Since its introduction in the year 2000 by Ahlswede et al., the network coding paradigm has revolutionized the way we understand information flows in networks. Traditionally, information transmitted in a communication network was treated as a commodity in a transportation network, much like cars on highways or fluids in pipes. This approach, however, fails to capture the very nature of information, which in contrast to material goods, can be coded and decoded. The network coding techniques take full advantage of the inherent properties of information, and allow the nodes in a network, not only to store and forward, but also to "mix", i.e., encode, their received data. This approach was shown to result in a substantial throughput gain over the traditional routing and tree packing techniques. In this dissertation, we study applications of network coding for guarantying reliable and secure information transmission in networks with compromised edges. First, we investigate the construction of robust network codes for achieving network resilience against link failures. We focus on the practical important case of unicast networks with non-uniform edge capacities where a single link can fail at a time. We demonstrate that these networks exhibit unique structural properties when they are minimal, i.e., when they do not contain redundant edges. Based on this structure, we prove that robust linear network codes exist for these networks over GF(2), and devise an efficient algorithm to construct them. Second, we consider the problem of securing a multicast network against an eavesdropper that can intercept the packets on a limited number of network links. We recast this problem as a network generalization of the classical wiretap channel of Type II introduced by Ozarow and Wyner in 1984. In particular, we demonstrate that perfect secrecy can be achieved by using the Ozarow-Wyner scheme of coset coding at the source, on top of the implemented network code. Consequently, we transparently recover important results available in the literature on secure network coding. We also derive new bounds on the required secure code alphabet size and an algorithm for code construction. In the last part of this dissertation, we study the connection between index coding, network coding, and matroid linear representation. We devise a reduction from the index coding problem to the network coding problem, implying that in the linear case these two problems are equivalent. We also present a second reduction from the matroid linear representability problem to index coding, and therefore, to network coding. The latter reduction establishes a strong connection between matroid theory and network coding theory. These two reductions are then used to construct special instances of the index coding problem where vector linear codes outperform scalar linear ones, and where non-linear encoding is needed to achieve the optimal number of transmission. Thereby, we provide a counterexample to a related conjecture in the literature and demonstrate the benefits of vector linear codes.

We consider different ways of describing a matroid to a Turing machine by listing the members of various families of subsets, and we construct an order on these different methods of description. We show that, under this scheme, several natural matroid problems are complete in classes thought not to be equal to P. We list various results linking parameters of basis graphs to parameters of their associated matroids. For small values of k we determine which matroids have the clique number, chromatic number, or maximum degree of their basis graphs bounded above by k. If P is a class of graphs that is closed under isomorphism and induced subgraphs, then the set of matroids whose basis graphs belong to P is closed under minors. We characterise the minor-closed classes that arise in this way, and exhibit several examples. One way of choosing a basis of a matroid at random is to select a total ordering of the ground set uniformly at random and use the greedy algorithm. We consider the class of matroids having the property that this procedure chooses a basis uniformly at random. Finally we consider a problem mentioned by Oxley. He asked if, for every two elements and n - 2 cocircuits in an n-connected matroid, there is a circuit that contains both elements and that meets every cocircuit. We show that a slightly stronger property holds for regular matroids.

The Kazhdan-Lusztig polynomial of a matroid M, denoted P_M( t ), was recently defined by Elias, Proudfoot, and Wakefield. These polynomials are analogous to the classical Kazhdan-Lusztig polynomials associated with Coxeter groups. For example, in both cases there is a purely combinatorial recursive definition. Furthermore, in the classical setting, if the Coxeter group is a Weyl group then the Kazhdan-Lusztig polynomial is a Poincare polynomial for the intersection cohomology of a particular variety; in the matroid setting, if M is a realizable matroid then the Kazhdan-Lusztig polynomial is also the intersection cohomology Poincare polynomial of a variety corresponding to M. (Though there are several analogies between the two types of polynomials, the theory is quite different.) Here we compute the Kazhdan-Lusztig polynomials of several graphical matroids, including thagomizer graphs, the complete bipartite graph K_{2,n}, and (conjecturally) fan graphs. Additionally, we investigate a conjecture by the author, Proudfoot, and Young on the real-rootedness for Kazhdan-Lusztig polynomials of these matroids as well as a conjecture on the interlacing behavior of these roots. We also show that the Kazhdan-Lusztig polynomials of uniform matroids of rank n − 1 on n elements are real-rooted. This dissertation includes both previously published and unpublished co-authored material.

Connectivity is a fundamental tool for matroid theorists, which has become increasingly important in the eventual solution of many problems in matroid theory. Loosely speaking, connectivity can be used to help describe a matroid's structure. In this thesis, we prove a series of results that further the knowledge and understanding in the field of matroid connectivity. These results fall into two parts. First, we focus on 3-connected matroids. A chain theorem is a result that proves the existence of an element, or elements, whose deletion or contraction preserves a predetermined connectivity property. We prove a series of chain theorems for 3-connected matroids where, after fixing a basis B, the elements in B are only eligible for contraction, while the elements not in B are only eligible for deletion. Moreover, we prove a splitter theorem, where a 3-connected minor is also preserved, resolving a conjecture posed by Whittle and Williams in 2013. Second, we consider k-connected matroids, where k >= 3. A certain tree, known as a k-tree, can be used to describe the structure of a k-connected matroid. We present an algorithm for constructing a k-tree for a k-connected matroid M. Provided that the rank of a subset of E(M) can be found in unit time, the algorithm runs in time polynomial in |E(M)|. This generalises Oxley and Semple's (2013) polynomial-time algorithm for constructing a 3-tree for a 3-connected 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.

We begin by introducing matroids in the context of finite collections of vectors from a vector space over a specified field, where the notion of independence is linear independence. Then we will introduce the concept of a matroid invariant. Specifically, we will look at the Tutte polynomial, which is a well-defined two-variable invariant that can be used to determine differences and similarities between a collection of given matroids. The Tutte polynomial can tell us certain properties of a given matroid (such as the number of bases, independent sets, etc.) without the need to manually solve for them. Although the Tutte polynomial gives us significant information about a matroid, it does not uniquely determine a matroid. This thesis will focus on non-isomorphic matroids that have the same Tutte polynomial. We call such matroids Tutte-equivalent, and we will study the characteristics needed for two matroids to be Tutte-equivalent. Finally, we will demonstrate methods to construct families of Tutte-equivalent matroids.

The rapid developments in the sensor and its related technology have made automation possible in many processes in diverse fields. Also sensor-based fault diagnosis and quality improvements have been made possible. These tasks depend highly on the sensor network for the accurate measurements. The two major problems that affect the reliability of the sensor system/network are sensor failures and sensor anomalies. The usage of redundant sensors offers some tolerance against these two problems. Hence the redundancy analysis of the sensor system is essential in order to clearly know the robustness of the system against these two problems. The degree of sensor redundancy defined in this thesis is closely tied with the fault-tolerance of the sensor network and can be viewed as a parameter related to the effectiveness of the sensor system design. In this thesis, an efficient algorithm to determine the degree of sensor redundancy for linear sensor systems is developed. First the redundancy structure is linked with the matroid structure, developed from the design matrix, using the matroid theory. The matroid problem equivalent to the degree of sensor redundancy is developed and the mathematical formulation for it is established. The solution is obtained by solving a series of l1-norm minimization problems. For many problems tested, the proposed algorithm is more efficient than other known alternatives such as basic exhaustive search and bound and decomposition method. The proposed algorithm is tested on problem instances from the literature and wide range of simulated problems. The results show that the algorithm determines the degree of redundancy more accurately when the design matrix is dense than when it is sparse. The algorithm provided accurate results for most problems in relatively short computation times.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-23T15:16:49Z No. of bitstreams: 1 JOSÉ EDER SALVADOR DE VASCONCELOS - DISSERTAÇÃO PPGMAT 2009..pdf: 634118 bytes, checksum: e65e70c702364b197a36f09e8d1ef296 (MD5) / Made available in DSpace on 2018-07-23T15:16:49Z (GMT). No. of bitstreams: 1 JOSÉ EDER SALVADOR DE VASCONCELOS - DISSERTAÇÃO PPGMAT 2009..pdf: 634118 bytes, checksum: e65e70c702364b197a36f09e8d1ef296 (MD5) Previous issue date: 2009-11 / Capes / O número de cocircuitos disjuntos em uma matróide é delimitado pelo seu posto. Existem, no entanto, matróides de posto arbitrariamente grande que não contêm dois cocircuitos disjuntos. Considere, por exemplo,M(Kn) eUn,2n. Além disso, a matróide bicircularB(Kn) pode ter posto arbitrariamente grande, mas não tem 3 cocircuitos disjuntos. Nós apresentaremos uma prova, obtida por Jim Geelen e Kasper Kabell em (5), para o seguinte fato: para cadak en, existe uma constantec tal que, seM é uma matróide com posto no mínimoc, entãoM temk cocircuitos disjuntos ou contém uma das seguintes matróides como menorUn,2n,M(Kn) ouB(Kn). / The number of disjoint cocircuits in a matroid is bounded by its rank. There are, however, matroids of rank arbitrarily large that do not contain two disjoint cocircuits. Consider, for example,M(kn) andUn,2n. Moreover, the bicircular matroidB(kn) may have arbitrarily large rank but do not have 3 disjoints cocircuits. We show a proof obtained by Jim Geelen and Kasper Kabell in (5) to the following fact: for everyk andn, there is a constantc such that ifM is a matroid with rank at leastc, thenM hask disjoint cocircuits orM contains one of the following matroids as a minorUn,2n, M(kn) orB(kn).