Bounded analytic functions on the open unit disk D = {z ∈ C | |z| < 1} are a fre-quent area of study in complex function theory. While it is easy to understand thebehavior of analytic functions on sequences with limit points inside D, the theorybecomes much more complicated as sequences converge to the boundary, ∂D. In thisthesis, we will explore boundary theorems, which can guarantee specific desired be-havior of these analytic functions. The thesis describes an elementary approach toproving Fatou’s Non-Tangential Limit Theorem, as well as proofs and discussion ofthe subsequent classical boundary theorems for specific points, Julia’s Theorem andthe Julia-Carathéodory Theorem. This thesis serves as a synthesis of these boundarytheorems in order to fill a gap in the overarching literature.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-10-30T11:09:54Z No. of bitstreams: 2 Dissertação - Braully Rocha da Silva - 2018.pdf: 1396149 bytes, checksum: 9a9145cb07e037a784d2d15d43cbd1ff (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-10-30T11:20:36Z (GMT) No. of bitstreams: 2 Dissertação - Braully Rocha da Silva - 2018.pdf: 1396149 bytes, checksum: 9a9145cb07e037a784d2d15d43cbd1ff (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-10-30T11:20:36Z (GMT). No. of bitstreams: 2 Dissertação - Braully Rocha da Silva - 2018.pdf: 1396149 bytes, checksum: 9a9145cb07e037a784d2d15d43cbd1ff (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-09-24 / Outro / In this work we present results and implementantions for hull and Carathéodory numbers in P3 convexity. We obtain results for graphs of diameter 2 having cut-vertex for both problems. Finally, entering more complex cases, we were able to determine a logarithmic limit, means of algorithm, for the hull number in case of graph diameter 2 and 2-connected. Exploring more restrictive cases, we determined a constant limit for some subclasses of graphs of diameter 2. We made also implementations and algorithms for these parameters. Implementations algorithms heuristic, parallel, and brute force. Finally, although not directly related, we developed an algorithm for Moore's graphs generation, which may be one of the ways to find Moore missinge graph, if it exists, a question that remains unknown for 55 years. And finally, we conclude with some conjectures interesting, for limits to the hull and Carathéodory numbers, in other classes of graphs, that were not explored in this work, but was identified by the implementations, and can be better explored in future works. / Nesta dissertação, tratamos de limites para o número envoltório e o número de Carathéodory na Convexidade P3. Aferimos resultados para grafos de diâmetro 2 com vértice de corte para ambos os problemas. Adentrando em casos mais complexos, conseguimos determinar um limite logarítmico, por meio de algoritmo pseudo-polimonial, para o número envoltório de grafos de diâmetro 2 biconexos. Explorando um pouco mais restritivamente, conseguimos determinar um limite constante para algumas subclasses de grafos de diâmetro 2, os grafos maximais sem triângulo. Não atendo somente aos resultados teóricos, realizamos também implementações e algoritmos para esses parâmetros. As implementações perfazem algoritmos heurísticos, paralelos e força bruta. Por fim, embora não diretamente relacionado, desenvolvemos uma algoritmo para geração de grafos de Moore, que pode ser um dos caminhos para encontrar o ultimo grafo de Moore, caso ele exista. Questão que remanesce desconhecido e procurada por 55 anos.

Orientador: Márcio Ricardo Alves Gouveia / Resumo: O presente trabalho trata de conceitos relacionados com a medida generalizada de Hausdorff, onde o principal objetivo consiste na obtenção de conjuntos cuja dimensão seja um número positivo não inteiro. Ele começa com uma definição sobre as propriedades que uma função de conjunto deve satisfazer para ser considerada uma medida de Carathéodory, suas implicações e consequências. Após a explicação destes conceitos iniciais, dá-se alguns exemplos de funções de conjunto contínuas e monótonas com a apresentação da função de escala logarítmica, que é peça chave para o desenvolvimento de conjuntos de medidas positivas não inteiras, além da introdução da medida de Hausdorff com seus desdobramentos. Algumas hipóteses sobre funções côncavas são apresentadas juntamente com fórmulas deduzidas com bases nestas hipóteses e na concavidade da função. Utiliza-se a função de escala logarítima para a determinação da dimensão de vários conjuntos, inclusive o conjunto de Cantor. Posteriormente, há uma adaptação dos conceitos trabalhados para o tratamento de dimensões relacionadas à números diádicos irracionais. Por fim, os conceitos tratados sobre a reta real são estendidos para produtos cartesianos, com especial enfoque para conjuntos planares. / Abstract: The present work deals with concepts related to the generalized Hausdorff measure, where the main objective is to obtain sets whose dimension is a positive non integer number. It begins with a definition of the properties that a set function must satisfy to be considered a Carathéodory measure, their implications and consequences. Following the explanation of these initial concepts, some examples of continuous and monotonous set functions are given with the presentation of the logarithmic scale function, which is key to the development of non-integer positive measure sets, in addition to the introduction of the Hausdorff measure with its developments. Some assumptions about concave functions are presented together with formulas derived from these assumptions and the concavity of the function. The logarithmic scale function is used to determine the dimension of various sets, including the Cantor set. Later, there is an adaptation of the concepts worked for the treatment of dimensions related to irrational dyadic numbers. Finally, the concepts treated on the real line are extended to Cartesian products, with special focus on planar sets. / Mestre

Le théorème de Carathéodory coloré, prouvé en 1982 par Bárány, énonce le résultat suivant. Etant donnés d Å1 ensembles de points S1,SdÅ1 dans Rd , si chaque Si contient 0 dans son enveloppe convexe, alors il existe un sous-ensemble arc-en-ciel T µ SdÅ1 iÆ1 Si contenant 0 dans son enveloppe convexe, i.e. un sous-ensemble T tel que jT \Si j • 1 pour tout i et tel que 0 2 conv(T ). Ce théorème a donné naissance à de nombreuses questions, certaines algorithmiques et d’autres plus combinatoires. Dans ce manuscrit, nous nous intéressons à ces deux aspects. En 1997, Bárány et Onn ont défini la programmation linéaire colorée comme l’ensemble des questions algorithmiques liées au théorème de Carathéodory coloré. Parmi ces questions, deux ont particulièrement retenu notre attention. La première concerne la complexité du calcul d’un sous-ensemble arc-en-ciel comme dans l’énoncé du théorème. La seconde, en un sens plus générale, concerne la complexité du problème de décision suivant. Etant donnés des ensembles de points dans Rd , correspondant aux couleurs, il s’agit de décider s’il existe un sous-ensemble arc-en-ciel contenant 0 dans son enveloppe convexe, et ce en dehors des conditions du théorème de Carathéodory coloré. L’objectif de cette thèse est de mieux délimiter les cas polynomiaux et les cas “difficiles” de la programmation linéaire colorée. Nous présentons de nouveaux résultats de complexités permettant effectivement de réduire l’ensemble des cas encore incertains. En particulier, des versions combinatoires du théorème de Carathéodory coloré sont présentées d’un point de vue algorithmique. D’autre part, nous montrons que le problème de calcul d’un équilibre de Nash dans un jeu bimatriciel peut être réduit polynomialement à la programmation linéaire coloré. En prouvant ce dernier résultat, nous montrons aussi comment l’appartenance des problèmes de complémentarité à la classe PPAD peut être obtenue à l’aide du lemme de Sperner. Enfin, nous proposons une variante de l’algorithme de Bárány et Onn, calculant un sous ensemble arc-en-ciel contenant 0 dans son enveloppe convexe sous les conditions du théorème de Carathéodory coloré. Notre algorithme est clairement relié à l’algorithme du simplexe. Après une légère modification, il coïncide également avec l’algorithme de Lemke, calculant un équilibre de Nash dans un jeu bimatriciel. La question combinatoire posée par le théorème de Carathéodory coloré concerne le nombre de sous-ensemble arc-en-ciel contenant 0 dans leurs enveloppes convexes. Deza, Huang, Stephen et Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597–604 (2006)) ont formulé la conjecture suivante. Si jSi j Æ d Å1 pour tout i 2 {1, . . . ,d Å1}, alors il y a au moins d2Å1 sous-ensemble arc-en-ciel contenant 0 dans leurs enveloppes convexes. Nous prouvons cette conjecture à l’aide d’objets combinatoires, connus sous le nom de systèmes octaédriques, dont nous présentons une étude plus approfondie / The colorful Carathéodory theorem, proved by Bárány in 1982, states the following. Given d Å1 sets of points S1, . . . ,SdÅ1 µ Rd , each of them containing 0 in its convex hull, there exists a colorful set T containing 0 in its convex hull, i.e. a set T µ SdÅ1 iÆ1 Si such that jT \Si j • 1 for all i and such that 0 2 conv(T ). This result gave birth to several questions, some algorithmic and some more combinatorial. This thesis provides answers on both aspects. The algorithmic questions raised by the colorful Carathéodory theorem concern, among other things, the complexity of finding a colorful set under the condition of the theorem, and more generally of deciding whether there exists such a colorful set when the condition is not satisfied. In 1997, Bárány and Onn defined colorful linear programming as algorithmic questions related to the colorful Carathéodory theorem. The two questions we just mentioned come under colorful linear programming. This thesis aims at determining which are the polynomial cases of colorful linear programming and which are the harder ones. New complexity results are obtained, refining the sets of undetermined cases. In particular, we discuss some combinatorial versions of the colorful Carathéodory theorem from an algorithmic point of view. Furthermore, we show that computing a Nash equilibrium in a bimatrix game is polynomially reducible to a colorful linear programming problem. On our track, we found a new way to prove that a complementarity problem belongs to the PPAD class with the help of Sperner’s lemma. Finally, we present a variant of the “Bárány-Onn” algorithm, which is an algorithmcomputing a colorful set T containing 0 in its convex hull whose existence is ensured by the colorful Carathéodory theorem. Our algorithm makes a clear connection with the simplex algorithm. After a slight modification, it also coincides with the Lemke method, which computes a Nash equilibriumin a bimatrix game. The combinatorial question raised by the colorful Carathéodory theorem concerns the number of positively dependent colorful sets. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597–604 (2006)) conjectured that, when jSi j Æ d Å1 for all i 2 {1, . . . ,d Å1}, there are always at least d2Å1 colourful sets containing 0 in their convex hulls. We prove this conjecture with the help of combinatorial objects, known as the octahedral systems. Moreover, we provide a thorough study of these objects

La thèse est consacrée à une étude d'interpolation complexe "semi-libre" dans le sens suivant: étant donné un ensemble "sigma" dans le disque unité D et une fonction f holomorphe dans D appartenant à une certaine classe X, on cherche g dans une autre classe Y (plus petite que X) qui minimise la norme de g dans Y parmi toutes les fonctions g satisfaisant g=f sur l'ensemble "sigma". Plus précisément, nous nous intéressons aux estimations de la constante d'interpolation suivante: c(sigma, X, Y ) = sup{ inf{||g||_Y: g=f sur sigma}: ||f||_X<=1} Dans la thèse, nous étudions le cas où Y = H^infini et où l'espace des contraintes X est choisi parmi les espaces suivants: les espaces de Hardy, les espaces de Bergman pondérés à poids radial ou encore les espaces de fonctions holomorphes ayant leurs coefficients de Taylor dans lp(w) (w étant un poids). La thèse contient également certaines applications aux nombres conditionnés des matrices de Toeplitz. / The thesis is devoted to a "semi-free" interpolation problem in the following way. Let sigma be a finite set of the unit disc D and f an holomorphic function in D which belongs to a certain class X, we search for g in another class Y (smaller than X) which minimize the norm of g in Y among all the functions g such that g=f on the set "sigma". More precisely, we are interested in the following interpolation constant : c(sigma, X, Y ) = sup{ inf{||g||_Y: g=f sur sigma}: ||f||_X<=1}. We study in the thesis the case where Y=H^\infinity and the space of constrains X is chosen among the following spaces: Hardy spaces, weighted Bergman spaces (with radial waights), and holomorphic functions which Taylor coefficients are in lp(w) (w being a weight). The thesis also contains an application to the condition numbers of Toeplitz matrices.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-09T16:42:35Z No. of bitstreams: 1 EMANUELA RÉGIA DE SOUSA COELHO - DISSERTAÇÃO PPGMAT 2014..pdf: 722841 bytes, checksum: 35c227da51e1da58846c5d7bdb22cd7f (MD5) / Made available in DSpace on 2018-08-09T16:42:35Z (GMT). No. of bitstreams: 1 EMANUELA RÉGIA DE SOUSA COELHO - DISSERTAÇÃO PPGMAT 2014..pdf: 722841 bytes, checksum: 35c227da51e1da58846c5d7bdb22cd7f (MD5) Previous issue date: 2014-02 / Capes / Para ler o reumo deste trabalho recomendamos o download do arquivo, pois o mesmo possui fórmulas e caracteres matemáticos que não foram possíveis transcreve-los. / To read the progress of this work we recommend downloading the file, as it has formulas and mathematical characters that could not be transcribed.

The validity of the von-Neumann inequality for commuting $n$ - tuples of $3\times 3$ matrices remains open for $n\geq 3$. We give a partial answer to this question, which is used to obtain a necessary condition for the Carathéodory-Fejérinterpolation problem on the polydisc$\D^n. $ in the special case of $n=2$ (which follows from Ando's theorem as well), this necessary condition is made explicit. We discuss an alternative approach to the Carathéodory-Fejérinterpolation problem, in the special case of $n=2$, adapting a theorem of Korányi and Pukánzsky. As a consequence, a class of polynomials are isolated for which a complete solution to the Carathéodory-Fejér interpolation problem is easily obtained. Many of our results remain valid for any $n\in \mathbb N$, however the computations are somewhat cumbersome. Recall the well known inequality due to Varopoulos, namely, $\lim{n\to \infty}C_2(n)\leq 2 K^\C_G$, where $K^\C_G$ is the complex Grothendieck constant and \[C_2(n)=sup\{\|p(\boldsymbolT)\|:\|p\|_{\D^n,\infty}\leq 1, \|\boldsymbol T\|_{\infty} \leq 1\}.\] Here the supremum is taken over all complex polynomials $p$ in $n$ variables of degree at most $2$ and commuting $n$ - tuples$\boldsymbolT:=(T_1,\ldots,T_n)$ of contractions. We show that \[\lim_{n\to \infty} C_2 (n)\leq \frac{3\sqrt{3}}{4} K^\C_G\] obtaining a slight improvement in the inequality of Varopoulos. We also discuss several finite and infinite dimensional operator space structures on $\ell^1(n) $, $n>1. $

In this thesis we study embeddings of spaces of functions defined on Carnot- Carathéodory spaces. Main results of this work consist of conditions for Sobolev- type embeddings of higher order between rearrangement-invariant spaces. In a special case when the underlying measure space is the so-called X-PS domain in the Heisenberg group we obtain full characterization of a Sobolev embedding. The next set of main results concerns compactness of the above-mentioned em- beddings. In these cases we obtain sufficient conditions. We apply the general results to important particular examples of function spaces. In the final part of the thesis we present a new algorithm for approximation of the least concave majorant of a function defined on an interval complemented with the estimate of the error of such approximation. 1

Muitas relações min-max em otimização combinatória podem ser demonstradas através de total dual integralidade de sistemas lineares. O conceito algébrico de bases de Hilbert foi originalmente introduzido com o objetivo de melhor compreender a estrutura geral dos sistemas totalmente dual integrais. Resultados apresentados posteriormente mostraram que bases de Hilbert também são relevantes para a otimização combinatória em geral e para a caracterização de certas classes de objetos discretos. Entre tais resultados, foram provadas, a partir dessas bases, versões do teorema de Carathéodory para programação inteira. Nesta dissertação, estudamos aspectos estruturais e computacionais de bases de Hilbert e relações destas com programação inteira e otimização combinatória. Em particular, consideramos versões inteiras do teorema de Carathéodory e conjecturas relacionadas. / There are several min-max relations in combinatorial optimization that can be proved through total dual integrality of linear systems. The algebraic concept of Hilbert basis was originally introduced with the objective of better understanding the general structure of totally dual integral systems. Some results that were proved later have shown that Hilbert basis are also relevant to combinatorial optimization in a general manner and to characterize certain classes of discrete objects. Among such results, there are versions of Carathéodory\'s theorem for integer programming that were proved through those basis. In this dissertation, we study structural and computational aspects of Hilbert basis and their relations to integer programming and combinatorial optimization. In particular, we consider integer versions of Carathéodory\'s theorem and related conjectures.

Ce mémoire s'organise autour de deux cadres d'étude : d'une part, celui des espaces symétriques riemanniens non compacts X = G/K, pour lesquels nous prouvons un encadrement optimal et global en les variables d'espace et de temps, du noyau de la chaleur associé à l'opérateur de Laplace-Beltrami L ; d'autre part, dans le cas d'un groupe de Lie semi-simple G, nous montrons que tous les sous-laplaciens sur G qui induisent l'action de L sur X = G/K présentent des analogies avec L vis-à-vis de l'équation de la chaleur : le bas de leur spectre L^2 est le même, les distances de Carnot-Carathéodory associées sont comparables à la métrique riemannienne sur X et, surtout, les noyaux de la chaleur sont tous comparables (en temps grand) au noyau de la chaleur sur X. Nous en déduisons en particulier des encadrements très précis des noyaux de la chaleur dans ce cadre, ainsi que des fonctions de Green correspondantes.