First introduced by Forman in 1998, discrete Morse theory has become a standard tool in topological combinatorics. The main idea of discrete Morse theory is to pair cells in a cellular complex in a manner that permits cancellation via elementary collapses, reducing the complex under consideration to a homotopy equivalent complex with fewer cells. In chapter 1, we introduce the relevant background for discrete Morse theory. In chapter 2, we define a discrete Morse matching for a family of independence complexes that generalize the matching complexes of suitable "small" grid graphs. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups. Furthermore, we determine the Euler characteristic for these complexes and prove that several of their homology groups are non-zero. In chapter 3, we introduce the notion of a homomorphism complex for partially ordered sets, placing particular emphasis on maps between chain posets and the Boolean algebras. We extend the notion of folding from general graph homomorphism complexes to the poset case, and we define an iterative discrete Morse matching for these Boolean complexes. We provide formulas for enumerating the number of critical cells arising from this matching as well as for the Euler characteristic. We end with a conjecture on the optimality of our matching derived from connections to 3-equal manifolds

This thesis is comprised of three main parts: The Hermite-Hadamard inequality on discrete time scales, the fractional Hermite-Hadamard inequality, and Karush-Kuhn- Tucker conditions on higher dimensional discrete domains. In the first part of the thesis, Chapters 2 & 3, we define a convex function on a special time scale T where all the time points are not uniformly distributed on a time line. With the use of the substitution rules of integration we prove the Hermite-Hadamard inequality for convex functions defined on T. In the fourth chapter, we introduce fractional order Hermite-Hadamard inequality and characterize convexity in terms of this inequality. In the fifth chapter, we discuss convexity on n{dimensional discrete time scales T = T1 × T2 × ... × Tn where Ti ⊂ R , i = 1; 2,…,n are discrete time scales which are not necessarily periodic. We introduce the discrete analogues of the fundamental concepts of real convex optimization such as convexity of a function, subgradients, and the Karush-Kuhn-Tucker conditions. We close this thesis by two remarks for the future direction of the research in this area.

The main purpose of this thesis is to define the stability of a system of linear difference equations of the form, ∇y(t) = Ay(t), and to analyze the stability theory for such a system using the eigenvalues of the corresponding matrix A in nabla discrete calculus and nabla fractional discrete calculus. Discrete exponential functions and the Putzer algorithms are studied to examine the stability theorem. This thesis consists of five chapters and is organized as follows. In the first chapter, the Gamma function and its properties are studied. Additionally, basic definitions, properties and some main theorem of discrete calculus are discussed by using particular example. In the second chapter, we focus on solving the linear difference equations by using the undetermined coefficient method and the variation of constants formula. Moreover, we establish the matrix exponential function which is the solution of the initial value problems (IVP) by the Putzer algorithm.

This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.

Hamming-distance graphs arise naturally in the study of error-correcting codes and have been utilized by several authors to provide new proofs for (and in some cases improve) known bounds on the size of block codes. We study various standard graph properties of the Hamming-distance graphs with special emphasis placed on the chromatic number. A notion of robustness is defined for colorings of these graphs based on the tolerance of swapping colors along an edge without destroying the properness of the coloring, and a complete characterization of the maximally robust colorings is given for certain parameters. Additionally, explorations are made into subgraph structures whose identification may be useful in determining the chromatic number.

The game of Subset Take-Away is an impartial combinatorial game posed by David Gale in 1974. The game can be played on various discrete structures, including but not limited to graphs, hypergraphs, polygonal complexes, and partially ordered sets. While a universal winning strategy has yet to be found, results have been found in certain cases. In 2003 R. Riehemann focused on Subset Take-Away on bipartite graphs and produced a complete game analysis by studying nim-values. In this work, we extend the notion of Take-Away on a bipartite graph to Take-Away on particular hypergraphs, namely oddly-uniform hypergraphs and evenly-uniform hypergraphs whose vertices satisfy a particular coloring condition. On both structures we provide a complete game analysis via nim-values. From here, we consider different discrete structures and slight variations of the rules for Take-Away to produce some interesting results. Under certain conditions, polygonal complexes exhibit a sequence of nim-values which are unbounded but have a well-behaved pattern. Under other conditions, the nim-value of a polygonal complex is bounded and predictable based on information about the complex itself. We introduce a Take-Away variant which we call “Take-As-Much-As-You-Want”, and we show that, again, nim-values can grow without bound, but fortunately they are very easily described for a given graph based on the total number of vertices and edges of the graph. Finally we consider Take-Away on a specific type of partially ordered set which we call a rank-complete poset. We have results, again via an analysis of nim-values, for Take-Away on a rank-complete poset for both ordinary play as well as misère play.

Knot nullification is an unknotting operation performed on knots and links that can be used to model DNA recombination moves of circular DNA molecules in the laboratory. Thus nullification is a biologically relevant operation that should be studied. Nullification moves can be naturally grouped into two classes: coherent nullification, which preserves the orientation of the knot, and incoherent nullification, which changes the orientation of the knot. We define the coherent (incoherent) nullification number of a knot or link as the minimal number of coherent (incoherent) nullification moves needed to unknot any knot or link. This thesis concentrates on the study of such nullification numbers. In more detail, coherent nullification moves have already been studied at quite some length. This is because the preservation of the previous orientation of the knot, or link, makes the coherent operation easier to study. In particular, a complete solution of coherent nullification numbers has been obtained for the torus knot family, (the solution of the torus link family is still an open question). In this thesis, we concentrate on incoherent nullification numbers, and place an emphasis on calculating the incoherent nullification number for the torus knot and link family. Unfortunately, we were unable to compute the exact incoherent nullification numbers for most torus knots. Instead, our main results are upper and lower bounds on the incoherent nullification number of torus knots and links. In addition we conjecture what the actual incoherent nullification number of a torus knot will be.

At the intersection of algebraic geometry, number theory, and combinatorics, an interesting problem is counting points on an algebraic curve over a finite field. When specialized to the case of elliptic curves, this question leads to a surprising connection with a particular family of graphs. In this document, we present some of the underlying theory and then summarize recent results concerning the aforementioned relationship between elliptic curves and graphs. A few results are additionally further elucidated by theory that was omitted in their original presentation.

In 1961, P. W. Kasteleyn provided a baffling-looking solution to an apparently simple tiling problem: how many ways are there to tile a rectangular region with dominos? We examine his proof, simplifying and clarifying it into this nearly self-contained work.

Even if an encryption algorithm is mathematically strong, humans inevitably make for a weak link in most security protocols. A sufficiently threatening adversary will typically be able to force people to reveal their encrypted data. Methods of deniable encryption seek to mend this vulnerability by allowing for decryption to alternate data which is plausible but not sensitive. Existing schemes which allow for deniable encryption are best suited for use by parties who wish to communicate with one another. They are not, however, ideal for personal data storage. This paper develops a plausibly-deniable encryption system for use with personal data storage, such as hard drive encryption. This is accomplished by narrowing the encryption algorithm’s message space, allowing different plausible plaintexts to correspond to one another under different encryption keys.