Brands of cumulants in non-commutative probability, and relations between them

The study of non-commutative probability revolves around the different notions of independeces, such as free, Boolean and monotone. To each type of independence one can associate a notion of cumulants that linearize the addition of independent random variables. These notions of cumulants are a clear analogue of classic cumulants that linearize the addition of independent random variables. The family of set partitions P plays a key role in the combinatorial study of probability because several formulas relating moments to a brand of cumulants can be expressed as a sum indexed by set partitions. An intriguing fact observed in the recent research literature was that non-commutative cumulants sometimes have applications to other areas of non-commutative probability than the one they were designed for. Thus one wonders if there are nice combinatorial formulas to directly transition from one brand of cumulants to another. This thesis is concerned with the study of interrelations between different brands of cumulants associated to classic, Boolean, free and monotone independences. My development is naturally divided in four topics. For the first topic I focus on free cumulants, and I use them in the study of the distribution of the anti-commutator ab + ba of two free random variables a and b. This follows up on some questions raised a while ago by Nica and Speicher in the 1990’s. I next consider the notion of convolution in the framework of a family of lattices, which goes back to work of Rota and collaborators in the 1970’s. I focus on the lattices of non-crossing partitions NC and put into evidence a certain group of semi-multiplicative functions, which encapsulate the moment-cumulant formulas and the inter-cumulant transition formulas for several known brands of cumulants in non-commutative probability. I next extend my considerations to the framework of P, which allows us to include the classical cumulants in the picture. Moreover, I do this in a more structured fashion by introducing a notion of iterative family of partitions S in P. This unifies the considerations related to NC and P, and also provides a whole gallery of new examples. Finally, it is important to make the observation that the group of semi-multiplicative functions which arise in connection to an iterative family is in fact a dual structure. Namely, it appears as the group of characters for a certain Hopf algebra over the partitions in S. In particular, the antipode promises to serve as a universal inversion tool for moment-cumulant formulas and for transition formulas between different brands of cumulants

Novel procedures for graph edge-colouring

Orientador: Dr. Renato CarmoCoorientador: Dr. André Luiz Pires GuedesTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Informática. Defesa : Curitiba, 05/12/2018Inclui referências e índiceÁrea de concentração: Ciência da ComputaçãoResumo: O índice cromático de um grafo G é o menor número de cores necessário para colorir as arestas de G de modo que não haja duas arestas adjacentes recebendo a mesma cor. Pelo célebre Teorema de Vizing, o índice cromático de qualquer grafo simples G ou é seu grau máximo , ou é ? + 1, em cujo caso G é dito Classe 1 ou Classe 2, respectivamente. Computar uma coloração de arestas ótima de um grafo ou simplesmente determinar seu índice cromático são problemas NP-difíceis importantes que aparecem em aplicações notáveis, como redes de sensores, redes ópticas, controle de produção, e jogos. Neste trabalho, nós apresentamos novos procedimentos de tempo polinomial para colorir otimamente as arestas de grafos pertences a alguns conjuntos grandes. Por exemplo, seja X a classe dos grafos cujos maiorais (vértices de grau ?) possuem soma local de graus no máximo ?2 ?? (entendemos por 'soma local de graus' de um vértice x a soma dos graus dos vizinhos de x). Nós mostramos que quase todo grafo está em X e, estendendo o procedimento de recoloração que Vizing usou na prova para seu teorema, mostramos que todo grafo em X é Classe 1. Nós também conseguimos resultados em outras classes de grafos, como os grafos-junção, os grafos arco-circulares, e os prismas complementares. Como um exemplo, nós mostramos que um prisma complementar só pode ser Classe 2 se for um grafo regular distinto do K2. No que diz respeito aos grafos-junção, nós mostramos que se G1 e G2 são grafos disjuntos tais que |V(G1)| _ |V(G2)| e ?(G1) _ ?(G2), e se os maiorais de G1 induzem um grafo acíclico, então o grafo-junção G1 ?G2 é Classe 1. Além desses resultados em coloração de arestas, apresentamos resultados parciais em coloração total de grafos-junção, de grafos arco-circulares, e de grafos cobipartidos, bem como discutimos um procedimento de recoloração para coloração total. Palavras-chave: Coloração de grafos e hipergrafos (MSC 05C15). Algoritmos de grafos (MSC 05C85). Teoria dos grafos em relação à Ciência da Computação (MSC 68R10). Graus de vértices (MSC 05C07). Operações de grafos (MSC 05C76).Abstract: The chromatic index of a graph G is the minimum number of colours needed to colour the edges of G in a manner that no two adjacent edges receive the same colour. By the celebrated Vizing's Theorem, the chromatic index of any simple graph G is either its maximum degree ? or it is ? + 1, in which case G is said to be Class 1 or Class 2, respectively. Computing an optimal edge-colouring of a graph or simply determining its chromatic index are important NP-hard problems which appear in noteworthy applications, like sensor networks, optical networks, production control, and games. In this work we present novel polynomial-time procedures for optimally edge-colouring graphs belonging to some large sets of graphs. For example, let X be the class of the graphs whose majors (vertices of degree ?) have local degree sum at most ?2 ? ? (by 'local degree sum' of a vertex x we mean the sum of the degrees of the neighbours of x). We show that almost every graph is in X and, by extending the recolouring procedure used by Vizing's in the proof for his theorem, we show that every graph in X is Class 1. We further achieve results in other graph classes, such as join graphs, circular-arc graphs, and complementary prisms. For instance, we show that a complementary prism can be Class 2 only if it is a regular graph distinct from the K2. Concerning join graphs, we show that if G1 and G2 are disjoint graphs such that |V(G1)| _ |V(G2)| and ?(G1) _ ?(G2), and if the majors of G1 induce an acyclic graph, then the join graph G1 ?G2 is Class 1. Besides these results on edge-colouring, we present partial results on total colouring join graphs, cobipartite graphs, and circular-arc graphs, as well as a discussion on a recolouring procedure for total colouring. Keywords: Colouring of graphs and hypergraphs (MSC 05C15). Graph algorithms (MSC 05C85). Graph theory in relation to Computer Science (MSC 68R10). Vertex degrees (MSC 05C07). Graph operations (MSC 05C76)

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

M2-Edge Colorings Of Cacti And Graph Joins

An edge coloring φ of a graph G is called an M2-edge coloring if |φ(v)| ≤ 2 for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let 2(G) denote the maximum number of colors used in an M2-edge coloring of G. In this paper we determine 2(G) for trees, cacti, complete multipartite graphs and graph joins

CIMODE 2016: 3º Congresso Internacional de Moda e Design: proceedings

O CIMODE 2016 é o terceiro Congresso Internacional de Moda e Design, a decorrer de 9 a 12 de maio de 2016 na cidade de Buenos Aires, subordinado ao tema : EM--‐TRAMAS. A presente edição é organizada pela Faculdade de Arquitetura, Desenho e Urbanismo da Universidade de Buenos Aires, em conjunto com o Departamento de Engenharia Têxtil da Universidade do Minho e com a ABEPEM – Associação Brasileira de Estudos e Pesquisa em Moda.info:eu-repo/semantics/publishedVersio