Diese Arbeit entwickelt aufbauend auf bekannten Resultaten die Modultheorie über der äußeren Algebra in Teilen weiter, insbesondere werden die Tiefe eines Moduls und Moduln mit linearer injektiver Auflösung untersucht. Angewendet werden die Resultate auf die Orlik-Solomon Algebra eines Matroids.

Consider a network which consists of noiseless point to point channels. In this network, the source node wants to send messages to a specific set of sink nodes. If an intermediate node v has just one input channel then the received symbol by that node can be replicated and sent to the outgoing channels from v. If v has at least two incoming channels then it has two options. It can either send the received symbols one-by-one, one symbol in each time unit, or v can transmit a combination of the received symbols. The former choice takes more time compared to the latter option, which is called network coding. In the literature, it has been shown that in a single source finite acyclic network the maximum throughput can be achieved by using linear network codes. Significant effort has been made to efficiently construct good network codes. In addition, a polynomial time algorithm for constructing a linear network code on a given network was introduced. Also an algorithm for constructing a linear multicast code on an acyclic network was introduced. Finally, a method for finding a representation matrix for the network matroid of a given network G was also introduced. This matrix can be used to construct a generic code. In this thesis we first provide a review of some known methods for constructing linear multicast, broadcast and dispersion codes for cyclic and acyclic networks. We then give a method for normalization of a non-normal code, and also give a new algorithm for constructing a linear multicast code on a cyclic network. The construction of generic network codes is also addressed. / Graduate / 0984 / 0544 / esmaeili@uvic.ca

In the thesis "On Boundaries of Statistical Models" problems related to a description of probability distributions with zeros, lying in the boundary of a statistical model, are treated. The distributions considered are joint distributions of finite collections of finite discrete random variables. Owing to this restriction, statistical models are subsets of finite dimensional real vector spaces. The support set problem for exponential families, the main class of models considered in the thesis, is to characterize the possible supports of distributions in the boundaries of these statistical models. It is shown that this problem is equivalent to a characterization of the face lattice of a convex polytope, called the convex support. The main tool for treating questions related to the boundary are implicit representations. Exponential families are shown to be sets of solutions of binomial equations, connected to an underlying combinatorial structure, called oriented matroid. Under an additional assumption these equations are polynomial and one is placed in the setting of commutative algebra and algebraic geometry. In this case one recovers results from algebraic statistics. The combinatorial theory of exponential families using oriented matroids makes the established connection between an exponential family and its convex support completely natural: Both are derived from the same oriented matroid. The second part of the thesis deals with hierarchical models, which are a special class of exponential families constructed from simplicial complexes. The main technical tool for their treatment in this thesis are so called elementary circuits. After their introduction, they are used to derive properties of the implicit representations of hierarchical models. Each elementary circuit gives an equation holding on the hierarchical model, and these equations are shown to be the "simplest", in the sense that the smallest degree among the equations corresponding to elementary circuits gives a lower bound on the degree of all equations characterizing the model. Translating this result back to polyhedral geometry yields a neighborliness property of marginal polytopes, the convex supports of hierarchical models. Elementary circuits of small support are related to independence statements holding between the random variables whose joint distributions the hierarchical model describes. Models for which the complete set of circuits consists of elementary circuits are shown to be described by totally unimodular matrices. The thesis also contains an analysis of the case of binary random variables. In this special situation, marginal polytopes can be represented as the convex hulls of linear codes. Among the results here is a classification of full-dimensional linear code polytopes in terms of their subgroups. If represented by polynomial equations, exponential families are the varieties of binomial prime ideals. The third part of the thesis describes tools to treat models defined by not necessarily prime binomial ideals. It follows from Eisenbud and Sturmfels' results on binomial ideals that these models are unions of exponential families, and apart from solving the support set problem for each of these, one is faced with finding the decomposition. The thesis discusses algorithms for specialized treatment of binomial ideals, exploiting their combinatorial nature. The provided software package Binomials.m2 is shown to be able to compute very large primary decompositions, yielding a counterexample to a recent conjecture in algebraic statistics.

In the thesis "On Boundaries of Statistical Models" problems related to a description of probability distributions with zeros, lying in the boundary of a statistical model, are treated. The distributions considered are joint distributions of finite collections of finite discrete random variables. Owing to this restriction, statistical models are subsets of finite dimensional real vector spaces. The support set problem for exponential families, the main class of models considered in the thesis, is to characterize the possible supports of distributions in the boundaries of these statistical models. It is shown that this problem is equivalent to a characterization of the face lattice of a convex polytope, called the convex support. The main tool for treating questions related to the boundary are implicit representations. Exponential families are shown to be sets of solutions of binomial equations, connected to an underlying combinatorial structure, called oriented matroid. Under an additional assumption these equations are polynomial and one is placed in the setting of commutative algebra and algebraic geometry. In this case one recovers results from algebraic statistics. The combinatorial theory of exponential families using oriented matroids makes the established connection between an exponential family and its convex support completely natural: Both are derived from the same oriented matroid. The second part of the thesis deals with hierarchical models, which are a special class of exponential families constructed from simplicial complexes. The main technical tool for their treatment in this thesis are so called elementary circuits. After their introduction, they are used to derive properties of the implicit representations of hierarchical models. Each elementary circuit gives an equation holding on the hierarchical model, and these equations are shown to be the "simplest", in the sense that the smallest degree among the equations corresponding to elementary circuits gives a lower bound on the degree of all equations characterizing the model. Translating this result back to polyhedral geometry yields a neighborliness property of marginal polytopes, the convex supports of hierarchical models. Elementary circuits of small support are related to independence statements holding between the random variables whose joint distributions the hierarchical model describes. Models for which the complete set of circuits consists of elementary circuits are shown to be described by totally unimodular matrices. The thesis also contains an analysis of the case of binary random variables. In this special situation, marginal polytopes can be represented as the convex hulls of linear codes. Among the results here is a classification of full-dimensional linear code polytopes in terms of their subgroups. If represented by polynomial equations, exponential families are the varieties of binomial prime ideals. The third part of the thesis describes tools to treat models defined by not necessarily prime binomial ideals. It follows from Eisenbud and Sturmfels'' results on binomial ideals that these models are unions of exponential families, and apart from solving the support set problem for each of these, one is faced with finding the decomposition. The thesis discusses algorithms for specialized treatment of binomial ideals, exploiting their combinatorial nature. The provided software package Binomials.m2 is shown to be able to compute very large primary decompositions, yielding a counterexample to a recent conjecture in algebraic statistics.

Le travail général de cette thèse consiste à étendre les outils analytiques et algébriques usuellement employés dans la résolution de problèmes combinatoires déterministes à un cadre combinatoire stochastique. Deux cadres distincts sont étudiés : les problèmes combinatoires stochastiques discrets et les problèmes stochastiques continus. Le cadre discret est abordé à travers le problème de la forêt couvrante de poids maximal dans une formulation Two-Stage à multi-scénarios. La version déterministe très connue de ce problème établit des liens entre la fonction de rang dans un matroïde et la formulation duale, via l'algorithme glouton. La formulation stochastique discrète du problème de la forêt maximale couvrante est transformée en un problème déterministe équivalent, mais du fait de la multiplicité des scénarios, le dual associé est en quelque sorte incomplet. Le travail réalisé ici consiste à comprendre en quelles circonstances la formulation duale atteint néanmoins un minimum égal au problème primal intégral. D'ordinaire, une approche combinatoire classique des problèmes de graphes pondérés consiste à rechercher des configurations particulières au sein des graphes, comme les circuits, et à explorer d'éventuelles recombinaisons. Pour donner une illustration simple, si on change d'une manière infinitésimale les valeurs de poids des arêtes d'un graphe, il est possible que la forêt couvrante de poids maximal se réorganise complètement. Ceci est vu comme un obstacle dans une approche purement combinatoire. Pourtant, certaines grandeurs analytiques vont varier de manière continue en fonction de ces variations infinitésimales, comme la somme des poids des arêtes choisies. Nous introduisons des fonctions qui rendent compte de ces variations continues, et nous examinons dans quels cas les formulations duales atteignent la même valeur que les formulations primales intégrales. Nous proposons une méthode d'approximation dans le cas contraire et nous statuons sur la NP complétude de ce type de problème.Les problèmes stochastiques continus sont abordés via le problème de sac à dos avec contrainte stochastique. La formulation est de type ``chance constraint'', et la dualisation par variable lagrangienne est adaptée à une situation où la probabilité de respecter la contrainte doit rester proche de $1$. Le modèle étudié est celui d'un sac à dos où les objets ont une valeur et un poids déterminés par des distributions normales. Dans notre approche, nous nous attachons à appliquer des méthodes de gradient directement sur la formulation en espérance de la fonction objectif et de la contrainte. Nous délaissons donc une possible reformulation classique du problème sous forme géométrique pour détailler les conditions de convergence de la méthode du gradient stochastique. Cette partie est illustrée par des tests numériques de comparaison avec la méthode SOCP sur des instances combinatoires avec méthode de Branch and Bound, et sur des instances relaxées. / The global purpose of this thesis is to study the conditions to extend analytical and algebraical properties commonly observed in the resolution of deterministic combinatorial problems to the corresponding stochastic formulations of these problems. Two distinct situations are treated : discrete combinatorial stochastic problems and continuous stochastic problems. Discrete situation is examined with the Two Stage formulation of the Maximum Weight Covering Forest. The well known corresponding deterministic formulation shows the connexions between the rank function of a matroid, the greedy algorithm , and the dual formulation. The discrete stochastic formulation of the Maximal Covering Forest is turned into a deterministic equivalent formulation, but, due to the number of scenarios, the associated dual is not complete. The work of this thesis leads to understand in which cases the dual formulation still has the same value as the primal integer formulation. Usually, classical combinatorial approaches aim to find particular configurations in the graph, as circuits, in order to handle possible reconfigurations. For example, slight modifications of the weights of the edges might change considerably the configuration of the Maximum Weight Covering Forest. This can be seen as an obstacle to handle pure combinatorial proofs. However, some global relevant quantities, like the global weight of the selected edges during the greedy algorithm, have a continuous variation in function of slight modifications. We introduce some functions in order to outline these continuous variations. And we state in which cases Primal integral problems have the same objective values as dual formulations. When it is not the case, we propose an approximation method and we examine the NP completeness of this problem.Continuous stochastic problems are presented with the stochastic Knapsack with chance constraint. Chance constraint and dual Lagrangian formulation are adapted in the case where the expected probability of not exceeding the knapsack capacity is close to $1$. The introduced model consists in items whose costs and rewards follow normal distributions. In our case, we try to apply direct gradient methods without reformulating the problem into geometrical terms. We detail convergence conditions of gradient based methods directly on the initial formulation. This part is illustrated with numerical tests on combinatorial instances and Branch and Bound evaluations on relaxed formulations.

My thesis is mostly concerned with algebraic and combinatorial aspects of the theory of hyperplane arrangements. More specifically, I study the Orlik-Terao algebra of a hyperplane arrangement and the broken circuit complex of a matroid. The Orlik-Terao algebra is a useful tool for studying hyperplane arrangements, especially for characterizing some non-combinatorial properties. The broken circuit complex, on the one hand, is closely related to the Orlik-Terao algebra, and on the other hand, plays a crucial role in the study of many combinatorial problem: the coefficients of the characteristic polynomial of a matroid are encoded in the f-vector of the broken circuit complex of the matroid. Among main results of the thesis are characterizations of the complete intersection and Gorenstein properties of the broken circuit complex and the Orlik-Terao algebra. I also study the h-vector of the broken circuit complex of a series-parallel network and relate certain entries of that vector to ear decompositions of the network. An application of the Orlik-Terao algebra in studying the relation space of a hyperplane arrangement is also included in the thesis.