In pursuit of negatively associated measures, this thesis focuses on certain negative correlation properties in matroids. In particular, the results presented contribute to the search for matroids which satisfy $$P(\{X:e,f\in X\}) \leq P(\{X:e\in X\})P(\{X:f\in X\})$$ for certain measures, $P$, on the ground set. Let $\mathcal M$ be a matroid. Let $(y_g:g\in E)$ be a weighting of the ground set and let $${Z = \sum_{X}\left( \prod_{x\in X} y_x\right) }$$ be the polynomial which generates Z-sets, were Z $\in \{$ B,I,S $\}$. For each of these, the sum is over bases, independent sets and spanning sets, respectively. Let $e$ and $f$ be distinct elements of $E$ and let $Z_e$ indicate partial derivative. Then $\mathcal M$ is Z-Rayleigh if $Z_eZ_f-ZZ_{ef}\geq 0$ for every positive evaluation of the $y_g$s. The known elementary results for the B, I and S-Rayleigh properties and two special cases called negative correlation and balance are proved. Furthermore, several new results are discussed. In particular, if a matroid is binary on at most nine elements or paving or rank three, then it is I-Rayleigh if it is B-Rayleigh. Sparse paving matroids are B-Rayleigh. The I-Rayleigh difference for graphs on at most seven vertices is a sum of monomials times squares of polynomials and this same special form holds for all series parallel graphs.

In pursuit of negatively associated measures, this thesis focuses on certain negative correlation properties in matroids. In particular, the results presented contribute to the search for matroids which satisfy $$P(\{X:e,f\in X\}) \leq P(\{X:e\in X\})P(\{X:f\in X\})$$ for certain measures, $P$, on the ground set. Let $\mathcal M$ be a matroid. Let $(y_g:g\in E)$ be a weighting of the ground set and let $${Z = \sum_{X}\left( \prod_{x\in X} y_x\right) }$$ be the polynomial which generates Z-sets, were Z $\in \{$ B,I,S $\}$. For each of these, the sum is over bases, independent sets and spanning sets, respectively. Let $e$ and $f$ be distinct elements of $E$ and let $Z_e$ indicate partial derivative. Then $\mathcal M$ is Z-Rayleigh if $Z_eZ_f-ZZ_{ef}\geq 0$ for every positive evaluation of the $y_g$s. The known elementary results for the B, I and S-Rayleigh properties and two special cases called negative correlation and balance are proved. Furthermore, several new results are discussed. In particular, if a matroid is binary on at most nine elements or paving or rank three, then it is I-Rayleigh if it is B-Rayleigh. Sparse paving matroids are B-Rayleigh. The I-Rayleigh difference for graphs on at most seven vertices is a sum of monomials times squares of polynomials and this same special form holds for all series parallel graphs.

Smart devices with multiple on-board sensors, networked through wired or wireless links, are distributed in physical systems and environments. Broad applications of such sensor networks include manufacturing quality control and wireless sensor systems. In the operation of sensor systems, robust methods for retrieving reliable information from sensor systems are crucial in the presence of potential sensor failures. Existence of sensor redundancy is one of the main drivers for the robustness or fault tolerance capability of a sensor system. The redundancy degree of sensors plays two important roles pertaining to the robustness of a sensor network. First, the redundancy degree provides proper parameter values for robust estimator; second, we can calculate the fault tolerance capability of a sensor network from the redundancy degree. Given this importance of the redundancy degree, this dissertation presents efficient algorithms based on matroid theory to compute the redundancy degree of a clustered sensor network. In the efficient algorithms, a cluster pattern of a sensor network allows us to decompose a large sensor network into smaller sub-systems, from which the redundancy degree can be found more efficiently. Finally, the robustness analysis as well as its algorithm procedure is illustrated using examples of a multi-station assembly process and calibration of wireless sensor networks.

A binary clutter is cycling if its packing and covering linear program have integral optimal solutions for all Eulerian edge capacities. We prove that the clutter of odd st- walks of a signed graph is cycling if and only if it does not contain as a minor the clutter of odd circuits of K5 nor the clutter of lines of the Fano matroid. Corollaries of this result include, of many, the characterization for weakly bipartite signed graphs, packing two- commodity paths, packing T-joins with small |T|, a new result on covering odd circuits of a signed graph, as well as a new result on covering odd circuits and odd T-joins of a signed graft.

<p>In this thesis we generalize the well-known connections between closure operators, closure systems and complete lattices. We introduce a special kind of a partial closure operator, named sharp partial closure operator, and show that each sharp partial closure operator uniquely corresponds to a partial closure&nbsp; system. We further introduce a special kind of a partial clo-sure system, called principal partial closure system, and then prove the representation theorem for ordered sets with respect to the introduced partial closure operators and partial closure systems.<br />Further, motivated by a well-known connection between matroids and geometric lattices, given that the notion of matroids can be naturally generalized to partial matroids (by dening them with respect to a partial closure operator instead of with respect to a closure operator), we dene geometric poset, and show that there is a same kind of connection between partial matroids and geometric posets as there is between matroids and geometric lattices. Furthermore, we then dene semimod-ular poset, and show that it is indeed a generalization of semi-modular lattices, and that there is a same kind of connection between semimodular and geometric posets as there is between<br />semimodular and geometric lattices.</p><p>Finally, we note that the dened notions can be applied to im-plicational systems, that have many applications in real world,particularly in big data analysis.</p> / <p>U ovoj tezi uop&scaron;tavamo dobro poznate veze između operatora zatvaranja, sistema zatvaranja i potpunih mreža. Uvodimo posebnu vrstu parcijalnog operatora zatvaranja, koji nazivamo o&scaron;tar parcijalni operator zatvaranja, i pokazujemo da svaki o&scaron;tar parcijalni operator zatvaranja jedinstveno korespondira parcijalnom sistemu zatvaranja. Dalje uvodimo posebnu vrstu parcijalnog sistema zatvaranja, nazvan glavni parcijalni sistem zatvaranja, a zatim dokazujemo teoremu reprezentacije za posete u odnosu na uvedene parcijalne operatore zatvaranja i parcijalne sisteme zatvaranja. Dalje, s obzirom na dobro poznatu vezu između matroida i geometrijskih mreža, a budući da se pojam matroida može na prirodan nacin uop&scaron;titi na parcijalne&nbsp; matroide (defini&scaron;ući ih preko parcijalnih operatora zatvaranja umesto preko operatora&nbsp; zatvaranja), defini&scaron;emo geometrijske uređene skupove i pokazujemo da su povezani sa parcijalnim matroidima na isti način kao &scaron;to su povezani i matroidi i&nbsp; geometrijske mreže. Osim toga, defini&scaron;emo polumodularne uređene skupove i pokazujemo da su oni zaista uop&scaron;tenje polumodularnih mreža i da ista veza postoji&nbsp; između polumodularnih i geometrijskih poseta kao &scaron;to imamo između polumodularnih i geometrijskih mreža. Konačno, konstatujemo da definisani pojmovi&nbsp; mogu biti primenjeni na implikacione sisteme, koji imaju veliku primenu u realnom svetu, posebno u analizi velikih podataka.</p>

Making use of algebraic and combinatorial techniques, we study two topics: the arithmetic degree of squarefree strongly stable ideals and the h-vectors of matroid complexes. For a squarefree monomial ideal, I, the arithmetic degree of I is the number of facets of the simplicial complex which has I as its Stanley-Reisner ideal. We consider the case when I is squarefree strongly stable, in which case we give an exact formula for the arithmetic degree in terms of the minimal generators of I as well as a lower bound resembling that from the Multiplicity Conjecture. Using this, we can produce an upper bound on the number of minimal generators of any Cohen-Macaulay ideals with arbitrary codimension extending Dubreil’s theorem for codimension 2. A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible h-vectors of such complexes. In the case when the complex has dimension 1 we completely resolve this question and we give some partial results for higher dimensions. We also prove the 1-dimensional case of a conjecture of Stanley that all matroid h-vectors are pure O-sequences. Finally, we completely characterize the Stanley-Reisner ideals of matroid complexes.

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.