergodique de Birkhoff, de l'épiconvergence et des ensembles aléatoires : aspects théoriques et applications." Paris 9, 2003. https://portail.bu.dauphine.fr/fileviewer/index.php?doc=2003PA090008.
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Pereira, Miriam da Silva [UNESP]. "Teoria de singularidades e classificação de problemas de bifurcação Z2-equivariantes de Corank 2." Universidade Estadual Paulista (UNESP), 2006. http://hdl.handle.net/11449/94214.
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Neste trabalho classificamos problemas de bifurcação Z2-equivariantes de corank 2 até co- dimensão 3 via técnicas da Teoria de Singularidades. A abordagem para classificar tais problemas é baseada no processo de redução à forma normal de Birkhoff para estudar a interação de modos Hopf-Pontos de Equilíbrio. O comportamento geométrico das soluções dos desdobramentos das formas normais obtidas é descrito pelos diagramas de bifurcação e estudamos a estabilidade assintótica desses ramos.
In this work we classify the Z2-equivariant corank 2 bifurcation problems up to codimension 3 via Singularity Theory techniques. The approach to classify such problems is based on the Birkhoff normal form to study Hopf-Steady- State mode interaction. The geometrical behavior of the solutions of the unfolding of the normal forms is described by the bifurcation diagrams and we study the asymptotic stability of such branches.
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Li, Yi. "Variable-step variable-order 3-stage Hermite-Birkhoff ODE solver of order 5 to 15 with a C++ program." Thesis, University of Ottawa (Canada), 2008. http://hdl.handle.net/10393/28001.
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Variable-step variable-order 3-stage Hermite-Birkhoff (HB) methods HB( p)3 of order p = 5 to 15 are constructed for solving nonstiff differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge-Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems of HB type. Fast algorithms are developed for solving these systems in O(p2) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The order and stepsize of these methods are controlled by four local error estimators. These methods, when programmed in Matlab, are superior to Matlab's ode113 in solving several problems often used to test higher order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. On the other hand, HB(5-15)3 are programmed in object-oriented C++ and the Dormand-Prince 13-stage nested Runge-Kutta pair DP(8,7)13M are programmed in C. DP(8,7) is found to use less CPU time, have smaller maximum global error but require a larger number of function evaluations than HB(5-15)3. However, for expensive equations, such as the Cubicwave, HB(5-15)3 is superior. In the C++ program, array and matrix are considered to be new objects. Algorithms, testing programs and new objects are structured separately as header files.
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Zhang, Yu. "Variable-step variable-order 3-stage Hermite-Birkhoff-Obrechkoff ODE solver of order 4 to 14 with a C program." Thesis, University of Ottawa (Canada), 2007. http://hdl.handle.net/10393/27500.
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Variable-step variable-order 3-stage Hermite-Birkhoff-Obrechkoff methods of order 4 to 14, denoted by HBO(4-14)3, are constructed for solving nonstiff systems of first-order differential equations of the form y' = f (x, y), y( x0) = y0. These methods use y' and y" as in Obrechkoff's method. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep- and Runge-Kutta-type order conditions which are reorganized into linear Vandermonde-type systems. Fast algorithms are developed for solving these systems to obtain Hermite-Birkhoff interpolation polynomials in terms of generalized Lagrange basis functions. The new methods have larger regions of absolute stability than Adams-Bashforth-Moulton methods of comparable orders in PECE mode. The order and stepsize of these methods are controlled by four local error estimators. When programmed in Matlab, HBO(4-14)3 are superior to Matlab's ode113 in solving several problems often used to test higher order ODE solvers on the basis of the number of function evaluations, CPU time, and maximum global error. It is also superior to the variable-step 3-stage HBO(14)3 of order 14 on some problems. When programmed in C, HBO(4-14)3 is superior to the Dormand-Prince Runge-Kutta nested pair DP(8,7)13M in solving expensive equations over a long period of time. HBO(4-14)3 has been implemented in C. The C program for HBO(4-14)3 is an important contribution of this thesis.
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Zhuang, Yuchuan. "Variable-step variable-order 2-stage Hermite-Birkhoff-Obrechkoff ODE solver of order 3 to 14 with a C program." Thesis, University of Ottawa (Canada), 2008. http://hdl.handle.net/10393/27746.
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Variable-step variable-order 2-stage Hermite-Birkhoff-Obrechkoff (HBO) methods, HBO(p)2, of order p = 3 to 14, named HBO(3-14)2, are constructed for solving nonstiff first-order differential equations. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to multistep and Runge-Kutta type order conditions which are reorganized into linear Vandermonde-type systems of HBO type. Fast algorithms are developed for solving these systems in O( p2) operations to obtain Hermite-Birkhoff interpolation polynomials in terms of generalized Lagrange basis functions. The order and step size of these methods are controlled by four local error estimators. For numerical computation the lower order 3 is raised to 4 since HBO(4-14)2 produces better results. When programmed in Matlab, HBO (4-14) 2 is superior to Matlab's ode113 in solving several problems often used to test higher order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. On the other hand, HBO (4-14)2 and the Dormand-Prince 13-stage nested Runge-Kutta pair DP(8,7)13M are programmed in C. In this case, DP(8,7) uses less CPU time, have smaller maximum global error but require a larger number of function evaluations than HBO(4-14)2 on nonexpensive problems. However, for expensive equations, such as the Cubicwave, HBO(4-14)2 is superior. Compared with previous results obtained by the 3-stage HBO(4-14)3 on Van der Pol equations with increasing value of epsilon, the new HBO(4-14)2 finally dominates.
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Pittman-Polletta, Benjamin Rafael. "Factorization in unitary loop groups and reduced words in affine Weyl groups." Diss., The University of Arizona, 2010. http://hdl.handle.net/10150/194348.
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The purpose of this dissertation is to elaborate, with specific examples and calculations, on a new refinement of triangular factorization for the loop group of a simple, compact Lie group K, first appearing in Pickrell & Pittman-Polletta 2010. This new factorization allows us to write a smooth map from the unit circle into K (having a triangular factorization) as a triply infinite product of loops, each of which depends on a single complex parameter. These parameters give a set of coordinates on the loop group of K.The order of the factors in this refinement is determined by an infinite sequence of simple generators in the affine Weyl group associated to K, having certain properties. The major results of this dissertation are examples of such sequences for all the classical Weyl groups.We also produce a variation of this refinement which allows us to write smooth maps from the unit circle into the special unitary group of n by n matrices as products of 2n+1 infinite products. By analogy with the semisimple analog of our factorization, we suggest that this variation of the refinement has simpler combinatorics than that appearing in Pickrell & Pittman-Polletta 2010.
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Abbondanza, Nicola. "Trasformazioni che conservano la misura." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/9688/.
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Si consideri un insieme X non vuoto su cui si costruisce una sigma-algebra F, una trasformazione T dall'insieme X in se stesso F-misurabile si dice che conserva la misura se, preso un elemento della sigma-algebra, la misura della controimmagine di tale elemento è uguale a quella dell'elemento stesso. Con questa nozione si possono costruire vari esempi di applicazioni che conservano la misura, nell'elaborato si presenta la trasformazione di Gauss. Questo tipo di trasformazioni vengono utilizzate nella teoria ergodica dove ha senso considerare il sistema dinamico a tempi discreti T^j x; dove x = T^0 x è un dato iniziale, e studiare come la dinamica dipende dalla condizione iniziale x. Il Teorema Ergodico di Von Neumann afferma che dato uno spazio di Hilbert H su cui si definisce un'isometria U è possibile considerare, per ogni elemento f dello spazio di Hilbert, la media temporale di f che converge ad un elemento dell'autospazio relativo all'autovalore 1 dell'isometria. Il Teorema di Birkhoff invece asserisce che preso uno spazio X sigma-finito ed una trasformazione T non necessariamente invertibile è possibile considerare la media temporale di una funzione f sommabile, questa converge sempre ad una funzione f* misurabile e se la misura di X è finita f* è distribuita come f. In particolare, se la trasformazione T è ergodica si avrà che la media temporale e spaziale coincideranno.
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Pereira, Miriam da Silva. "Teoria de singularidades e classificação de problemas de bifurcação Z2-equivariantes de Corank 2 /." São José do Rio Preto : [s.n.], 2006. http://hdl.handle.net/11449/94214.
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Orientador: Angela Maria SittaBanca: Maria Aparecida Soares Ruas
Banca: Claudio Aguinaldo Buzzi
Resumo: Neste trabalho classificamos problemas de bifurcação Z2-equivariantes de corank 2 até co- dimensão 3 via técnicas da Teoria de Singularidades. A abordagem para classificar tais problemas é baseada no processo de redução à forma normal de Birkhoff para estudar a interação de modos Hopf-Pontos de Equilíbrio. O comportamento geométrico das soluções dos desdobramentos das formas normais obtidas é descrito pelos diagramas de bifurcação e estudamos a estabilidade assintótica desses ramos.
Abstract: In this work we classify the Z2-equivariant corank 2 bifurcation problems up to codimension 3 via Singularity Theory techniques. The approach to classify such problems is based on the Birkhoff normal form to study Hopf-Steady- State mode interaction. The geometrical behavior of the solutions of the unfolding of the normal forms is described by the bifurcation diagrams and we study the asymptotic stability of such branches.
Mestre
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Rodríguez, Ruiz José. "Integración en espacios de Banach." Doctoral thesis, Universidad de Murcia, 2006. http://hdl.handle.net/10803/10963.
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Esta tesis doctoral se enmarca dentro de la teoría de integración de funciones con valores en espacios de Banach. Analizamos con detalle la integral de Birkhoff de funciones vectoriales, así como sus correspondientes versiones dentro de los contextos de la integración respecto de medidas vectoriales y la integración de multi-funciones. Comparamos estos métodos de integración con otros bien conocidos (integrales de Bochner, Pettis, McShane, Debreu, etc.). Caracterizamos, en términos de integración vectorial, algunas propiedades de los espacios de Banach donde las (multi-) funciones toman valores.The general framework of this memoir is the theory of integration of functions with values in Banach spaces. We analyze in detail the Birkhoff integral of vector-valued functions, as well as its corresponding versions within the settings of integration with respect to vector measures and integration of multi-valued functions. We compare these methods of integration with others which are well known (Bochner, Pettis, McShane, Debreu, etc.). We characterize, in terms of vector integration, some properties of the Banach spaces where the (multi-) functions take their values.
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Shannon, Mario. "Dehn surgeries and smooth structures on 3-dimensional transitive Anosov flows." Thesis, Bourgogne Franche-Comté, 2020. http://www.theses.fr/2020UBFCK035.
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Cette thèse porte sur les chirurgies de Dehn et les structures différentielles associées aux flots d'Anosov transitifs en dimension trois. Les flots d'Anosov constituent une classe très importante des systèmes dynamiques, par leurs propriétés chaotiques persistantes par perturbations, autant que par leur riche interaction avec la topologie de la variété ambiante. Bien que beaucoup soient connus sur le comportement dynamique et ergodique de ces flots, il n'y a pas une compréhension assez claire sur la classification de ses différentes classes d'équivalence orbitale. Jusqu'à ce moment, les plus grands progrès ont été fait en dimension trois, où il y une famille de techniques pour la construction d'exemples de flot d'Anosov connue comme chirurgies.Pendant la réalisation de cette thèse, dans un premier temps nous nous sommes intéressés à une chirurgie en particulier, connue comme la chirurgie de Goodman. Cette procédure consiste à choisir une orbite périodique du flot et réaliser une chirurgie de Dehn autour de cette orbite, adaptée au flot d'une façon telle qu'on obtient une nouvelle variété munie d'un flot d'Anosov. La problématique que soulève cette technique est que, pour la réalisation de la chirurgie, un des paramètres à choisir est une surface plongée dans la 3-variété et un difféomorphisme défini sur elle. De ce fait, l'espace de paramètres est, a priori, de dimension infinie et, pourtant, ce n'est pas facile d'avoir un contrôle sur la classe d'équivalence du flot obtenu par cette méthode. Il existe une deuxième procédure, qui peut-être interprétée comme une version infinitésimale de celle qui précède, connue comme la chirurgie de Fried. Celle-ci consiste à éclater l'orbite périodique, obtenant de ce fait un flot défini sur une variété à bord, puis collapser cette composante de bord d'une façon non-triviale et produire un nouveau flot. Cette chirurgie produit des flots univoquement définis, mais ceux-ci ne sont pas munis d'une structure hyperbolique naturelle. Ils sont, par construction, flots topologiquement d'Anosov.Notre contribution consiste à montrer que, si on assume de plus que les flots sont transitifs, alors une chirurgie de Goodman et une chirurgie de Fried autour de la même orbite périodique produisent des flots équivalents, à égal élection de paramètres entiers.Dans un second temps nous avons travaillé sur une question un peu plus abstraite, mais qui est naturellement liée à certaines procédures techniques dans la construction de flots hyperboliques. C'est le problème de savoir si tout flot dit topologiquement d'Anosov (i.e. expansif et qui satisfait la propriété de shadowing de Bowen correspond à un flot hyperbolique différentiable, à équivalence orbitale près. Dans le cas particulier où le flot est transitif, il est connu depuis très longtemps qu'il peut être muni d'une structure non-uniformément hyperbolique définie dans le complémentaire d'un ensemble fini d'orbites périodiques. La plus grande difficulté est de construire des modèles (globalement) hyperboliques associés au flot original.Dans ce contexte, notre contribution consiste à montrer que tout flot topologiquement d'Anosov et transitif, défini dans une variété de dimension trois, est orbitalement équivalent à un flot d'Anosov lisseThe present thesis is about Dehn surgeries and smooth structures associated with transitive Anosov flows in dimension three. Anosov flows constitute a very important class of dynamical systems, because of its persistent chaotic behaviour, as well as for its rich interaction with the topology of the ambient space. Even if a lot is known about the dynamical and ergodic properties of these systems, there is not a clear understanding about how to classify its different orbital equivalence classes. Until now, the biggest progress has been done in dimension three, where there is a family of techniques intended for the construction of Anosov flows called surgeries.During the realization of this thesis, in a first time we have been interested in a particular surgery method, known as the Goodman surgery. This method consists in make a Dehn surgery on a chosen periodic orbit, but adapted to the flow, in such a way to obtain a new manifold equipped with an Anosov flow. For making this surgery, one of the parameters that has to be chosen is an embedded surface in the 3-manifold and a diffeomorphism defined on it. Thus, the parameter space is, a priori, of infinite dimension and it is not easy to have control on the orbital equivalence class of the obtained flow. There exists a second method, that can be interpreted as an infinitesimal version of the previous one, known as the Fried surgery. It consists in making a blow-up of the flow along the periodic orbit, obtaining in this way a flow in a manifold with boundary, for then blowing-down the boundary component in a non-trivial way and produce a new flow. This surgery produces flows defined in a unique way, but they are not equipped with a natural uniformly hyperbolic structure. They are, by construction, topological Anosov flows.Our contribution is to show that, if we assume that the flow is transitive, then a Goodman surgery or a Fried surgery performed on a periodic orbit produce orbitally equivalent flows, for the same choice of integer parameters.In a second time, we have been interested for a more abstract question, but which is also related to some technical issues in the construction of hyperbolic flows. It is the problem of determining if every topologically Anosov flow (i.e. expansive and satisfying the Bowen shadowing property) correspond to a smooth hyperbolic flow, up to orbital equivalence. In the particular case that the flow is transitive, it has been known that there exists a non-uniformly hyperbolic structure defined in the complement of a finite set of periodic orbits. The main difficulty is the construction of (global) hyperbolic models associated to the original flow.In this setting, our contribution is to show that every transitive topologically Anosov flow on a closed manifold is orbital equivalent to a smooth Anosov flow
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Shonoda, Emad N. Naseem. "On Ruled Surfaces in three-dimensional Minkowski Space." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2010. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-63555.
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In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common
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Ducatez, Raphaël. "Analyse mathématique de divers systèmes de particules en milieu désordonné." Thesis, Paris Sciences et Lettres (ComUE), 2018. http://www.theses.fr/2018PSLED013/document.
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Cette thèse est consacrée à l’étude mathématique de divers systèmes de particules classiques et quantiques, en milieu désordonné. Elle comprend quatre travaux publiés ou soumis. Dans le premier nous fournissons une nouvelle formule permettant de prouver la localisation d’Anderson en une dimension d’espace et de caractériser la décroissance des fonctions propres à l’infini. Le second contient l’une des premières preuves de la localisation pour une infinité de particules en intéraction, dans l’approximation d’Hartree-Fock. Le troisième est dédié au modèle d’Anderson soumis à une perturbation périodique en temps. Sous certaines conditions sur la fréquence d’oscillation nous prouvons l’absence de diffusion. Dans le dernier travail nous montrons la décroissancedes corrélations pour le modèle du Jellium en une dimension dans un fond inhomogène, en utilisant la distance de Hilbert sur les cônes et le théorème de Birkhoff-HopfThis thesis is devoted to the mathematical study of some systems of classical and quantum particles, in a disordered medium. It comprises four published or submitted works. In the first one we provide a new formula allowing to prove Anderson localisation in one space dimension and to characterise the decay at infinity of the eigenfunctions. The second contains one of the first proofs of localisation for infinitely many particles in interaction, in the Hartree-Fock approximation. The third work is dedicated to the Anderson model in a time-periodic perturbation. Under certain conditions on the oscillation frequency we prove the absence of diffusion. In the last work we show the decay of correlations for the one-dimensional Jellium model in an inhomogeneous background, using the Hilbert distance on cones and the Birkhoff-Hopf theorem
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Imekraz, Rafik. "Etude dynamique de quelques équations aux dérivées partielles hamiltoniennes non linéaires à potentiel confinant." Nantes, 2010. http://archive.bu.univ-nantes.fr/pollux/show.action?id=f78473aa-7d4c-4a95-a2c8-e6d600ac58cd.
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L'objet de la thèse est l'étude de la stabilité des solutions de certaines équations aux dérivées partielles (EDP) non linéaires de type Schrödinger à potentiel confinant sur Rn et à condition initiale régulière. Les deux potentiels étudiés sont l'oscillateur harmonique multidimensionnel et le potentiel polynomial confinant unidimensionnel. La stabilité en question peut se résumer ainsi : il existe un intervalle temporel d'existence de la solution telle que sa longueur dépend de facon polynomiale de la petitesse de la condition initiale (existence presque globale) et sur lequel la solution reste dynamiquement proche de la solution d'une équation explicite complètement intégrable (avec même condition initiale). Nous utilisons la théorie des formes normales de Birkhoff pour aborder notre problème. Le point clé est le caractère hamiltonien des EDP concernées. Nous créons un modèle différentiel abstrait (qui comprend l'EDP étudiée) et l'on y démontre l'existence de formes normales de Birkhoff à tout ordre, c'est-à-dire des renormalisations adéquates de l'hamiltonien qui en l'occurrence impliquent la stabilitéThis thesis is concerned by stability of solutions of some non linear Schroedinger partial differential equations (PDE) on Rn with a confining potential and a regular initial condition. Two potentials are studied : the harmonic oscillator multidimensional and the polynomial confining potential unidimensional. In our context, the stability means roughly the following : the solution exists on a time-interval whose length depends polynomially on the smallness of the initial condition (almost global existence) and stays near the solution of an explicit completely integrable equation with the same initial condition. We use the Birkhoff's normal forms theory to handle our issue. The key point is the Hamiltonian structure of our PDE. We create an abstract differential model (which encompasses our PDE) and prove that it has a Birkhoff's normal form of all order, ie a proper renormalization of the Hamiltonian which ensures in particular the stability
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Herlemont, Basile. "Differential calculus on h-deformed spaces." Thesis, Aix-Marseille, 2017. http://www.theses.fr/2017AIXM0377/document.
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L'anneau $\Diff(n)$ des opérateurs différentiels $\h$-déformés apparaît dans la théorie des algèbres de réduction.Dans cette thèse, nous construisons les anneaux des opérateurs différentiels généralisés sur les espaces vectoriels $\h$-déformés de type $\gl$. Contrairement aux espaces vectoriels $q$-déformés pour lequel l'anneau des opérateurs différentiels est unique \`a isomorphisme pr\`es, l'anneau généralisé des opérateurs différentiels $\h$-déformés $\Diffs(n)$ est indexée par une fonction rationnelle $\sigma$ en $n$ variables, solution d'un syst\`eme d\'eg\'en\'er\'e d'\'equations aux diff\'erences finies. Nous obtenons la solution g\'en\'erale de ce syst\`eme. Nous montrons que le centre de $\Diffs(n)$ est un anneau des polynômes en $n$ variables. Nous construisons un isomorphisme entre des localisations de l'anneau $\Diffs(n)$ et de l’algèbre de Weyl $\text{W}_n$ l’étendue par $n$ indéterminés. Nous présentons des conditions irréductibilité des modules de dimension fini de $\Diffs(n)$. Finalement, nous discutons des difficultés a trouver les constructions analogues pour l'anneau $\Diff(n,N)$ correspondant \`a $N$ copies de $\Diff(n)$The ring $\Diff(n)$ of $\h$-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the $\h$-deformed vector spaces of $\gl$-type. In contrast to the $q$-deformed vector spaces for which the ring of differential operators is unique up to an isomorphism, the general ring of $\h$-deformed differential operators $\Diffs(n)$ is labeled by a rational function $\sigma$ in $n$ variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system. We show that the center of $\Diffs(n)$ is a ring of polynomials in $n$ variables. We construct an isomorphism between certain localizations of $\Diffs(n)$ and the Weyl algebra $\W_n$ extended by $n$ indeterminates. We present some conditions for the irreducibility of the finite dimensional $\Diffs(n)$-modules. Finally, we discuss difficulties for finding analogous constructions for the ring $\Diff(n, N)$ formed by several copies of $\Diff(n)$
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Belhaj, Mohamed Mohamed. "Renormalisation dans les algèbres de HOPF graduées connexes." Thesis, Clermont-Ferrand 2, 2014. http://www.theses.fr/2014CLF22515/document.
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Dans cette thèse, nous nous intéressons à la renormalisation de Connes et Kreimer dans le contexe des algèbres de Hopf de graphes de Feynman spécifiés. Nous construisons une structure d'algèbre de Hopf $\mathcal{H}_\mathcal{T}$ sur l'espace des graphes de Feynman spécifié d'une théorie quantique des champs $\mathcal{T}$. Nous définissons encore un dédoublement $\wt\mathcal{D}_\mathcal{T}$ de la bigèbre de graphes de Feynman spécifiés, un produit de convolution \divideontimes et un groupe de caractères de cette algèbre de Hopf à valeurs dans une algèbre commutative qui prend en compte la dépendance en les moments extérieurs. Nous mettons en place alors la renormalisation décrite par A. Connes et D. Kreimer et la décomposition de Birkhoff pour deux schémas de renormalisation : le schéma minimal de renormalisation et le schéma de développement de Taylor. Nous rappelons la définition des intégrales de Feynman associées à un graphe. Nous montrons que ces intégrales sont holomorphes en une variable complexe D dans le cas des fonctions de Schwartz, et qu'elles s'étendent en une fonction méromorphe dans le cas des fonctions de types Feynman. Nous pouvons alors déterminer les parties finies de ces intégrales en utilisant l'algorithme BPHZ après avoir appliqué la procédure de régularisation dimensionnelleIn this thesis, we study the renormalization of Connes-Kreimer in the contex of specified Feynman graphs Hopf algebra. We construct a Hopf algebra structure $\mathcal{H}_\mathcal{T}$ on the space of specified Feynman graphs of a quantum field theory $\mathcal{T}$. We define also a doubling procedure for the bialgebra of specified Feynman graphs, a convolution product and a group of characters of this Hopf algebra with values in some suitable commutative algebra taking momenta into account. We then implement the renormalization described by A. Connes and D. Kreimer and the Birkhoff decomposition for two renormalization schemes: the minimal subtraction scheme and the Taylor expansion scheme.We recall the definition of Feynman integrals associated with a graph. We prove that these integrals are holomorphic in a complex variable D in the case oh Schwartz functions, and that they extend in a meromorphic functions in the case of a Feynman type functions. Finally, we determine the finite parts of Feynman integrals using the BPHZ algorithm after dimensional regularization procedure
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Eloy, Anton. "Classification et géométrie des équations aux q-différences : étude globale de q-Painlevé, classification non isoformelle et Stokes à pentes arbitraires." Thesis, Toulouse 3, 2016. http://www.theses.fr/2016TOU30223/document.
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Cette thèse s'intéresse à la classification géométrique, locale et globale, des équations aux q-différences. Dans un premier temps nous réalisons une étude globale de certains systèmes dérivés des équations de q-Painlevé et introduits par Murata, en proposant une correspondance de Riemann-Hilbert-Birkhoff entre de tels systèmes et leurs matrices de connexion. Dans un second temps nous nous intéressons à la classification locale, en construisant un fibré vectoriel équivariant sur l'espace des classes formelles à deux pentes dont la fibre au dessus d'une classe formelle est l'espace de ses classes analytiques isoformelles. Ceci fait, voyant que l'action du groupe des automorphismes du gradué s'impose naturellement dans l'étude de ce fibré, nous nous intéressons à l'espace des classes analytiques, soit des classes analytiques isoformelles modulo cette action, dont nous proposons dans un cas restreint une première approche de classification via l'utilisation de variétés toriques. Dans un troisième temps nous construisons, via des transformations de q-Borel et de q-Laplace, des q-Stokes, soit des solutions méromorphes de systèmes, dans le cadre des systèmes à deux pentes dont une non entière et une nulleThis thesis falls within the context of global and local geometric classification of q-difference equations. In a first part we study the global behaviour of some systems derived from q-Painlevé equations and introduced by Murata. We do so by constructing a Riemann-Hilbert-Birkhoff correspondence between such systems and their connexion matrices. In a second part we work on local classification by providing a construction of an equivariant vector bundle over the space of all formal classes with two slopes, the fibre over a formal class being the space of its isoformal analytic classes. As the action of the group of automorphisms of the graded module arises naturally when we study this bundle, we take an interest in the study of the space of analytic classes, which is the space of isoformal analytic classes modulo this action. We propose a first approach of such a classification by using toric varieties. In a third part we construct q-Stokes, i.e. meromorphic solutions of systems, in the context of systems with one non-integral slope and one equal to zero, this by using q-Borel and q-Laplace transforms
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Nguyen, Tien Zung. "A la recherche des tores perdus." Habilitation à diriger des recherches, Université Montpellier II - Sciences et Techniques du Languedoc, 2001. http://tel.archives-ouvertes.fr/tel-00001283.
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C'est l'histoire d'un mathématicien qui est allé à la recherche des tores perdusdans la jungle des systèmes complètement intégrables. Il a trouvé des feuilles
particulières et des tores pour construire une petite cabane qui donne une vue
topologique sur la jungle.
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Yan, Jingzhi. "Utilisation de feuilletages transverse à l'étude d'homéomorphismes préservant l'aire de surfaces." Thesis, Paris 6, 2014. http://www.theses.fr/2014PA066458/document.
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Cette thèse concerne les homéomorphismes de surfaces.Soit f un difféomorphisme d'une surface M préservant l'aire et isotope à l'identité. Si f a un point fixe contractile isolé et dégénéré z0 avec un indice de Lefschetz égal à 1, et si l'aire de M est finie, nous prouverons au chapitre 3 que z0 est accumulé non seulement par des points périodiques mais aussi par des orbites périodiques au sens de la mesure. Plus précisément, la mesure de Dirac en z0 est la limite en topologie faible-étoile d'une suite de probabilités invariantes supportées par des orbites périodiques. Notre preuve est totalement topologique et s'applique au cas d'homéomorphismes en considérant l'ensemble de rotation local.Au chapitre 4, nous étudierons des homéomorphismes préservant l’aire et isotope à l’identité. Nous prouverons l’existence d'isotopies maximales particulières: les isotopies maximales à torsion faible. En particulier, lorsque f est un difféomorphisme ayant un nombre fini de points fixes tous non-dégénérés, une isotopie I joignant l'identité à f est à torsion faible si et seulement si pour tout point z fixé le long de I, le nombre de rotation (réel) ρ(I,z), qui est bien défini quand on éclate f en z, est contenu dans (-1,1). Nous démontrerons l'existence d'isotopies maximales à torsion faible, et nous étudierons la dynamique locale de feuilletages transverses à l'isotopie près des singularités isolées.Au chapitre 5, nous énoncerons une généralisation d'un théorème de Poincaré-Birkhoff local au cas où il existe des points fixes au bordThis thesis concerns homeomorphisms of surfaces.Let f be an area preserving diffeomorphism of an oriented surface M isotopic to the identity. If f has an isolated degenerate contractible fixed point z0 with Lefschetz index one, and if the area of M is finite, we will prove in Chapter 3 that z0 is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at z0 is the limit in weak-star topology of a sequence of invariant probability measures supported on periodic orbits. Our proof is purely topological and will works for homeomorphisms and is related to the notion of local rotation set.In chapter 4, we will define a kind of identity isotopies: torsion-low isotopies. In particular, when f is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity isotopy I of f is torsion-low if and only if for every point z fixed along the isotopy, the (real) rotation number ρ(I,z), which is well defined when one blows-up f at z, is contained in (-1,1). We will prove the existence of torsion-low maximal identity isotopies, and we will deduce the local dynamics of the transverse foliations of any torsion-low maximal isotopy near any isolated singularity.In chapter 5, we will generalize a local Poincaré-Birkhoff theorem to the case where there exist fixed points on the boundary
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Deneufchâtel, Matthieu. "Intégrales Itérées en Physique Combinatoire." Phd thesis, Université Paris-Nord - Paris XIII, 2012. http://tel.archives-ouvertes.fr/tel-00736727.
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Nous présentons différents résultats liés par les outils et les structures qu'ils font intervenir (intégrales itérées, produits de mélange). Dans la première partie, nous considérons le calcul de certaines intégrales de type Selberg et leurs limites lorsque le nombre de variables tend vers l'infini. Dans le cas général, on montre que le résultat s'exprime comme un produit dont le nombre de facteurs ne dépend pas du nombre de variables (sous certaines conditions). Si la puissance du déterminant de Vandermonde vaut 2, il est possible de calculer la limite de ces intégrales lorsque le nombre de variables tend vers l'infini à l'aide d'opérateurs liés à l'interpolation de Newton. Dans la seconde partie, nous étudions les propriétés de dépendance linéaire de familles de fonctions obtenues par intégrales itérées et donnons un critère qui permet d'assurer l'indépendance linéaire d'une famille infinie de fonctions à partir de l'étude des relations entre les fonctions obtenues par intégrales simples. Nous montrons comment construire effectivement les corps de germes de fonctions analytiques nécessaires et en donnons quelques exemples qui permettent d'étendre les résultats connus sur les hyperlogarithmes. Ensuite, nous étudions certaines bases de l'algèbre libre dans le but d'appliquer la factorisation de Schützenberger. Nous rappelons quelques résultats classiques, puis nous intéressons à la famille obtenue à partir des mots de Lyndon. Celle-ci ne permet pas d'écrire la factorisation qui nous intéresse mais nous précisons les caractéristiques de sa famille duale. Enfin, nous donnons un critère relatif à deux familles en dualité assurant que l'on peut écrire cette factorisation.
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Decaens, Simon. "Une histoire de la théorie des treillis au sein de l'American Mathematical Society entre 1933 et 1948." Thesis, Sorbonne Paris Cité, 2018. http://www.theses.fr/2018USPCC308.
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Introduite en 1933 par Garrett Birkhoff, la théorie des treillis semble s’établir, en une quinzaine d’années, comme un domaine autonome des mathématiques, dont l’essor se situe dans un contexte de circulation de l’algèbre moderne aux États-Unis. Ce travail questionne l’apparition et le développement d’une théorie des treillis, ses liens avec l’algèbre moderne et le rôle de l’American Mathematical Society (AMS) dans ce processus. Après avoir problématisé la catégorie historiographique de théorie, nous envisagerons la théorie des treillis selon trois biais différents. Premièrement, nous l’aborderons à travers les articles de G. Birkhoff et Øystein Ore, souvent considérés comme fondateurs de la théorie. Ici, la théorie est un objet explicitement reconnu par les acteurs pour désigner et relier leurs travaux. Comme catégorie d’analyse, elle masque cependant leur diversité en les agrégeant sous une même dénomination non-problématisée. Deuxièmement, la théorie sera envisagée à une échelle plus large, à partir de publications de membres de l’AMS s’intéressant aux treillis. Elle apparaît alors comme un ensemble de pratiques partagées par un collectif de mathématicien·ne·s. Enfin, dans un dernier chapitre nous aborderons la promotion de la théorie des treillis au sein de l’AMS. Je tenterai de montrer qu’elle profite à la fois d’un statut d’« algèbre abstraite américaine » et des positions de ses promoteur·rice·s dans la sociétéIntroduced in 1933 by Garrett Birkhoff, Lattice Theory seems to settle, in about fifteen years, as an autonomous domain of mathematics, whose rise takes place in a context of circulation of modern algebra in the United-States. The current work questions the appearance and development of a theory of lattices, its links to modern algebra and the role of the American Mathematical Society (AMS) in this process. After problematizing the historiographical category of a theory, we will consider the theory of lattices through three different biases. First, we will approach it through the article of G. Birkhoff and Øystein Ore, often considered as founders for the theory. Here, the theory is an object explicitly identified by the actors to designate and link their works together. However, as an analytical category, it hides their diversity by joining them into a same non-problematized denomination. Secondly, the theory will be considered at a larger scale, from the publications of members of the AMS interested in lattices. From here, it appears as a set of practices shared by a collective of mathematicians. Finally, in a last chapter we will approach the promotion of Lattice Theory within the AMS. I will try to show that it benefits from both the status of an « american abstract algebra » and the positions of its promoters within the society
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Wu, Senlin. "Geometry of Minkowski Planes and Spaces -- Selected Topics." Doctoral thesis, [S.l. : s.n.], 2009. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200900226.
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