Rose-Hulman Undergraduate Mathematics JournalCopyright (c) 2023 Rose-Hulman Institute of Technology All rights reserved.
https://scholar.rose-hulman.edu/rhumj
Recent documents in Rose-Hulman Undergraduate Mathematics Journalen-usWed, 29 Nov 2023 01:46:13 PST3600Eigenvalue Algorithm for Hausdorff Dimension on Complex Kleinian Groups
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/12
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/12Mon, 27 Nov 2023 13:55:44 PST
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.
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Jacob Linden et al.Further Generalizations of Happy Numbers
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/11
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/11Mon, 30 Oct 2023 09:36:55 PDT
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, we consider a variety of special cases, in which the existence of certain fixed points and cycles of infinite families of generalized happy functions can be proven.
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E. Simonton WilliamsDivisibility Probabilities for Products of Randomly Chosen Integers
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/10
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/10Fri, 27 Oct 2023 10:03:00 PDT
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.
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Noah Y. FineElliptic triangles which are congruent to their polar triangles
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/9
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/9Fri, 27 Oct 2023 09:36:46 PDT
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.
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Jarrad S. Epkey et al.Structure of a Total Independent Set
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/8
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/8Thu, 26 Oct 2023 08:21:05 PDT
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.
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Lewis StantonA Model for the Multi-Virus Contact Process
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/7
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/7Fri, 20 Oct 2023 15:20:57 PDT
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 of infection rate $\lambda$ will result in the death of the process almost surely. In the case of $d$-regular infinite trees, We also give a lower bound on the infection rate in order for the process to survive, and an upper bound for the process to die out.
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Xu Huangk-Distinct Lattice Paths
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/6
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/6Fri, 29 Sep 2023 09:01:26 PDT
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(n^{k}).
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Eric J. Yager et al.Utilizing graph thickness heuristics on the Earth-moon Problem
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/5
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/5Thu, 28 Sep 2023 10:31:20 PDT
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.
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Robert C. WeaverNumber of Regions Created by Random Chords in the Circle
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/4
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/4Thu, 07 Sep 2023 09:56:43 PDT
In this paper we discuss the number of regions in a unit circle after drawing n i.i.d. random chords in the circle according to a particular family of distribution. We find that as n goes to infinity, the distribution of the number of regions, properly shifted and scaled, converges to the standard normal distribution and the error can be bounded by Stein's method for proving Central Limit Theorem.
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Shi FengThe Mean Sum of Squared Linking Numbers of Random Piecewise-Linear Embeddings of $K_n$
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/3
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/3Thu, 07 Sep 2023 08:35:48 PDT
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 throughout. In particular, we give a model of random piecewise-linear embeddings of complete graphs where the number of line segments between vertices is given by a random variable. We find further that in our model of the random piecewise-linear embeddings, the order of the expected sum of squared linking numbers is still $\Theta(n (n!))$.
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Yasmin Aguillon et al.On Solutions of First Order PDE with Two-Dimensional Dirac Delta Forcing Terms
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/2
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/2Tue, 11 Jul 2023 14:51:05 PDT
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.
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Ian RobinsonThe Existence of Solutions to a System of Nonhomogeneous Difference Equations
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/1
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/1Mon, 03 Jul 2023 15:10:58 PDT
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, solutions to this class of boundary value problems exist and are not unique.
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Stephanie WalkerA Characterization of Complex-Valued Random Variables With Rotationally-Invariant Moments
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/8
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/8Wed, 14 Jun 2023 14:37:10 PDT
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," moment and cumulant generating functions, and polar decomposition. In the second part, we consider random variables whose moments are not necessarily finite, but which have a density. In this setting, we prove two characterizations that are equivalent to rotational invariance, one involving polar decomposition, and the other involving entropy. If a random variable has both a density and moments which determine it, all of these characterizations are equivalent.
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Michael L. MaielloSome Thoughts on The 3 × 3 Magic Square of Squares Problem
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/7
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/7Fri, 09 Jun 2023 13:35:58 PDT
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.
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Desmond WeisenbergMotion Planning Algorithm in a Y-Graph
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/6
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/6Tue, 23 May 2023 15:30:50 PDT
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.
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David BaldiConstructing Spanning Sets of Affine Algebraic Curvature Tensors
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/5
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/5Fri, 05 May 2023 13:46:42 PDT
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.
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Stephen J. KellyA Note On The Involutive Concordance Invariants For Certain (1,1)-Knots
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/4
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/4Fri, 05 May 2023 13:46:33 PDT
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 map iota on CFK(K) induced by a conjugation operation, and extracts from this data two new numerical invariants of knot concordance. These new invariants are especially interesting because, unlike many other concordance invariants from Heegaard Floer homology, they do not necessarily vanish on knots of finite order in the group of concordance classes of knots. The chain map iota is in general difficult to compute, and computations have been carried out for relatively few knots. We give a complete computation of iota for 10 and 11-crossing knots satisfying a certain simplicity condition, called the (1,1)-knots.
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Anna Antal et al.Optimal Monohedral Tilings of Hyperbolic Surfaces
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/3
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/3Thu, 02 Mar 2023 06:20:08 PST
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 area π/3 for 3≤n≤10. However, for n>10, it is difficult to rule out non-convex n-gons that tile irregularly.
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Leonardo DiGiosia et al.Strong Recovery In Group Synchronization
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/2
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/2Thu, 02 Mar 2023 06:20:00 PST
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 observations (i.e., strong recovery). We prove two main results, one positive and one negative. In the positive direction, we prove that for a sequence of synchronization problems containing the complete digraph along with a relatively well behaved prior distribution and observation kernel, with high probability we can recover the correct edge labeling. Our negative result provides conditions on a sequence of sparse graphs under which it is impossible to recover the correct edge labeling with high probability.
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Bradley StichThe Determining Number and Cost of 2-Distinguishing of Select Kneser Graphs
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/1
https://scholar.rose-hulman.edu/rhumj/vol24/iss1/1Thu, 02 Mar 2023 06:19:53 PST
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 the cost of $2$-distinguishing families of Kneser graphs $K_{n:k}$ by using optimal determining sets of those families. We show the determining number of $\kntwo$ is equal to $\left\lceil{ \frac{2n-2}{3}}\right\rceil$and give linear bounds on $\rho(\kntwo)$ when $n$ is sufficiently sized.
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James E. Garrison