A positive integer n is defined to be happy if iteration of the function taking the sum of the squares of the digits of n eventually reaches 1. In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple and establish further results extending the known structure of happy numbers to higher powers. Then we...
We find a formula for the probability that the product of n positive integers, chosen at random, is divisible by some integer d. We do this via an inductive application of the Chinese Remainder Theorem, generating functions, and several other combinatorial arguments. Additionally, we apply this formula to find a unique, but slow, probabilistic primality test.
We prove that an elliptic triangle is congruent to its polar triangle if and only if six specific Wallace-Simson lines of the triangle are concurrent. (If a point projected onto a triangle has the three feet of its projections collinear, that line is called a Wallace-Simson line.) These six lines would be concurrent at the orthocenter. The six lines come from projecting a vertex...
Let $G$ be a simple, connected and finite graph with order $n$. Denote the independence number, edge independence number and total independence number by $\alpha(G), \alpha'(G)$ and $\alpha
We study one specific version of the contact process on a graph. Here, we allow multiple infections carried by the nodes and include a probability of removing nodes in a graph. The removal probability is purely determined by the number of infections the node carries at the moment when it gets another infection. In this paper, we show that on any finite graph, any positive value...
Lattice paths can be used to model scheduling and routing problems, and, therefore, identifying maximum sets of k-distinct paths is of general interest. We extend the work previously done by Gillman et. al. to determine the order of a maximum set of k-distinct lattice paths. In particular, we disprove a conjecture by Gillman that a greedy algorithm gives this maximum order and...
This paper utilizes heuristic algorithms for determining graph thickness in order to attempt to find a 10-chromatic thickness-2 graph. Doing so would eliminate 9 colors as a potential solution to the Earth-moon Problem. An empirical analysis of the algorithms made by the author are provided. Additionally, the paper lists various graphs that may or nearly have a thickness of 2...
DNA and other polymer chains in confined spaces behave like closed loops. Arsuaga et al. \cite{AB} introduced the uniform random polygon model in order to better understand such loops in confined spaces using probabilistic and knot theoretical techniques, giving some classification on the mean squared linking number of such loops. Flapan and Kozai \cite{flapan2016linking...
We provide solutions of a first order, linear partial differential equation of two variables where the nonhomogeneous term is a two-dimensional Dirac delta function. Our results are achieved by applying the unilateral Laplace Transform, solving the subsequently transformed PDE, and reverting back to the original space-time domain. A discussion of existence and uniqueness of...
This article will demonstrate a process using Fixed Point Theory to determine the existence of multiple positive solutions for a type of system of nonhomogeneous even ordered boundary value problems on a discrete domain. We first reconstruct the problem by transforming the system so that it satisfies homogeneous boundary conditions. We then create a cone and an operator...
A complex-valued random variable Z is rotationally invariant if the moments of Z are the same as the moments of W=e^{i*theta}Z. In the first part of the article, we characterize such random variables, in terms of "vanishing unbalanced moments,
A magic square is a square grid of numbers where each row, column, and long diagonal has the same sum (called the magic sum). An open problem popularized by Martin Gardner asks whether there exists a 3×3 magic square of distinct positive square numbers. In this paper, we expand on existing results about the prime factors of elements of such a square, and then provide a full list...
We present an explicit algorithm for two robots to move autonomously and without collisions on a track shaped like the letter Y. Configuration spaces are of practical relevance in designing safe control schemes for automated guided vehicles. The topological complexity of a configuration space is the minimal number of continuous instructions required to move robots between any...
In this paper, we construct two spanning sets for the affine algebraic curvature tensors. We then prove that every 2-dimensional affine algebraic curvature tensor can be represented by a single element from either of the two spanning sets. This paper provides a means to study affine algebraic curvature tensors in a geometric and algebraic manner similar to previous studies of...
A knot K is a smooth embedding of the circle into the three-dimensional sphere; two knots are said to be concordant if they form the boundary of an annulus properly embedded into the product of the three-sphere with an interval. Heegaard Floer knot homology is an invariant of knots introduced by P. Ozsváth and Z. Szabó in the early 2000's which associates to a knot a filtered...
The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric
The group synchronization problem is to estimate unknown group elements at the vertices of a graph when given a set of possibly noisy observations of group differences at the edges. We consider the group synchronization problem on finite graphs with size tending to infinity, and we focus on the question of whether the true edge differences can be exactly recovered from the...
A graph $G$ is said to be \emph{d-distinguishable} if there exists a not-necessarily proper coloring with $d$ colors such that only the trivial automorphism preserves the color classes. For a 2-distinguishing labeling, the \emph{ cost of $2$-distinguishing}, denoted $\rho(G),$ is defined as the minimum size of a color class over all $2$-distinguishing colorings of $G$. Our work...
The jump graph J(G) of a simple graph G has vertices which represent edges in G where two vertices in J(G) are adjacent if and only if the corresponding edges in G do not share an endpoint. In this paper, we examine sequences of graphs generated by iterating the jump graph operation and characterize the behavior of this sequence for all initial graphs. We build on work by...
The goal of graph edge coloring is to color a graph G with as few colors as possible such that each edge receives a color and that adjacent edges, that is, different edges incident to a common vertex, receive different colors. The chromatic index, denoted χ′(G), is the minimum number of colors required for such a coloring to be possible. There are two important lower bounds for...
Young tableaux are combinatorial objects related to the partitions of an integer and have various applications in representation theory. They are particularly useful in the study of the fibers arising from the Springer resolution. In recent work of Graham-Precup-Russell, an association has been made between a given row-strict tableau and three disjoint subsets of {1,2,...,n...
In this paper we investigate actions of SAut(Fn), the unique index 2 subgroup of Aut(Fn), on small sets, improving upon results by Baumeister--Kielak--Pierro for several small values of n. Using a computational approach for n ⩾ 5, we show that every action of SAut(Fn) on a set containing fewer than 20 elements is trivial.
The probability that two elements commute in a non-Abelian finite group is at most 5 8 . We prove several generalizations of this result for dihedral groups. In particular, we give specific values for the probability that a product of an arbitrary number of dihedral group elements is equal to its reverse, and also for the probability that a product of three elements is equal to a...
We study a special class of generalized continuous fractions, both in real and complex settings, and show that in many cases, the set of numbers that can be represented by a continued fraction for that class form a Cantor set. Specifically, we study generalized continued fractions with a fixed absolute value and a variable coefficient sign. We ask the same question in the complex...
The spatially implicit Tilman-Levins ODE model helps to explain why so many plant species can coexist in grassland communities. This now-classic modeling framework assumes a trade-off between colonization and competition traits and predicts that habitat destruction can lead to long transient declines called ``extinction debts.