This paper will discuss the analogues between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle by utilizing mathematical proving techniques like partial sums, committees, telescoping, mathematical induction and applying George Polya's perspective. The topics presented in this paper will show that Pascal's triangle and Leibniz's triangle both have hockey stick type patterns, patterns of sums within shapes, and have the natural numbers, triangular numbers, tetrahedral numbers, and pentatope numbers hidden within. In addition, this paper will show how Pascal's Arithmetic Triangle can be used to construct Leibniz's Harmonic Triangle and show how both triangles relate to combinatorics and arithmetic through the coefficients of the binomial expansion. Furthermore, combinatorics plays an increasingly important role in mathematics, starting when students enter high school and continuing on into their college years. Students become familiar with the traditional arguments based on classical arithmetic and algebra, however, methods of combinatorics can vary greatly. In combinatorics, perhaps the most important concept revolves around the coefficients of the binomial expansion, studying and proving their properties, and conveying a greater insight into mathematics.

Let d > 0 be a fixed integer and let A ⊆ ℝd be a collection of n ≥ d + 2 points which we lift into ℝd+1. Further let k be an integer satisfying 0 ≤ k ≤ n-(d+2) and assign to each k-subset of the points of A a (regular) triangulation obtained by deleting the specified k-subset and projecting down the lower hull of the convex hull of the resulting lifting. Next, for each triangulation we form the characteristic vector defined by Gelfand, Kapranov, and Zelevinsky by assigning to each vertex the sum of the volumes of all adjacent simplices. We then form a vector for the lifting, which we call the k-compound GKZ-vector, by summing all the characteristic vectors. Lastly, we construct a polytope Σk(A) ⊆ ℝ|A| by taking the convex hull of all obtainable k-compound GKZ-vectors by various liftings of A, and note that $\Sigma_0(\A)$ is the well-studied secondary polytope corresponding to A. We will see that by varying k, we obtain a family of polytopes with interesting properties relating to Minkowski sums, Gale transforms, and Lawrence constructions, with the member of the family with maximal k corresponding to a zonotope studied by Billera, Fillamen, and Sturmfels. We will also discuss the case k = d = 1, in which we can provide a combinatorial description of the vertices allowing us to better understand the graph of the polytope and to obtain formulas for the numbers of vertices and edges present.

Consider a probability distribution on the permutations of n elements. If the probability of each permutation is proportional to θK, where K is the number of cycles in the permutation, then we say that the distribution generates a θ-biased random permutation. A random permutation is a special θ-biased random permutation with θ = 1. The mth moment of the rth longest cycle of a random permutation is Θ(nm), regardless of r and θ. The joint moments are derived, and it is shown that the longest cycles of a permutation can either be positively or negatively correlated, depending on θ. The mth moments of the rth shortest cycle of a random permutation is Θ(nm−θ/(ln n)r−1) when θ < m, Θ((ln n)r) when θ = m, and Θ(1) when θ > m. The exponent of cycle lengths at the 100qth percentile goes to q with zero variance. The exponent of the expected cycle lengths at the 100qth percentile is at least q due to the Jensen’s inequality, and the exact value is derived.

The main purpose of this thesis to examine the controllability and observability of the linear discrete fractional systems. First we introduce the problem and continue with the review of some basic definitions and concepts of fractional calculus which are widely used to develop the theory of this subject. In Chapter 3, we give the unique solution of the fractional difference equation involving the Riemann-Liouville operator of real order between zero and one. Additionally we study the sequential fractional difference equations and describe the way to obtain the state-space repre- sentation of the sequential fractional difference equations. In Chapter 4, we study the controllability and observability of time-invariant linear nabla fractional systems.We investigate the time-variant case in Chapter 5 and we define the state transition matrix in fractional calculus. In the last chapter, the results are summarized and directions for future work are stated.

An Italian dominating function on a graph $G = (V,E)$ is a function such that $f : V \to \{0,1,2\}$, and for each vertex $v \in V$ for which $f(v) = 0$, we have $\sum_{u\in N(v)}f(u) \geq 2$. The weight of an Italian dominating function is $f(V) = \sum_{v\in V(G)}f(v)$. The minimum weight of all such functions on a graph $G$ is called the Italian domination number of $G$. In this thesis, we will consider Italian domination in various types of products of a graph $G$ with the complete graph $K_2$. We will find the value of the Italian domination number for ladders, specific families of prisms, mobius ladders and related products including categorical products $G\times K_2$ and lexicographic products $G\cdot K_2$. Finally, we will conclude with open problems.

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In this thesis, we introduce Roman domination cover rubbling as an extension of domination cover rubbling. We define a parameter on a graph $G$ called the \textit{Roman domination cover rubbling number}, denoted $\rho_{R}(G)$, as the smallest number of pebbles, so that from any initial configuration of those pebbles on $G$, it is possible to obtain a configuration which is Roman dominating after some sequence of pebbling and rubbling moves. We begin by characterizing graphs $G$ having small $\rho_{R}(G)$ value. Among other things, we also obtain the Roman domination cover rubbling number for paths and give an upper bound for the Roman domination cover rubbling number of a tree.

This study is an investigation of the performance of a discrete latent state model devised by Paulson (1982) to account for signed-number arithmetic test data gathered by Birenbaum and Tatsuoka (1980). One hundred twenty nine students took a test which consists of sixteen item types with four parallel arithmetic items of each type. The present study utilizes the five addition item types of four items each; hence, there are four parallel subtests. Responses to the addition items can be analyzed in terms of two components: the siqn component (is the sign correct?), and the absolute value component (is the size of the answer correct?). Paulson's model describes how students perform on the two components separately and how the component responses are related. This study examines the parallelism of the four subtests, in terms of equality of means, standard deviations, and correlations between all pairs of subtests. Decision consistency between subtests is another useful indicator of measurement reliability, particularly for tests of concept mastery. The model implies that the consistency between any two pairs of subtests should be equal; this implication is tested. The specific numerical values predicted by the model for the means, standard deviations, correlations, and decision consistency indices are tested against the corresponding observed statistics. All the analyses described so far are done separately for both the sign and the absolute value components of the responses. A method to synthesize overall correct response from estimated parameter values of two components is derived and tested against observed values. The results are that "parallel" items within item types are not all parallel and finer characterization would be needed to describe the items completely. However, the deviations from strict parallelism are slight. Paulson's model demonstrates good predictive ability; on both components and on the overall responses. Most of the deviations from the prediction can be attributed to not strictly parallel subtests and estimated parameter values not being the best possible estimates.

Let S ⊆ V be an arbitrary subset of vertices of a graph G = (V,E). The differential ∂(S) equals the difference between the cardinality of the set of vertices not in S but adjacent to vertices in S, and the cardinality of the set S. The differential of a graph G equals the maximum differential of any subset S of V . A set S is called a zero set if ∂(S) = 0. In this thesis we introduce the study of zero sets in graphs. We give proofs of the existence of zero sets in various kinds of graphs such as even order graphs, bipartite graphs, and graphs of maximum degree 3. We also give proofs regarding the existence of graphs which contain no zero sets and the construction of zero-free graphs from graphs which contain zero sets.

The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. One view of this problem is in edge-colorings of complete graphs. For any graphs G, H1, ..., Hk, we write G → (H1, ..., Hk), or G → (H)k when H1 = ··· = Hk = H, if every k-edge-coloring of G contains a monochromatic Hi in color i for some i ∈ {1,...,k}. The Ramsey number rk(H1, ..., Hk) is the minimum integer n such that Kn → (H1, ..., Hk), where Kn is the complete graph on n vertices. Computing rk(H1, ..., Hk) is a notoriously difficult problem in combinatorics. A weakening of this problem is to restrict ourselves to Gallai colorings, that is, edge-colorings with no rainbow triangles. From this we define the Gallai-Ramsey number grk(K3,G) as the minimum integer n such that either Kn contains a rainbow triangle, or Kn → (G)k . In this thesis, we determine the Gallai-Ramsey numbers for C7 with multiple colors. We believe the method we developed can be applied to find grk(K3, C2n+1) for any integer n ≥ 2, where C2n+1 denotes a cycle on 2n + 1 vertices.