Reziumė Disertacijoje nagrinėjamos atsitiktinių keitinių problemos yra priskirtinos tikimybinei kombinatorikai. Gauti rezultatai aprašo visiškai adityviųjų funkcijų, apibrėžtų simetrinėje grupėje, reikšmių asimptotinius skirstinius Evenso tikimybinio mato atžvilgiu, kai grupės eilė neaprėžtai didėja. Išvestos adityviųjų funkcijų laipsninių ir faktorialinių momentų formulės. Funkcijų, išreiškiančių atsitiktinio keitinio ciklų su bet kokiais apribojimais skaičius, atveju rastos būtinos ir pakankamos ribinių tikimybinių dėsnių egzistavimo sąlygos. Išsamiai išnagrinėtas konvergavimas į Puasono, quasi-Puasono, Bernulio, binominio ir kitus skirstinius, sukoncentruotus sveikųjų neneigiamų skaičių aibėje. Rezultatai apibendrinti sveikareikšmių visiškai adityviųjų funkcijų klasėje. Darbe įrodytas bendras silpnasis didžiųjų skaičių dėsnis, rastos būtinos ir pakankamos adityviųjų funkcijų sekų pasiskirstymo funkcijų konvergavimo į išsigimusį nuliniame taške dėsnį egzistavimo sąlygos. Sprendžiamos problemos yra susietos su tikimybiniais vektorių, turinčių sveikąsias neneigiamas koordinates, uždaviniais. Adicinėje tokių vektorių pusgrupėje išnagrinėti multiplikatyviųjų funkcijų vidurkiai tikimybinio mato, vadinamo Evanso atrankos formule, atžvilgiu. Gauti tikslūs viršutinieji ir apatinieji įverčiai. Iš jų išplaukia svarbios atsitiktinių keitinių tikimybių savybės. Disertacijoje plėtojami faktorialinių momentų ir kiti kombinatoriniai bei tikimybiniai metodai. / In the thesis the examining problems of random permutations are attributed to the probabilistic combinatorics. Obtained results describe asymptotical distributions of completely additive functions values defined on a symmetric group with respect to Ewens probability measure, if the group order unbounded increases. Power and factorial moments formulae of additive functions are derived. There are established necessary and sufficient conditions under which the distributions of a number of cycles with restricted lengths obey the limit probability laws. The convergence to the Poisson, quasi-Poisson, Bernoulli, binomial and other distributions, defined on the positive whole - number set are exhaustively investigated. The results are generalized on the class of whole - number completely additive functions. The general weak law of large numbers is proved in the thesis, necessary and sufficient existence conditions, under which the distributions of the sequences of additive functions converge to the degenerate at the point zero limit law are established. Examining problems are related to the probability tasks of the vectors, which have whole - numbered nonnegative coordinates. The mean values of multiplicative functions defined on those vectors’ additive semigroup with respect to the Ewens measure, called Ewens Sampling Formula, and investigated. Lower and upper sharp estimates are obtained. From the latter results follow important probabilities’ properties of random... [to full text]

Made available in DSpace on 2016-06-02T20:28:31Z (GMT). No. of bitstreams: 1 6450.pdf: 757192 bytes, checksum: 765b66ca6aed0686ecbcd10c145cefac (MD5) Previous issue date: 2014-11-28 / Financiadora de Estudos e Projetos / In this dissertation we ll present a discussion about the Representations of the Symmetric Group Sn and Alternating Group An. We ll study basics results of the Young s Theory about the representations of the Symmetric Group and discover the decomposition of the algebra FSn in simple subalgebras. After, we ll utilize this decomposition to find the decomposition of the algebra FAn in simple subalgebras. Finally, we ll use this decompositions, together with the PI Theory, for get the sequence of A-codimensions for the Grassmann Algebra (Exterior Algebra) infinitely generated. / Neste trabalho apresentamos uma discussão a respeito das Representações dos Grupos Simétrico Sn e do Grupo Alternante An. Estudaremos resultados básicos da Teoria de Young sobre as representações do grupo simétrico para encontrarmos a decomposição da álgebra de grupo FSn em subálgebras simples. Depois utilizaremos tal decomposição para encontrar a decomposição da álgebra de grupo FAn em subálgebras simples. Por fim empregaremos as informações a respeito das decomposições acima citadas, juntamente com a PI-Teoria, para obter a sequência de A-codimensões para a álgebra de Grassmann (álgebra exterior) infinitamente gerada.

Un système couvrant est un ensemble fini de progressions arithmétiques avec la propriété que chaque entier appartient à au moins une des progressions. L’étude des systèmes couvrants a été initié par Erdős dans les années 1950, et il posa dans les années qui suivirent plusieurs questions sur ces objets mathématiques. Une de ses questions les plus célèbres est celle du plus petit module : est-ce que le plus petit module de tous les systèmes couvrants avec modules distinct est borné uniformément? En 2015, Hough a montré que la réponse était affirmative, et qu’une borne admissible est 1016. En se basant sur son travail, mais en simplifiant la méthode, Balister, Bollobás, Morris, Sahasrabudhe et Tiba on réduit cette borne a 616, 000. Leur méthode a menée a plusieurs applications supplémentaires. Entre autres, ils ont compté le nombre de système couvrant avec un nombre fixe de module. La première partie de ce mémoire vise a étudier une question similaire. Nous allons essayer de compter le nombre de système couvrant avec un ensemble de module fixé. La technique que nous utiliserons nous mènera vers l’étude des symmétries de système couvrant. Dans la seconde partie, nous répondrons à des variantes du problème du plus petit module. Nous regarderons des bornes sur le plus petit module d’un système couvrant de multiplicité s, c’est-à-dire un système couvrant dans lequel chaque module apparait au plus s fois. Nous utiliserons ensuite ce résultat afin montrer que le plus petit module d’un système couvrant de multiplicité 1 d’une progression arithmétique est borné, ainsi que pour montrer que le n-eme plus petit module dans un système couvrant de multiplicité 1 est borné. / A covering system is a finite set of arithmetic progressions with the property that every integer belongs to at least one of them. The study of covering systems was started by Erdős in the 1950’s, and he asked many questions about them in the following years. One of the most famous questions he asked was if the minimum modulus of a covering system with distinct moduli is bounded uniformly. In 2015, Hough showed that it is at most 1016. Following on his work, but simplifying the method, Balister, Bollobás, Morris, Sahasrabudhe and Tiba showed that it is at most 616, 000. Their method led them to many further applications. Notably, they counted the number of covering systems with a fixed number of moduli. The first part of this thesis seeks to study a related question, that is to count the number of covering systems with a given set of moduli. The technique developped to do this for some sets will lead us to look at symmetries of covering systems. The second part of this thesis will look at variants of the minimum modulus problem. Notably, we will be looking at bounds on the minimum modulus of a covering system of multiplicity s, that is a covering system in which each moduli appears at most s times, as well as bounds on the minimum modulus of a covering system of multiplicity 1 of an arithmetic progression, and finally look at bounds for the n-th smallest modulus in a covering system.

Cette thèse est consacrée à l’étude des représentations modulaires des algèbres d’Ariki-Koike, et des liens avec la théorie des cristaux et des bases canoniques de Kashiwara via le théorème de catégorification d’Ariki. Dans un premier temps, on étudie, grâce à des outils combinatoires, les matrices de décomposition de ces algèbres en généralisant les travaux de Geck et Jacon. On classifie entièrement les cas d’existence et de non-existence d’ensembles basiques, en construisant explicitement ces ensembles lorsqu’ils existent. On explicite ensuite les isomorphismes de cristaux pour les représentations de Fock de l’algèbre affine quantique Uq(sle). On construit alors un isomorphisme particulier, dit canonique, qui permet entre autres une caractérisation non-récursive de n’importe quelle composante connexe du cristal. On souligne également les liens avec la combinatoire des mots sous-jacente à la structure cristalline des espaces de Fock, en décrivant notamment un analogue de la correspondance de Robinson-Schensted-Knuth pour le type A affine. / This thesis is devoted to the study of modular representations of Ariki-Koike algebras, and of the connections with Kashiwara’s crystal and canonical bases theory via Ariki’s categorification theorem. First, we study, using combinatorial tools, the decomposition matrices associated to these algebras, generalising the works of Geck and Jacon. We fully classify the cases of existence and non-existence of canonical basic sets, and we explicitely construct these sets when they exist. Next, we make explicit the crystal isomorphisms for Fock spaces representations of the quantum affine algebra Uq(sle). We then construct of a particular isomorphism, so-called canonical, which gives, inter alia, a non-recursive description of any connected component of the crystal. We also stress the links with the combinatorics of words underlying the crystal structure of Fock spaces, by describing notably an analogue of the Robinson-Schensted-Knuth correspondence for affine type A.

Let C be the class of finite nilpotent, solvable, symmetric, simple or semi-simple groups and n be a positive integer. We discuss the following question on group laws: What is the length of the shortest non-trivial law holding for all finite groups from the class C of order less than or equal to n?:Introduction 0 Essentials from group theory 1 The two main tools 1.1 The commutator lemma 1.2 The extension lemma 2 Nilpotent and solvable groups 2.1 Definitions and basic properties 2.2 Short non-trivial words in the derived series of F_2 2.3 Short non-trivial words in the lower central series of F_2 2.4 Laws for finite nilpotent groups 2.5 Laws for finite solvable groups 3 Semi-simple groups 3.1 Definitions and basic facts 3.2 Laws for the symmetric group S_n 3.3 Laws for simple groups 3.4 Laws for finite linear groups 3.5 Returning to semi-simple groups 4 The final conclusion Index Bibliography / Sei C die Klasse der endlichen nilpotenten, auflösbaren, symmetrischen oder halbeinfachen Gruppen und n eine positive ganze Zahl. We diskutieren die folgende Frage über Gruppengesetze: Was ist die Länge des kürzesten nicht-trivialen Gesetzes, das für alle endlichen Gruppen der Klasse C gilt, welche die Ordnung höchstens n haben?:Introduction 0 Essentials from group theory 1 The two main tools 1.1 The commutator lemma 1.2 The extension lemma 2 Nilpotent and solvable groups 2.1 Definitions and basic properties 2.2 Short non-trivial words in the derived series of F_2 2.3 Short non-trivial words in the lower central series of F_2 2.4 Laws for finite nilpotent groups 2.5 Laws for finite solvable groups 3 Semi-simple groups 3.1 Definitions and basic facts 3.2 Laws for the symmetric group S_n 3.3 Laws for simple groups 3.4 Laws for finite linear groups 3.5 Returning to semi-simple groups 4 The final conclusion Index Bibliography