Cette thèse vise à contribuer à l’approche DDL (Data-Driven Learning) dans l’enseignement du lexique en FLE. Dans le cadre de l’approche DDL, on fait appel aux corpus pour enseigner différentes composantes d’une langue. Le lexique, étant un des premiers besoins d’un apprenant, car il donne accès à la communication en langue étrangère, fait l’objet de nombreux travaux de recherche actuels en linguistique et en didactique. L’idiomaticité, trait constitutif de toutes les langues, se manifeste sous forme d’expressions variées. Elle relève du champ de la lexicologie, et plus spécifiquement de la phraséologie. La linguistique de corpus permet d’observer ce fait de langue dans un cadre structure/sens. Les expressions idiomatiques en général, et en particulier les collocations, sont mises au centre de la démarche didactique dans cette thèse. Les collocations à verbe support restent une source d’erreurs importante même aux niveaux avancés d’apprentissage. Le matériel didactique présenté aux lecteurs de cette étude cherche à promouvoir l’exploitation directe des corpus bilingues par les apprenants en classe afin d’identifier ces collocations en L1 et en L2, de les comprendre, de trouver des correspondances et de les employer de manière appropriée. L’approche comparative français-russe renforcée par une observation des lignes de concordance issues de corpus authentiques devraient permettre une meilleure acquisition des faits linguistiques visés. Ce travail s’inscrit dans une perspective d’apprentissage déductif et d’autonomisation des apprenants. / This thesis aims to contribute to the DDL (Data-Driven Learning) approach in French vocabulary teaching. In the framework of the DDL approach we use corpora to teach different language phenomena. Vocabulary, one of the immediate needs of a language learner because it makes a communication in a foreign language possible, has become a popular research theme in the fields of linguistics and language teaching. Idiomaticity, an inherent part of all languages, manifests through various expressions. Phraseology studies different ways of expressing idiomaticity. Corpus linguistics permits to observe this language phenomenon in a structure/sense framework. Idiomatic expressions in general and collocations in particular are the heart and the main focus of the teaching perspective described in this thesis. Even advanced language learners make errors in light verb constructions. The teaching material presented in this study seeks to promote the search in bilingual corpora in the classroom in order to identify these collocations in L1 and in L2, to understand them, to find equivalents and finally, to use them correctly. A comparative French-Russian approach reinforced by a study of concordance lines from authentic corpora might contribute to better understanding of a particular language feature. This study falls in line with deductive learning practices and with the learners’ autonomisation perspective.

This dissertation is divided into four chapters. In Chapter 2 the equivariant homology groups of upper order ideals in the partition lattice are computed. The homology groups of these filters are written in terms of border strip Specht modules as well as in terms of links in an associated complex in the lattice of compositions. The classification is used to reproduce topological calculations of many well-studied subcomplexes of the partition lattice, including the d-divisible partition lattice and the Frobenius complex. In Chapter 3 the box polynomial B_{m,n}(x) is defined in terms of all integer partitions that fit in an m by n box. The real roots of the box polynomial are completely characterized, and an asymptotically tight bound on the norms of the complex roots is also given. An equivalent definition of the box polynomial is given via applications of the finite difference operator Delta to the monomial x^{m+n}. The box polynomials are also used to find identities counting set partitions with all even or odd blocks, respectively. Chapter 4 extends results from Chapter 3 to give combinatorial proofs for the ordinary generating function for set partitions with all even or all odd block sizes, respectively. This is achieved by looking at a multivariable generating function analog of the Stirling numbers of the second kind using restricted growth words. Chapter 5 introduces a colored variant of the ordered partition lattice, denoted Q_n^{\alpha}, as well an associated complex known as the alpha-colored permutahedron, whose face poset is Q_n^\alpha. Connections between the Eulerian polynomials and Stirling numbers of the second kind are developed via the fibers of a map from Q_n^{\alpha} to the symmetric group on n-elements

In this thesis, we study hash functions. We focus mainly on the famous Merkle-Damg˚ard construction and its generalisation. We show that even this generalised construction is not resistant to multicollision attacks. Combinatorics on words plays a fundamental role in the construction of our attack. We prove that regularities unavoidably appear in long words with bounded number of symbol occurences. We present our original results concerning regularities in long words. We lower some earlier published estimates, thus reducing the comlexity of the attack. Our results show that generalised iterated hash functions are interesting rather from the theoretical than practical point of view. 1

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Este trabalho aborda o uso de Objetos de Aprendizagem (OAs) voltado para o ensino de CombinatÃria e como esses objetos podem auxiliar no processo de aprendizagem dos alunos. Nesse sentido, o objetivo desse trabalho à verificar se a utilizaÃÃo de OAs nas aulas de MatemÃtica promovem uma melhor assimilaÃÃo dos conteÃdos relacionados à CombinatÃria, buscando despertar no aluno a curiosidade e a investigaÃÃo nesta Ãrea. Por isso, optou-se por um estudo experimental do qual participaram 20 alunos do 3o ano do Ensino MÃdio da Escola de Ensino Fundamental e MÃdio JoÃo Mattos, localizada em Fortaleza, CearÃ. Esses alunos foram escolhidos de forma aleatÃria e divididos entre dois grupos, experimental e controle, com 10 alunos em cada grupo. Para a coleta dos dados, foram utilizados, um questionÃrio socioeconÃmico e um teste de mÃltipla escolha sobre permutaÃÃo, arranjo e combinaÃÃo. Esses dados foram organizados e analisados com o auxÃlio do programa Microsoft Office Excel 2010. Os resultados mostram que o grupo que teve uma abordagem diferenciada (experimental) obteve um melhor desempenho em relaÃÃo ao grupo que teve uma abordagem tradicional (controle). / This work approaches the use of Learning Objects (LOs) facing the teaching of Combinatorics and how these objects can help in the learning process of students. Accordingly, the aim of this work is to check that the use of LOs in Math classes promotes a better assimilation of the contents related to Combinatorics, seeking to awaken in the student curiosity and research in this area. Therefore, we opted for an experimental study involving 20 students of the senior year of High School from School JoÃo Mattos, located in Fortaleza, CearÃ. These students were chosen at random and divided between two groups, experimental and control, with 10 students in each group. For the collect of data, were used, a socioeconomic questionnaire and a multiple choice test about permutation, arrangement and combination. These data were organized and analyzed with the help of Microsoft Office Excel 2010 program. The results show that the group that had a differentiated approach (experimental) obtained a better performance compared to the group that had a traditional approach (control). .

Esta dissertação apresenta o desenvolvimento de uma pesquisa referente ao ensino da Análise Combinatória, por meio da Resolução de Problemas, em uma turma de 6º ano do Ensino Fundamental. Para isso, elaborou-se uma sequência didática que buscava proporcionar aos educandos um contato com esse conteúdo antes do Ensino Médio. A partir dessa sequência analisou-se como a Resolução de Problemas, segundo Onuchic e Allevato, auxiliou os alunos a compreender os conceitos iniciais de Análise Combinatória, buscando também como referencial teórico o estudo referente ao Pensamento Matemático, de David Tall. Concluímos que a Resolução de Problemas auxiliou a expandir e modificar as Imagens dos Conceitos que os alunos possuíam com relação à Análise Combinatória. / This dissertation shows the development of research related to teaching Combinatorics, through Problem Solving, at a 6th grade level. A lesson plan was prepared and aimed to confront students of middle school with problems involving Combinatorics, allowing them to work with such concepts before high school. Based on this lesson plan, our intent was to verify how Problem Solving, according to Onuchic e Allevato, helped the students to understand initial concepts of Combinatorics. Also, using David Tall’s studies about Mathematical Thinking as reference. We could verify that the Problem Solving Theory helped the students to expand and modify their Concept Images related to Combinatorics.

Estudamos o comportamento da familia aresta-triangulo de grafos aleatorios exponenciais (ERG) usando metodos de Monte Carlo baseados em Cadeias de Markov. Comparamos contagens de subgrafos e correlacoes entre arestas de ergs as de Grafos Aleatorios Binomiais (BRG, tambem chamados de Erdos-Renyi). E um resultado teorico conhecido que para algumas parametrizacoes os limites das contagens de subgrafos de ERGs convergem para os de BRGs, assintoticamente no numero de vertices [BBS11, CD11]. Observamos esse fenomeno em grafos com poucos (20) vertices em nossas simulacoes. / We study the behavior of the edge-triangle family of exponential random graphs (ERG) using the Markov Chain Monte Carlo method. We compare ERG subgraph counts and edge correlations to those of the classic Binomial Random Graph (BRG, also called Erdos-Renyi model). It is a known theoretical result that for some parameterizations the limit ERG subgraph counts converge to those of BRGs, as the number of vertices grows [BBS11, CD11]. We observe this phenomenon on graphs with few (20) vertices in our simulations.

This thesis presents an introduction to Schur rings (S-rings) and their various properties. Special attention is given to S-rings that are commutative. A number of original results are proved, including a complete classification of the central S-rings over the simple groups PSL(2,q), where q is any prime power. A discussion is made of the counting of symmetric S-rings over cyclic groups of prime power order. An appendix is included that gives all S-rings over the symmetric group over 4 elements with basic structural properties, along with code that can be used, for groups of comparatively small order, to enumerate all S-rings and compute character tables for those S-rings that are commutative. The appendix also includes code optimized for the enumeration of S-rings over cyclic groups.

In a 2011 paper by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. An open problem from that paper is to determine the minimum number of edges necessary for a graph with a fixed number of vertices to be solvable. This thesis provides new bounds on this number. It also provides necessary and sufficient conditions for two families of graphs to be solvable, along with criticality results, and the maximum number of pegs that can be left in each of the two graph families.

A set S of vertices in a graph G=(V,E) is a dominating set if every vertex in V\S is adjacent to at least one vertex in S, and the minimum cardinality of a dominating set of G is the domination number of G. A vertex v in a dominating set S is said to be very cost effective if it is adjacent to more vertices in V\S than to vertices in S. A dominating set S is very cost effective if every vertex in S is very cost effective. The minimum cardinality of a very cost effective dominating set of G is the very cost effective domination number of G. We first give necessary conditions for a graph to have equal domination and very cost effective domination numbers. Then we determine an upper bound on the very cost effective domination number for trees in terms of their domination number, and characterize the trees which attain this bound. lastly, we show that no such bound exists for graphs in general, even when restricted to bipartite graphs.

In this paper, we consider decompositions of the complete graph on v vertices into 4-cycles with a pendant edge. In part, we will consider decompositions which admit automorphisms consisting of: (1) a single cycle of length v, (2) f fixed points and a cycle of length v − f, or (3) two disjoint cycles. The purpose of this thesis is to give necessary and sufficient conditions for the existence of cyclic, f-cyclic, and bicyclic Q-decompositions of Kv.