Finding the maximum size of a matching in an undirected graph and finding the maximum size of branching in a directed graph can be formulated as matrix rank problems. The Tutte matrix, introduced by Tutte as a representation of an undirected graph, has rank equal to the maximum number of vertices covered by a matching in the associated graph. The branching matrix, a representation of a directed graph, has rank equal to the maximum number of vertices covered by a branching in the associated graph. A mixed graph has both undirected and directed edges, and the matching forest problem for mixed graphs, introduced by Giles, is a generalization of the matching problem and the branching problem. A mixed graph can be represented by the matching forest matrix, and the rank of the matching forest matrix is related to the size of a matching forest in the associated mixed graph. The Tutte matrix and the branching matrix have indeterminate entries, and we describe algorithms that evaluate the indeterminates as rationals in such a way that the rank of the evaluated matrix is equal to the rank of the indeterminate matrix. Matroids in the context of graphs are discussed, and matroid formulations for the matching, branching, and matching forest problems are given.

This paper aims at presenting the necessary tools to prove that a scheme of finite type over Z exhibits the same singularities as those which occur on a Grassmann variety. First, basic theory regarding the combinatorial objects matroids is presented. Some important examples for the remainder of the paper are given, which also serve to aid the reader in intuition and understanding of matroids. Basic algebraic geometry is presented, and the building blocks affine varieties, projective varieties and general varieties are introduced. These object are generalised in the following subsection as affine schemes and schemes, which are the central object of study in modern algebraic geometry. Important results from the theory of algebraic groups are shown in order to better understand the formulation and proof of the Gelfand–MacPherson theorem, which in turn is utilised, together with Mnëv’s universality theorem, to prove the main result of the paper.

Dans cette thèse, nous proposons une approche combinatoire novatrice fondée sur les matroïdes orientés pour l'étude quantitative de la forme de structures anatomiques 3D. Nous nous basons sur des points de repère qui ont été préalablement localisés par des experts sur la structure anatomique étudiée. La nouveauté de cette méthode provient de l'utilisation de matroïdes orientés. Ces outils mathématiques nous permettent de coder la position relative des points de repère de façon purement combinatoire, c'est-à-dire sans utiliser de notions d'angles ou de distances, en associant un signe (0, + ou -) à chaque sous-ensemble de (d+1) points de repère où d est la dimension de l'espace (dans notre cas 2 ou 3). Dans une première partie, nous supposons qu'il existe des contraintes d'ordres sur chaque axe de coordonnée pour les points de repère. Nous obtenons alors une caractérisation (en dimension 2 et 3) des sous-ensembles de points de repère dont le signe associé est constant, quelles que soient les valeurs des coordonnées satisfaisant les contraintes d'ordre. Dans une deuxième partie, nous cherchons à classifier un ensemble de modèles 3D, en les codant au préalable par ces listes de signes. Nous analysons d'abord comment s'appliquent les algorithmes de clustering classiques, puis nous décrivons comment caractériser des classes de façon directe, à l'aide des signes associés à quelques sous-ensembles de points de repère. Dans une troisième partie, nous détaillons les algorithmes et l'implémentation en machine de cette nouvelle méthode de morphométrie afin de pouvoir l'appliquer à des données réelles. Dans la dernière partie, nous appliquons la méthode sur trois bases de données composées chacune de plusieurs dizaines de points de repères relevés sur plusieurs dizaines à plusieurs centaines de structures crâniennes pour des applications en anatomie comparée, en orthodontie et sur des cas cliniques d'enfants présentant des déformations cranio-faciales. / In this thesis, we propose an innovative combinatorial method based on oriented matroids for the quantitative study of the shape of 3D anatomical structures. We rely on landmarks which were previously defined by experts on the studied anatomical structure. The novelty of this method results from the use of oriented matroids. These mathematical tools allow us to encode the relative position of landmarks in a purely combinatorial way, that is without using concepts of angles or distances, by associating a sign (0, + or -) for each subset of (d+1) landmarks where d is the dimension of space (in our case 2 or 3). In the first part, we assume that there exist constraints of orders on each coordinate axis for the landmarks. We obtain a characterization (in dimension 2 and 3) of the subsets of landmarks of which the associated sign is constant, regardless of the values of the coordinates satisfying the constraints of order. In a second part, we try to classify a set of 3D models, encoding in advance by these lists of signs. We first analyze how to apply classic clustering algorithms, and then describe how to characterize the classes directly, using signs associated with some subsets of landmarks. In the third part, we explain the algorithms and the implementation of this new morphometry method in order to apply it to real data. In the last part, we apply the method to three databases each consisting of several dozens of points defined on several dozens to several hundreds of cranial structures for applications in comparative anatomy, in orthodontics and on clinical cases of children with craniofacial deformities.

Submitted by Angela Maria de Oliveira (amolivei@uepg.br) on 2017-10-31T13:25:24Z No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Dissertação - Rosilene Ales.pdf: 1659544 bytes, checksum: b146b0c0707ab6d3ca5c1e525b34793c (MD5) / Made available in DSpace on 2017-10-31T13:25:24Z (GMT). No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Dissertação - Rosilene Ales.pdf: 1659544 bytes, checksum: b146b0c0707ab6d3ca5c1e525b34793c (MD5) Previous issue date: 2017-09-29 / Whitney identificou as propriedades fundamentais de dependência, que são comuns entre grafos e matrizes dando origem a Teoria de Matróides em 1935. Neste trabalho será apresentada a construção de novas famílias de Matróides e a resolução de teoremas de maneira ampla e de fácil compreensão, pois o tema é definido de forma matemática puramente abstrata. Assim, para obter um novo Matróide utiliza-se um já dado, sendo definido em termos de seus conjuntos independentes. Destaca-se que Matróides é encontrado nas seguintes abrangências: em espaços vetoriais, ciclos em grafos, funções afins, circuitos, bases, rank, fecho e dualidade. Deste modo, para construir um código quântico, precisa compreender a teoria de informação e codificação quântica como os Postulados da Mecânica Quântica, estados de um ou vários qubits, operadores unitários, portas lógicas, medidas de estados quânticos, códigos estabilizadores e a classe dos códigos de Calderbank-Shor-Steane (CSS). Os códigos lineares são os códigos da Álgebra Linear, subespaços vetoriais definidos sobre corpos finitos. Os códigos quânticos tem a finalidade de proteger possíveis erros de um canal para detectar e corrigir tais erros. Com o fundamento teórico adquirido, pode se verificar a possibilidade de construir a conexão da teoria de Matróides e a teoria de codificação quântica, por meio da matriz de verificação de paridade de um código CSS e a matriz que gera um dado Matróide vetorial. / Whitney identified the fundamental properties of dependence, which are common among graphs and matrices giving rise to Matroid Theory in 1935. In this work will be presented the construction of new families of Matroid and the resolution of theorems in a comprehensive and easy to understand, since the theme is defined in purely abstract mathematical form. Thus, to obtain a new Matroid one uses an already given one, being defined in terms of its independent sets. It should be noted that Matroid is found in the following ranges: in vector spaces, cycles in graphs, related functions, circuits, bases, rank, closure and duality. Thus, to construct a quantum code, one must understand quantum information and coding theory such as the Postulates of Quantum Mechanics, single- or multi-qubit states, unit operators, logic gates, quantum state measures, stabilizing codes, and the class of codes of Calderbank-Shor-Steane (CSS). The linear codes are the codes of Linear Algebra, vector subspaces defined on finite bodies. Quantum codes are intended to protect potential errors from a channel to detect and correct such errors. With the acquired theoretical foundation, the possibility of constructing the connection of the Matroid theory and the theory of quantum codification can be verified by means of the parity check matrix of a CSS code and the matrix that generates a given vector matroid.

Two applications of Category Theory are considered. The link between them is applications to Physics and more specifically to Entropy. The first research chapter is broader in scope and not explicitly about Physics, although connections to Statistical Mechanics are made towards the end of the chapter. Matroids are abstract structures that describe dependence, and strong maps are certain structure-preserving functions between them with desirable properties. We examine properties of various categories of matroids and strong maps: we compute limits and colimits; we find free and cofree constructions of various subcategories; we examine factorisation structures, including a translation principle from geometric lattices; we find functors with convenient properties to/from vector spaces, multisets of vectors, geometric lattices, and graphs; we determine which widely used operations on matroids are functorial (these include deletion, contraction, series and parallel connection, and a simplification monad); lastly, we find a categorical characterisation of the greedy algorithm. In conclusion, this project determines which aspects of Matroid Theory are most and least conducive to categorical treatment. The purpose of the second research chapter is to provide a categorical framework for generalising and unifying notions of Entropy in various settings, exploiting the fact that Entropy is a monotone subadditive function. A categorical characterisation of Entropy through a category of thermodynamical systems and adiabatic processes is found. A modelling perspective (adiabatic categories) that directly generalises an existing model is compared to an axiomatisation through topological and linear structures (topological weak semimodules), where the latter is based on a categorification of semimodules. Properties of each class of categories are examined; most notably a cancellation property of adiabatic categories generalising an existing result, and an adjunction between the categories of weak semimodules and symmetric monoidal categories. An adjunction between categories of adiabatic categories and topological weak semimodules is found. We examine in which cases each of these classes of categories constitutes a traced monoidal category. Lastly, examples of physical applications are provided. In conclusion, this project uncovers a way of, and makes progress towards, retrieving the statistical formulation of Entropy from simple axioms.

A scenario where multiple entities interact with a common environment to achieve individual and common goals either co-operatively or competitively can be classified as a Multi-Agent System. In this thesis, we concentrate on the situations where the agents exhibit selfish, competitive and strategic behaviour, giving rise to interesting game theoretic and optimization problems. From a computational point of view, the presence of multiple agents introduces strategic and temporal issues, apart from enhancing the difficulty of optimization. We study the following natural mathematical models of such multi-agent problems faced in practice: a) combinatorial optimization problems with multi-agent submodular cost functions, b) combinatorial auctions with partially public valuations and c) online vertex-weighted bipartite matching and single bid budgeted allocations. We provide approximation algorithms, online algorithms and hardness of approximation results for these problems.

Orientadores: Reginaldo Palazzo Junior, Carlile Campos Lavor / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-11T22:44:54Z (GMT). No. of bitstreams: 1 LaGuardia_GiulianoGadioli_D.pdf: 1126065 bytes, checksum: c3ad65915db4e87e1752adbbbbef2841 (MD5) Previous issue date: 2008 / Resumo: Como principais contribuições desta tese, apresentamos novos métodos de construção que geram novas famílias de códigos quânticos CSS. As construções são baseadas em códigos cíclicos (clássicos) BCH, Reed-Solomon, Reed-Muller, Resíduos quadráticos e também nos códigos derivados do produto tensorial de dois códigos Reed-Solomon. Os principais códigos quânticos construídos neste trabalho, em termos de parâmetros, são os derivados dos códigos BCH clássicos. Além disso, estudamos as condições necessárias para analisar as situações nas quais os códigos cíclicos quânticos (clássicos) são códigos MDS (do inglês, Maximum- Distance-Separable codes). Apresentamos, também, novas conexões entre a teoria de matróides e a teoria dos códigos quânticos CSS, que acreditamos serem as primeiras conexões entre tais teorias. Mais especificamente, demonstramos que a função enumeradora de pesos de um código quântico CSS é uma avaliação do polinômio de Tutte da soma direta dos matróides originados a partir dos códigos clássicos utilizados na construção CSS. / Abstract: This thesis proposes, as the main contributions, constructions method of new families of quantum CSS codes. These constructions are based on classical cyclic codes of the types BCH, Reed-Solomon, Reed-Muller, Quadratic Residue and also are based on product codes of classical Reed-Solomon codes. The main family of quantum codes constructed in this work, i. e., quantum codes having better parameters, are the ones derived from classical BCH codes. Moreover, we present some new conditions in which quantum CSS cyclic codes are quantumMDS codes. In addition, we provide the elements to connect matroid theory and quantum coding theory. More specifically, we show that the weight enumerator of a CSS quantum code is equivalent to evaluating the Tutte polynomial of the direct sum of the matroid associated to the classical codes used in the CSS construction. / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

This thesis is divided into two parts, each part exploring a different topic within the general area of nonlinear algebra. In the first part, we study several applications of tensors. First, we study tensor networks, and more specifically: uniform matrix product states. We use methods from nonlinear algebra and algebraic geometry to answer questions about topology, defining equations, and identifiability of uniform matrix product states. By an interplay of theorems from algebra, geometry, and quantum physics we answer several questions and conjectures posed by Critch, Morton and Hackbusch. In addition, we prove a tensor version of the so-called quantum Wielandt inequality, solving an open problem regarding the higher-dimensional version of matrix product states. Second, we present new contributions to the study of fast matrix multiplication. Motivated by the symmetric version of matrix multiplication we study the plethysm S^k(sl_n) of the adjoint representation sl_n of the Lie group SL_n . Moreover, we discuss two algebraic approaches for constructing new tensors which could potentially be used to prove new upper bounds on the complexity of matrix multiplication. One approach is based on the highest weight vectors of the aforementioned plethysm. The other approach uses smoothable finite-dimensional algebras. Finally, we study the Hessian discriminant of a cubic surface, a recently introduced invariant defined in terms of the Waring rank. We express the Hessian discriminant in terms of fundamental invariants. This answers Question 15 of the 27 questions on the cubic surface posed by Bernd Sturmfels. In the second part of this thesis, we apply algebro-geometric methods to study matroids and flag matroids. We review a geometric interpretation of the Tutte polynomial in terms of the equivariant K-theory of the Grassmannian. By generalizing Grassmannians to partial flag varieties, we obtain a new invariant of flag matroids: the flag-geometric Tutte polynomial. We study this invariant in detail, and prove several interesting combinatorial properties.

Rigidity theory deals in problems of the following form: given a structure defined by geometric constraints on a set of objects, what information about its geometric behavior is implied by the underlying combinatorial structure. The most well-studied class of structures is the bar-joint framework, which is made of fixed-length bars connected by universal joints with full rotational degrees of freedom; the allowed motions preserve the lengths and connectivity of the bars, and a framework is rigid if the only allowed motions are trivial motions of Euclidean space. A remarkable theorem of Maxwell-Laman says that rigidity of generic bar-joint frameworks depends only on the graph that has as its edges the bars and as its vertices the joints. We generalize the "degree of freedom counts that appear in the Maxwell-Laman theorem to the very general setting of (k,l)-sparse and (k,l)-graded sparse hypergraphs. We characterize these in terms of their graph-graph theoretic and matroidal properties. For the fundamental algorithmic problems Decision, Extraction, Components, and Decomposition, we give efficient, implementable pebble game algorithms for all the (k,l)-sparse and (k,l)-graded-sparse families of hypergraphs we study. We then prove that all the matroids arising from (k,l)-sparse are linearly representable by matrices with a certain "natural" structure that captures the incidence structure of the hypergraph and the sparsity parameters k and l. Building on the combinatorial and linear theory discussed above, we introduce a new rigidity model: slider-pinning rigidity. This is an elaboration of the planar bar-joint model to include sliders, which constrain a vertex to move on a specific line. We prove the analogue of the Maxwell-Laman Theorem for slider pinning, using, as a lemma, a new proof of Whiteley's Parallel Redrawing Theorem. We conclude by studying the emergence of non-trivial rigid substructures in generic planar frameworks given by Erdos-Renyi random graphs. We prove that there is a sharp threshold for such substructures to emerge, and that, when they do, they are all linear size. This is consistent with experimental and simulation-based work done in the physics community on the formation of certain glasses.

In 2020, Huh, Matherne, Mészáros, and St. Dizier established the Lorentzian property of normalized Schur polynomials and conjectured the Lorentzian nature of other Schur-type symmetric polynomials. More recently in 2022, Matherne, Morales, and Selover proved that chromatic symmetric functions of indifference graphs of abelian Dyck paths are Lorentzian. In this thesis, we study the more general class of Lorentzian polynomials that is also invariant under the standard permutation action on variables. Throughout this work, we give exposition to the classical theory of symmetric polynomials and Lorentzian polynomials. Then we present several fundamental results on symmetric Lorentzian polynomials and highlight potential avenues for future research. / År 2020 bevisade Huh-Matherne-Mészáros-St.Dizier att normaliserade schur polynom är lorentziska och antog att andra symmetriska polynom av Schur-typ också är det. År 2022 bevisade Matherne-Morales-Selover att kromatiska symmetriska funktioner för indifferensgrafer av abeliska Dyck-paths är lorentziska. Motiverade av dessa resultat studerar vi den mer allmänna klassen av lorentziska polynom som också är invarianta under standardpermutationsverkan på variabler. I avhandlingen ger vi några grundläggande resultat om symmetriska lorentziska polynom och pekar på möjliga framtida riktningar.