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Articles 1 - 18 of 18
Full-Text Articles in Physical Sciences and Mathematics
Eigenvalue Algorithm For Hausdorff Dimension On Complex Kleinian Groups, Jacob Linden, Xuqing Wu
Eigenvalue Algorithm For Hausdorff Dimension On Complex Kleinian Groups, Jacob Linden, Xuqing Wu
Rose-Hulman Undergraduate Mathematics Journal
In this manuscript, we present computational results approximating the Hausdorff dimension for the limit sets of complex Kleinian groups. We apply McMullen's eigenvalue algorithm \cite{mcmullen} in symmetric and non-symmetric examples of complex Kleinian groups, arising in both real and complex hyperbolic space. Numerical results are compared with asymptotic estimates in each case. Python code used to obtain all results and figures can be found at \url{https://github.com/WXML-HausDim/WXML-project}, all of which took only minutes to run on a personal computer.
Further Generalizations Of Happy Numbers, E. Simonton Williams
Further Generalizations Of Happy Numbers, E. Simonton Williams
Rose-Hulman Undergraduate Mathematics Journal
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 construct a similar function expanding the definition of happy numbers to negative integers. Working with this function, we prove a range of results paralleling those already proven for traditional and generalized happy numbers. Finally, …
Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine
Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine
Rose-Hulman Undergraduate Mathematics Journal
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.
Elliptic Triangles Which Are Congruent To Their Polar Triangles, Jarrad S. Epkey, Morgan Nissen, Noelle K. Kaminski, Kelsey R. Hall, Nicholas Grabill
Elliptic Triangles Which Are Congruent To Their Polar Triangles, Jarrad S. Epkey, Morgan Nissen, Noelle K. Kaminski, Kelsey R. Hall, Nicholas Grabill
Rose-Hulman Undergraduate Mathematics Journal
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 of either triangle onto the given triangle. We describe how to construct such triangles and a dozen Wallace-Simson lines.
Structure Of A Total Independent Set, Lewis Stanton
Structure Of A Total Independent Set, Lewis Stanton
Rose-Hulman Undergraduate Mathematics Journal
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 $\alpha''(G)$ respectively. This paper establishes an upper bound for $\alpha''(G)$ in terms of $\alpha(G)$, $\alpha'(G)$ and $n$. We also describe the possible structures for a total independent set containing a given number of elements.
K-Distinct Lattice Paths, Eric J. Yager, Marcus Engstrom
K-Distinct Lattice Paths, Eric J. Yager, Marcus Engstrom
Rose-Hulman Undergraduate Mathematics Journal
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 also refine an upper bound given by Brewer et. al. We illustrate that brute force is an inefficient method to determine the maximum order, as it has time complexity O(nk).
Utilizing Graph Thickness Heuristics On The Earth-Moon Problem, Robert C. Weaver
Utilizing Graph Thickness Heuristics On The Earth-Moon Problem, Robert C. Weaver
Rose-Hulman Undergraduate Mathematics Journal
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, which may be solutions if one can find two planar subgraphs that partition all of the graph’s edges.
The Mean Sum Of Squared Linking Numbers Of Random Piecewise-Linear Embeddings Of $K_N$, Yasmin Aguillon, Xingyu Cheng, Spencer Eddins, Pedro Morales
The Mean Sum Of Squared Linking Numbers Of Random Piecewise-Linear Embeddings Of $K_N$, Yasmin Aguillon, Xingyu Cheng, Spencer Eddins, Pedro Morales
Rose-Hulman Undergraduate Mathematics Journal
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} extended these techniques to find the mean sum of squared linking numbers for random linear embeddings of complete graphs $K_n$ and found it to have order $\Theta(n(n!))$. We further these ideas by inspecting random piecewise-linear embeddings of complete graphs and give introductory-level summaries of the ideas …
On Solutions Of First Order Pde With Two-Dimensional Dirac Delta Forcing Terms, Ian Robinson
On Solutions Of First Order Pde With Two-Dimensional Dirac Delta Forcing Terms, Ian Robinson
Rose-Hulman Undergraduate Mathematics Journal
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 solutions, a derivation of solutions of the PDE coupled with a boundary and initial condition, as well as a few worked examples are provided.
The Existence Of Solutions To A System Of Nonhomogeneous Difference Equations, Stephanie Walker
The Existence Of Solutions To A System Of Nonhomogeneous Difference Equations, Stephanie Walker
Rose-Hulman Undergraduate Mathematics Journal
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 sufficient to apply the Guo-KrasnoselâA˘Zskii Fixed Point Theorem. The majority of the work involves developing the constraints ´ needed to utilized this fixed point theorem. The theorem is then applied three times, guaranteeing the existence of at least three distinct solutions. Thus, …
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem, Desmond Weisenberg
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem, Desmond Weisenberg
Rose-Hulman Undergraduate Mathematics Journal
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 of the ways a prime factor could appear in one. We also suggest a separate possible computational approach based on the prime signature of the center entry of the square.
Motion Planning Algorithm In A Y-Graph, David Baldi
Motion Planning Algorithm In A Y-Graph, David Baldi
Rose-Hulman Undergraduate Mathematics Journal
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 initial configuration to any final one without collisions. Using techniques from topological robotics, we calculate the topological complexity of two robots moving on a Y-track and exhibit an optimal algorithm realizing this exact number of instructions given by the topological complexity.
Constructing Spanning Sets Of Affine Algebraic Curvature Tensors, Stephen J. Kelly
Constructing Spanning Sets Of Affine Algebraic Curvature Tensors, Stephen J. Kelly
Rose-Hulman Undergraduate Mathematics Journal
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 canonical algebraic curvature tensors.
A Note On The Involutive Concordance Invariants For Certain (1,1)-Knots, Anna Antal, Sarah Pritchard
A Note On The Involutive Concordance Invariants For Certain (1,1)-Knots, Anna Antal, Sarah Pritchard
Rose-Hulman Undergraduate Mathematics Journal
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 chain complex CFK(K), which improves on classical invariants of the knot. Involutive Heegaard Floer homology is a variant theory introduced in 2015 by K. Hendricks and C. Manolescu which additionally considers a chain …
Optimal Monohedral Tilings Of Hyperbolic Surfaces, Leonardo Digiosia, Jahangir Habib, Jack Hirsch, Lea Kenigsberg, Kevin Li, Dylanger Pittman, Jackson Petty, Christopher Xue, Weitao Zhu
Optimal Monohedral Tilings Of Hyperbolic Surfaces, Leonardo Digiosia, Jahangir Habib, Jack Hirsch, Lea Kenigsberg, Kevin Li, Dylanger Pittman, Jackson Petty, Christopher Xue, Weitao Zhu
Rose-Hulman Undergraduate Mathematics Journal
The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric" problem differs for every given area, as solutions do not scale. Cox conjectured that a regular k-gonal tile with 120-degree angles is isoperimetric. For area π/3, the regular heptagon has 120-degree angles and therefore tiles many hyperbolic surfaces. For other areas, we show the existence of many tiles but provide no conjectured optima. On closed hyperbolic surfaces, we verify via a reduction argument using cutting and pasting transformations and convex hulls that the regular 7-gon is the optimal n-gonal tile of …
The Determining Number And Cost Of 2-Distinguishing Of Select Kneser Graphs, James E. Garrison
The Determining Number And Cost Of 2-Distinguishing Of Select Kneser Graphs, James E. Garrison
Rose-Hulman Undergraduate Mathematics Journal
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 also utilizes \emph{determining sets} of $G, $ sets of vertices $S \subseteq G$ such that every automorphism of $G$ is uniquely determined by its action on $S.$ The \emph{determining number} of a graph is the size of a smallest determining set. We investigate …
Iterated Jump Graphs, Fran Herr, Legrand Jones Ii
Iterated Jump Graphs, Fran Herr, Legrand Jones Ii
Rose-Hulman Undergraduate Mathematics Journal
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 Chartrand et al. who showed that a handful of jump graph sequences terminate and two sequences converge. We extend these results by showing that there are no non-trivial repeating sequences of jump …
The Chromatic Index Of Ring Graphs, Lilian Shaffer
The Chromatic Index Of Ring Graphs, Lilian Shaffer
Rose-Hulman Undergraduate Mathematics Journal
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 χ′(G) on every graph: maximum degree, denoted ∆(G), and density, denoted ω(G). Combining these two lower bounds, we know that every graph’s chromatic index must be at least ∆(G) or …