Showing papers in "Rose–Hulman Undergraduate Mathematics Journal in 2013"

Existence of the limit at infinity for a function that is integrable on the half line

Abstract: It is well known that for a function that is integrable on [0,∞), its limit at infinity may not exist. First we illustrated this statement with an example. Then, we present conditions that guarantee the existence of the limit in the following two cases: When the integrable function is non-negative, if the first, second, third, or fourth, derivative is bounded in a neighborhood of each local maximum of f , then the limit exists. Without the non-negative constraint, if an integrable function has a bounded derivative on the entire interval [0,∞), then the limit exists. Acknowledgements: I would like to thank the anonymous referee for his/her valuable comments that helped me improve this article. Also I want to thank my teachers at the San Marcos High School and at the Honors Math Camp at Texas State University for providing me with the knowledge and discipline to approach these problems. In addition, I want to thank my family their support and encouragement. Page 2 RHIT Undergrad. Math. J., Vol. 14, No. 1

Klein Links and Braids

Abstract: We introduce the construction of Klein links through an alteration to the orientation on the rectangular representation of a torus knot. We relate the resulting Klein links to their corresponding braid representations, and use these representations to understand the relationship between Klein links and torus knots as well as to prove relationships between several different Klein links. Acknowledgements: We want to thank our advisors Drs. Jennifer Bowen and John Ramsay for their support and guidance. Additionally, we wish to thank HHMI and Sophomore Research at the College of Wooster for funding our project as part of the AMRE program. Page 72 RHIT Undergrad. Math. J., Vol. 14, No. 1

The Three-Variable Bracket Polynomial for Reduced, Alternating Links

Abstract: We first show that the three-variable bracket polynomial is an invariant for reduced, alternating links. We then try to find what the polynomial reveals about knots. We find that the polynomial gives the crossing number, a test for chirality, and in some cases, the twist number of a knot. The extreme degrees of d are also studied. Acknowledgements: I would like to thank Dr. Rollie Trapp for advising me on this project, providing suggestions for what to investigate, and helping to complete several proofs. I would also like to thank Dr. Corey Dunn for his advice. This research was jointly funded by NSF grant DMS-1156608, and by California State University, San Bernardino. Page 98 RHIT Undergrad. Math. J., Vol. 14, no. 2

Directed Graphs of Commutative Rings

Abstract: The directed graph of a commutative ring is a graph representation of its additive and multiplicative structure. Using the mapping (a, b) → (a + b, a · b) one can create a directed graph for every finite, commutative ring. We examine the properties of directed graphs of commutative rings, with emphasis on the information the graph gives about the ring. Acknowledgements: We would like to acknowledge the University of St. Thomas for making this research possible and Dr. Michael Axtell for his guidance throughout. Page 168 RHIT Undergrad. Math. J., Vol. 14, no. 2

Rotationally Symmetric Rose Links

Abstract: This paper is an introduction to rose links and some of their properties. We used a series of invariants to distinguish some rose links that are rotationally symmetric. We were able to distinguish all 3-component rose links and narrow the bounds on possible distinct 4 and 5-component rose links to between 2 and 8, and 2 and 16, respectively. An algorithm for drawing rose links and a table of rose links with up to five components are included. Acknowledgements: The initial cursory observations on rose links were made during the 2011 Dr. Albert H. and Greta A. Bryan Summer Research Program at Simpson College in conjunction with fellow students Michael Comer and Jes Toyne, with Dr. William Schellhorn serving as research advisor. The research presented in this paper, including developing the definitions and related concepts about rose links and applying invariants, was conducted as a part of the author’s senior research project at Simpson College, again under the advisement of Dr. Schellhorn. The author would like to thank Dr. Schellhorn for his help and support, especially for his assistance with the definitions involving polygonal diagrams. Page 28 RHIT Undergrad. Math. J., Vol. 14, No. 1 Figure 1: The unknot is the simplest knot.

Subfield-Compatible Polynomials over Finite Fields

Abstract: Polynomial functions over finite fields are important in computer science and electrical engineering in that they present a mathematical representation of arithmetic circuits. This paper establishes necessary and sufficient conditions for polynomial functions with coefficients in a finite field and naturally restricted degrees to be compatible with given subfields. Most importantly, this is done for the case where the domain and codomain fields have differing cardinalities. These conditions, which are presented for polynomial rings in one and several variables, are developed via a universal permutation that depends only on the cardinalities of the given fields. Acknowledgements: The author would like to thank Florian Enescu, whose skill as an instructor is surpassed only by his dedication to the development of his students. The author would also like to thank the anonymous referee for providing excellent guidance toward improving this paper. This project was undertaken as a part of the Research Initiations in Mathematics, Mathematics Education, and Statistics (RIMMES) program at Georgia State University. Page 116 RHIT Undergrad. Math. J., Vol. 14, no. 2

Trees of Irreducible Numerical Semigroups

Abstract: A 2011 paper by Blanco and Rosales describes an algorithm for constructing a directed tree graph of irreducible numerical semigroups of fixed Frobenius numbers. This paper will provide an overview of irreducible numerical semigroups and the directed tree graphs. We will also present new findings and conjectures concerning the structure of these trees. Acknowledgements: We would like to give a special thanks to our mentor, Jeff Rushall, for his help and guidance, and to Ian Douglas for sharing his expertise in programming. Page 58 RHIT Undergrad. Math. J., Vol. 14, No. 1

Proof of Euler's φ (Phi) Function Formula

Abstract: Euler’s φ (phi) Function counts the number of positive integers not exceeding n and relatively prime to n. Traditionally, the proof involves proving the φ function is multiplicative and then proceeding to show how the formula arises from this fact. We ignore this fact, at least directly, and show a practical and sound method to calculate φ. We offer a proof of the closed form formula for this function relying on similar, but subtly different counting techniques. Acknowledgements: This collaboration would not have been possible without Dr. Harold Reiter. His encouragement for this collaborative effort, despite the great distance between the authors, is greatly appreciated. We would like to thank Dr. David Rader and the referees who reviewed the paper for the abundance of useful suggestions and comments. Thank you. Page 92 RHIT Undergrad. Math. J., Vol. 14, no. 2

Bounds for elements of the degree sequence of an unknown vertex set in a balanced bipartite graph

Abstract: Consider the set of all balanced bipartite graphs. Given the degree sequence of one vertex set in one of these graphs, we find bounds for any given position in the degree sequence of the unknown vertex set. Additionally, we establish bounds for the median of the unknown degree sequence, as well as bounds for any given percentile. We discuss the connection between this paper and the High School Prom Theorem. Acknowledgements: Supported in part by the Elmira College Summer Research Program. Thanks to Dr. Charlie Jacobson for his guidance, and to an anonymous referee for his helpful suggestions. PAGE 144 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1

Generic Polynomials for Transitive Permutation Groups of Degree 8 and 9

Abstract: We compute generic polynomials for certain transitive permuta- tion groups of degree 8 and 9, namely SL(2,3), the generalized dihedral group: C2 ⋉ (C3 × C3), and the Iwasawa group of order 16: M16. Rikuna proves the ex- istence of a generic polynomial for SL(2,3) in four parameters in (13); we extend a computation of Grobner in (5) to give an alternative proof of existence for this group's generic polynomial. We establish that the generic dimension and essential dimension of the generalized dihedral group are two. We establish over the rationals that the generic dimension and essential dimension of SL(2,3) and M16 are four.

Optimizing the Allocation of Vaccines in the Presence of Multiple Strains of the Influenza Virus

Abstract: During the annual flu season, multiple strains of the influenza virus are often present within a population. It is a significant challenge for health care administrators to determine the most effective allocation of multiple vaccines to combat the various strains when protecting the public. We employ a mathematical model, a system of differential equations, to find a strategy for vaccinating a population to minimize the number of infected individuals. We consider various strengths of transmission of the disease, availability of vaccine doses, vaccination rates, and other model parameters. This research may lead to more effective health care policies for vaccine administration. Acknowledgements: The authors would like to thank their research advisor, Dr. Alex Capaldi, for his guidance on this project. RHIT Undergrad. Math. J., Vol. 16, No. 1 Page 123

Directed Graphs of Commutative Rings with Identity

Abstract: The directed graph of a ring is a graphical representation of its additive and multiplicative structure. Using the directed edge relationship (a, b) → (a + b, a · b), one can create a directed graph for every ring. This paper focuses on the structure of the sources in directed graphs of commutative rings with identity, with special concentration in the finite and reduced cases. Acknowledgements: We would like to thank the Center for Applied Mathematics at the University of St. Thomas for funding our research. We would also like to thank Dr. Michael Axtell for his help and guidance, as well as Jared Skinner for his mathematica digraph package. Page 86 RHIT Undergrad. Math. J., Vol. 14, No. 1

Entanglement-Assisted Quantum Error-Correcting Codes from Generalized Quadrangles

Abstract: A generalized quadrangle GQ(s, t) is an incidence structure consisting of points and lines in which each line is incident with a fixed number of points, each point is incident with a fixed number of lines, and there is exactly one line connecting any point with a line not incident with the point. Entanglement-assisted quantum error-correcting codes provide a method for correcting data transmission errors in quantum computers. EAQECCs require entangled quantum states, called ebits, and it is desirable to minimize the number of ebits a code uses because ebits are difficult to manufacture. We use a binary incidence matrix N of a generalized quadrangle to create entanglement-assisted quantum error-correcting codes. The rank of NNT gives the number of ebits a code requires. Because incidence matrices of generalized quadrangles are highly structured and reflect the geometric properties of the quadrangles, we can examine the rank of N and NNT and write the parameters of quantum codes in terms of s and t. We identify a class of generalized quadrangles that produce quantum codes that require a low number of ebits, a class that produce quantum codes that require a large number of ebits, and a class that produces quantum codes that are too small to be useful. Acknowledgements: I would like to sincerely thank Dr. David Clark for all of his help as my advisor for this project, and I would like to thank the University of Minnesota Math Center for Educational Programs for funding my trip to present my results at a Pi Mu Epsilon conference. I am also very grateful for the anonymous referee’s feedback. Page 156 RHIT Undergrad. Math. J., Vol. 14, no. 2

The Broncho Tower: A Modification of the Tower of Hanoi Puzzle

Abstract: s from the 2013 Oklahoma Research Day Held at the University of Central Oklahoma 05. Mathematics and Science

Iterated functions and the Cantor set in one dimension

Abstract: In this paper we consider the long-term behavior of points in R under iterations of continuous functions. We show that, given any Cantor set Λ∗ embedded in R, there exists a continuous function F ∗ : R → R such that the points that are bounded under iterations of F ∗ are just those points in Λ∗. In the course of this, we find a striking similarity between the way in which we construct the Cantor middle-thirds set, and the way in which we find the points bounded under iterations of certain continuous functions. Acknowledgements: Thanks to Elizabeth Stanhope for some helpful comments and discussions. Page 80 RHIT Undergrad. Math. J., Vol. 14, no. 2

Category Theory and Galois Theory

Abstract: Galois theory translates questions about fields into questions about groups. The fundamental theorem of Galois theory states that there is a bijection between the intermediate fields of a field extension and the subgroups of the corresponding Galois group. After a basic introduction to category and Galois theory, this project recasts the fundamental theorem of Galois theory using categorical language and illustrates this theorem and the structure it preserves through an example. Acknowledgements: I would like to sincerely thank Professor Thomas Fiore for his continual support, encouragement, and invaluable guidance throughout this entire project. Page 134 RHIT Undergrad. Math. J., Vol. 14, No. 1

Equilateral Dimension of Riemannian Manifolds with Bounded Curvature

Abstract: The equilateral Dimension of a riemannian manifold is the maximum number of distinct equidistant points. In the first half of this paper we will give upper bounds for the equilateral dimension of certain Riemannian Manifolds. In the second half of the paper we will introduce a new metric invariant, called the equilateral length, which measures size of the equilateral dimension. This will then be used in the recognition program in Riemannian geometry, which seeks to identify certain Riemannian manifolds by way of metric invariants such as diameter, extent, or packing radius. Acknowledgements: I would like to thank my research advisor, Dr. Jian Ge for all his help with this paper. Page 102 RHIT Undergrad. Math. J., Vol. 14, No. 1 An equilateral set in a given metric space, X, is a set E such that the distance between any pair of distinct points in E is r, where r is some positive constant. The equilateral dimension, denoted e(X) can thus be defined to be the maximum cardinality of all equilateral sets. It is important to note that for general metric spaces the equilateral dimension need not be finite. A simple example of this is Z, with the discrete metric: d(x, y) = { 1 if x 6= y; 0 if x=y The equilateral dimension of R is simply n + 1. This can be seen by noting that every point of an equilateral set with m members must lie on the intersection of the boundary of m− 1 metric balls. Similiarly, the equilateral dimension of S, with metric induced from its canonical Riemannian metric, is n+ 2. Here we see that while S locally ”looks” like R, its equilateral dimension is nonetheless greater. This simple example illlustrates how geometric invariants of Riemannian manifolds, such as sectional or Ricci curvature, can manifest in the very coarse notion of the equilateral dimension, even if the two manifolds are diffeomorphic. Figure 1: Four equidistant points in S These considerations naturally lead to the question: Can the equilateral dimension of an arbitrary Riemannian manifold be bounded by some constant? The case of R, where e(R) = n+1, demonstrates that this bound will have to depend on the topological dimension of the manifold. The difficulty of answering this question arises because the equilateral dimension is an inherently global notion, and while the distance may be smooth inside the injectivity radius, we cannot always form uniform lower bounds on the injectivity radius. Hence a proof similiar to the case of R will fail for general manifolds, in which the boundary of open balls need not be, for example, smooth submanifolds. The problem of the equilateral dimension of a Riemannian manifold was initally posed by Richard Kusner in [Gu]. Kusner conjectured that the equilateral dimension of a Riemannian manifold of dimension n will always be ≤ n + 2. This paper will give sufficient conditions for this upper bound to hold, as well as other upper bounds which depend on the manifold’s RHIT Undergrad. Math. J., Vol. 14, No. 1 Page 103 dimension. Our main tool will be various comparison theorems, and a key ingredient is the assumption that the manifold is complete. This paper is divided into three parts. First, we will examine the case of nonnegative Ricci curvature using the Bishop-Gromov Volume Comparison Theorem to derive an upper bound of e(M). The fact that the Ricci curvature is nonnegative is necessary for the upper bound to depend solely on the dimension. A weaker result holds for Ricci curvature bounded below, but in this case the upper bound will depend on the distance between the points in the equilateral sets. The second part will be devoted to the case of non-negative sectional curvature, and in this case, Topogonov’s Theorem will be used for the same purpose. These upper bounds will depend solely on the dimension of the manifold, and will depend on the metric structure of S. Last, we will obtain various results which translate Grove’s rigidity results concerning the packing radius of Alexandrov Spaces to equilateral sets, and introduce a metric invariant which measures the “length” of equilateral sets of a particular order. In other words, we will discuss the repercussions of the existence of various types of equilateral sets on the geometric and topological structure of the manifold. Results concerning the packing radius will be used extensively throughout the last section. This paper will assume a knowledge of elementary Riemannian Geometry that may be found in [Pe],[dC], [Le], or [CE]. 1 The Case of Non-negative Ricci Curvature In this section, we derive an upper bound for the equilateral dimension of complete manifolds whose Ricci curvature is non-negative. Our main tool will be Gromov’s generalization of Bishop’s Volume Comparison Theorem. For a more general statement of the theorem the reader should consult [Z]. In general, volume of metric balls in Riemannian manifolds can be extremely difficult to compute, but the Bishop-Gromov Volume Comparison restricts the growth of these balls, which will be the pith of the proof of our first main theorem. The following theorem gives an upper bound for the growth of balls centered at a given point using spaces whose volumes can be more easily computed: the simply connected space forms of constant curvature, such as H,R, and S. In this paper, any expression involving a superscript H denotes the analogous quantity in the simply connected space forms of constant curvature H. Gromov’s Relative Volume Comparison Theorem 1. Let (M, g) be a complete Riemannian manifold of dimension n, and Ric(M) ≥ (n− 1)H, then for any p ∈ M and any 0 < r ≤ R vol(Bp(R)) vol(Bp(r)) ≤ vol (B(R)) vol(B(r)) This theorem comes from a comparison theorem concerning the distance function from a fixed point. The comparison theorem gives an upper bound for the rate of change of the area Page 104 RHIT Undergrad. Math. J., Vol. 14, No. 1 form of geodesic spheres along radial geodesics in terms of the corresponding quantity in the model space. The assumption that the manifold is complete is necessary in order to obtain the previous global result. The classical Hopf-Rinow Theorem states that completeness, in the metric sense, is equivalent to any two points being connected by a length-minimizing geodesic. This allows one to express the volume of a ball by integrating the area form along radial geodesics, while avoiding the points in which the distance function may not be smooth, thereby yielding a global result. Using this result, we may obtain the following upper bound for the equilateral dimension. We cannot say if this upper bound is sharp. The proof should be compared with the proof of Gromov’s Packing Lemma in [Z],[Gr], or [Pe]. Main Lemma 1. Let (M, g) be a complete Riemannian manifold of dimension n ≥ 2 and Ric(M) ≥ 0. Then given any equilateral set E = {p1, p2, . . . , pm}, then m ≤ 3. Proof. Let E be a equilateral set of distinct points p1, p2, . . . , pm such that d(pi, pj) = r for all distinct 1 ≤ i, j ≤ m. Then by the triangle inequality we have that for all 1 ≤ i, j ≤ m Bpi( r 2 ) ⊆ Bpj( 2 ), and when i, j are distinct Bpi( r 2 ) ∩Bpj( r 2) = ∅. Without loss of generality we may assume that vol(Bp1(r)) = min1≤i≤m vol(Bpi(r)). vol(Bp1 ( 3r 2 ) ) ≥ m ∑ i=1 vol(Bpi( r 2 )) ≥ m · vol(Bp1( r 2)) As Ric(M) ≥ 0, by the previous theorem we can bound m using the volume of balls in R. Therefore, m ≤ vol(Bp1( 3r 2 )) vol(Bp1( r 2 )) ≤ ωn( 3r 2 ) ωn( r 2 )n = 3 where ωn denotes the volume of a ball of radius one in R. Here we should note that the proof of this theorem relies solely on the packing type of complete Riemannian manifolds of dimension n and nonnegative Ricci curvature. For more concerning the packing type of a metric space the reader may consult [Gr]. Our main theorem is a simple corollary of this lemma: Main Theorem 1. Let (M, g) be a complete Riemannian manifold of dimension n ≥ 2 and Ric(M) ≥ 0, then e(M) ≤ 3 = C(n). 2 The Case of Positive Sectional Curvature We next shift our attention to the investigation of the equilateral dimension of manifolds with nonnegative and positive sectional curvature. Throughout this section our main tool will be the classical Topogonov’s Theorem, which is a global generalization of the Rauch Comparison Theorems. For a detailed proof the reader should consult [CE]. RHIT Undergrad. Math. J., Vol. 14, No. 1 Page 105 In the proceeding definition all indices are to be taken modulo 3. Let M be a Riemannian manifold, then a geodesic triangle in M is a triplet of geodesic segments, parameterized by arc length (γ1, γ2, γ3) of lengths (l1, l2, l3) such that γi(li) = γi+1(0) and li+1 ≥ li+2, for all i = 1, 2, 3. We will let αi = ](−γ′ i(li), γ′ i+1(0)) We now state part Topogonov’s Theorem: Topogonov’s Theorem 1. Let M be a complete manifold, with sec(M) ≥ H. A. Let (γ1, γ2, γ3) be a geodesic triangle in M n such that all geodesics are minimizing and if H > 0 then li ≤ π √ (H) . Let M denote the 2-dimensional space form of constant curvature H. Then there exists a geodesic triangle (γ̄1, γ̄2, γ̄3), such that for all i = 1, 2, 3 li = l̄i and αi ≤ ᾱi. B. Let γ1, γ2 be minimizing normal geodesics in M n such that γ1(l1) = γ2(0), of lengths l1, l2 and α = ](−γ′ 1(li), γ′ 2(0)). We call such a configuration a hinge. Additionally, if H > 0 assume that l2 ≤ π √ H Then there exists a hinge (γ̄1, γ̄2, α) of length l1, l2 in M H , such that d(γ1(0), γ2(l2)) ≤ d(γ̄1(0), γ̄2(l2)) C If equality holds in the above equation, then γ1, γ2 spans a surface with a totally geodesic interior which is isometric to the triangular surface spanned by γ̄1, γ̄2 Roughly speaking, the first part states that the angles of any geodesic triangle in a complete Riemannian manifold with sectional curvature ≥ H will always be greater than the corresponding angles formed by a triangle of the same lengths in the corresponding model space. The second part states that the distance between the endpoints of two geodesics (emanating from a common point with a given angle) will grow no faster than two

Sectioning Angles Using Hyperbolic Curves

Abstract: In the following paper, we will derive an equation of a hyperbola which passes through a point that, when connected to the foci of the hyperbola, creates a triangle such that measure of one base angle is twice of the other. Consequently, all points on this hyperbolic curve create triangles with this characteristic. Applying this to a specific condition, it is possible to show that with this hyperbola, a straight edge, and a compass, any angle can be trisected. 1. History of the Problem of Trisection The problem of trisecting an angle goes back to the time of the Ancient Greeks. Plato was the first to think of constructions as a process that must only be done with a straightedge, used to connect two points, and a compass, used to create circles and arcs. Trisecting the angle is one of three problems of Greek antiquity, the other two being creating a square with identical area as a circle and constructing a cube twice the volume of a given cube. Ultimately, the Greeks strived to trisect angles of arbitrary measure. They believed it was possible simply because some angles, such as 90 degrees, were shown to be easily trisected. Furthermore, processes had been discovered to both bisect and trisect line segments, so could any arbitrary angle be trisected? This fact encouraged mathematicians to believe that a general angle trisection method with just a straight edge and compass was within reach. Hippocrates (460-380 BC), the first person to attempt the trisection of an angle, was unsuccessful but contributed to the world of geometry by labeling points and lines with letters. Hippias (460-399 BC) formed the first successful trisection with a curve called quadratix (this was the first curve introduced in geometry besides lines and circles). Menaechmus (380-320 BC) discovered conic sections. Archimedes (287-217 BC) was able to produce an uncomplicated solution but it required the use of a marked straightedge twice in order to complete the trisection. Nicomedes (280-210 BC) also used a mark straightedge in one of his Date: July 6, 2012. 1 2 NICHOLAS MOLBERT, TIA BURDEN, AND JULIE FINK techniques, but succeeded in the trisection using a conchoid. Apolonius (250-175 BC) discovered that by using conic sections, trisection was possible. Both Pappus (early fourth century) and Descartes (15961650) used Apolonius’ discovery to trisect an angle with a hyperbola and parabola, respectively. However, the problem of trisecting an angle with only an unmarked straightedge and compass remained. Beginning in the sixteenth century, mathematicians utilized the concepts of the Greek constructions used in trisection and doubling the volume of the cube to solve cubic and quartic equations. Francios Viete (1540-1603) noticed a link between the equations and trigonometry when working with irreducible polynomials, which were at the heart of the proof of planar constructible impossibility. Pierre Wantzel (18141848) proved the problem of trisection impossible with planar constructions in his 1837 article titled Research on the Means of Knowing If a Problem of Geometry Can Be Solved with Compass and Straight Edge. Essentially, Wantzel’s proof was based on the fact that an angle trisection was equivalent to being able to construct roots of a cubic equation. This provided a definitive answer to the problem posed 2200 years later. The impossibility of the trisection using planar constructions then gave birth to modern abstract algebra. 2. Basic Constructions Constructions are divided planar and solid constructions. Planar constructions are done with a compass and straight edge. Solid constructions add conic sections to the set of things solidly constructible (i.e. hyperbolas). In this project, we will not explain how to perform solid constructions. We will simply state, ”construct the hyperbola.” 2.1. Notation. Before we begin listing constructions needed in trisecting an angle, let us define some notation needed in the following proofs of these constructions: • ←→ XY denotes the line through points X and Y . • XY denotes the segment from point X to point Y . • C(X, Y ) denotes a circle with center X and radius XY . • XY denotes the magnitude of segment XY • ∠XY Z denotes the measure or name of an angle, depending on the context. Let us start with the following axioms of planar constructions: (1) Given two points A and B, we can construct the line and the segment that passes through points A and B (i.e. ←→ AB and AB respectively), SECTIONING ANGLES USING HYPERBOLIC CURVES 3 (2) Given two points A and B, we can construct the circle with center A and radius AB. Denote the circle by C(A,B), (3) Given a line through the points A and B and an angle θ, we can construct angle θ such that one of its rays lies on the line through points A and B. Now we can proceed to proving propositions needed to produce the trisection of an angle using a hyperbolic curve, Proposition 2.1. (Rusty Compass Theorem) Given points A, B, and C, we wish to construct a circle centered at point A with radius equal to BC [8]. Proof. First, draw C(A,B) and C(B,A) and obtain point D which forms equilateral 4ABD. Then, construct C(B,C). Extend DB past point B and call DB ∩ C(B,C), point E. Construct C(D,E). Then extend DA past point A and label DA∩C(D,E), point F . Construct C(A,F ). Because E lies on C(B,C), BE = BC. Then, because 4ABD is equilateral, DA = DB. Also, because E and F lie on a circle with center D, DE = DF . Therefore, AF = BE = BC [2]. Figure 1. Rusty Compass Theorem Proposition 2.2. (Copying an Angle) Given ∠ABC and a line l containing a point D, we can find E on l and a point F such that ∠ABC = ∠EDF [8]. 4 NICHOLAS MOLBERT, TIA BURDEN, AND JULIE FINK Proof. Using Proposition 2.1 we construct the point E on line l with BA = DE. Construct a circle with center D with radius length BC. Construct another circle with center E with radius length AC. We get the point F . Therefore, 4ABC ∼= 4EDF by SSS. So, ∠ABC = ∠EDF . Figure 2. Copying an Angle Proposition 2.3. (Bisecting an Angle) Given ∠ABC, there is a point D such that ∠ABD ∼= ∠DBC [8]. Proof. Extend AB to get the line l. Construct C(B,C) to get the point E on l. Construct C(E,B) and C(C,B) to get D. Then ∠ABD ∼= ∠DBC. Therefore, BD bisects ∠ABC. SECTIONING ANGLES USING HYPERBOLIC CURVES 5 Figure 3. Bisecting an Angle Proposition 2.4. (Dropping a Perpendicular) Given a line l and a point p not on l, we can construct a line l′ which is perpendicular to l and passes through p [8]. Proof. There exists a point A on l. If ←→ pA ⊥ l, we are done. If not, construct C(p,A) to get B. Next, construct C(A,B) and C(B,A) to get C. Then ←→ pC is perpendicular to l. Figure 4. Dropping a Perpendicular Proposition 2.5. (Parallel Postulate, Playfair) Given a line l and P not on l, we can construct l′ through P and parallel to l [8]. Proof. Construct l′′ such that it is perpendicular to l and passes through P by Proposition 4. Now construct l′ through P and perpendicular to l′′ by Proposition 6. l′ is parallel to l through P . 6 NICHOLAS MOLBERT, TIA BURDEN, AND JULIE FINK Figure 5. Parallel Postulate, Playfair Proposition 2.6. (Raising a Perpendicular) Given a point p on a line l, then you can construct l′ through point p perpendicular to line l [8]. Proof. There exists a point A on line l distinct from point p. Construct C(p,A) to obtain l ∩C(p,A) = B. Construct C(A,B) and C(B,A) to get C. Draw ←→ pC. This line is l′. Figure 6. Raising a Perpendicular SECTIONING ANGLES USING HYPERBOLIC CURVES 7

Tangent circles in the hyperbolic disk

Abstract: Constructions of tangent circles in the hyperbolic disk, interpreted in Euclidean geometry, give us examples of four mutually tangent circles. These are shown to satisfy Descartes’s Theorem for tangent circles. We also show that the Archimedes twin circles in the hyperbolic arbelos are usually not hyperbolic congruent, even though they are Euclidean congruent. We include a few construction instructions because all items under consideration require surprisingly few steps. Acknowledgements: We would like to thank those who sponsored the Mohler-Thompson Summer Research Grant for making this research experience possible. I would also like to thank Dr. Elizabeth Jensen and Dr. Chad Gunnoe for organizing the free on-campus housing for the summer’s student researchers, Dr. Michael McDaniel for including me in this project and giving me this opportunity, the referee who reviewed this paper, and David Rader, editor of the Rose-Hulman Undergraduate Mathematics Journal. PAGE 14 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1

Analytic Extension and Conformal Mapping in the Dual and the Double Planes

Abstract: Many theorems in the complex plane have analogues in the dual (x + jy, j2 = 0) and the double (x + ky, k2 = 1) planes. In this paper, we prove that Schwarz reflection principle holds in the dual and the double planes. We also show that in these two planes the domain of an analytic function can usually be extended analytically to a larger region. In addition, we find that a certain class of regions can be mapped conformally to the upper half plane, which is analogous to the Riemann mapping theorem. Acknowledgements: The first three authors were supported by the National Science Foundation grant No. DNS-1002453. The fourth author was supported by a Jack and Lois Kuipers Applied Mathematics Fellowship. Page 136 RHIT Undergrad. Math. J., Vol. 14, no. 2

On the Coarse Geometry of Lp([a,b])

Abstract: Coarse geometry deals with the large-scale geometry of a space as opposed to its small-scale structure. This paper investigates the concept of coarse geometry and specifically studies the coarse geometry of spaces regularly encountered in real analysis. We construct a non-separable space that is coarse equivalent to the separable space L1 ([a, b] ,m). Acknowledgements: I would like to thank the WCSU Student Government Association for their financial support and the WCSU Honors Program for making this research possible. I’m indebted to professors David Burns, Samuel Lightwood, and Barry Mittag for all of their help and advice along the way. I would also like to thank the referee and David Rader for their suggestions on improving this paper. Page 2 RHIT Undergrad. Math. J., Vol. 14, no. 2

A Non-geometric Switch Toggling Problem

Abstract: Switch toggling games such as Lights Out and the σ-game are widely studied in mathematics and have been applied to model a variety of situations such as genetic networks and cellular automata. This paper introduces a class of toggling games where at each iteration a fixed number of switches is chosen to be toggled where the only switches changed are the switches chosen to be toggled. The switches all operate independently of each other and do not depend on the proximity or the position relative to any other switch. This paper classifies the conditions necessary and the steps taken to transition from all switches in the on state to all switches in the off state. Further results include the conditions required of the parity between the number of switches in the system and the fixed number of switches toggled at each step in order to transition from a given initial state to a specified terminal state. Acknowledgements: I would like to thank Dr. Richard Daquila for all of his help and support during this project as a part of the Muskie Fellowship program at Muskingum University. I would also like to thank Rose-Hulman Undergraduate Math Journal for giving me this opportunity as well as my parents for their continued love and support. Page 60 RHIT Undergrad. Math. J., Vol. 14, no. 2

Counting Maximal Chains in Weighted Voting Posets

Abstract: Weighted voting is built around the idea that voters have differing amounts of influence in elections, with familiar examples ranging from company shareholder meetings to the United States Electoral College. We examine the idea that each voter has a uniquely determined weight, paying particular attention to how voters leverage this weight to get their way on a specific yes/no motion (for example, by forming coalitions). After some more background on weighted voting, we describe a natural partial order relation between these coalitions of voters. This ordering can be modeled by a partially ordered set (poset), which we call a coalitions poset. Using this poset, we derive another important poset via a natural ordering on collections of coalitions. Our results begin by detailing a method for counting the number of maximal chains in the derived poset. After employing this method to find the number of maximal chains in the derived poset with 5 voters, we extend our method for use in the coalitions poset. Finally, we conjecture a formula for the number of maximal chains in the coalitions poset with n voters. Acknowledgements: The author would like to thank his thesis advisors, Dr. Jason Parsley and Dr. Sarah Mason, for all of their help on this paper. Page 158 RHIT Undergrad. Math. J., Vol. 14, No. 1 1 An Introduction to Weighted Voting Systems The focus of this paper is the study of weighted voting systems. A weighted voting system is defined as a collection of n voters, v1, v2, v3, ..., vn, who vote on a yes or no motion. Each voter vi has some given weight wi. In order for a motion to pass, the sum of the weights of all voters voting for the motion must meet or exceed some fixed quota q. Otherwise, the motion is said to fail. We define a coalition as a subset of our n voters who all vote the same way on a motion. A coalition can contain any number of voters, from no voters to all voters in the system. The weight of a coalition is the sum of the weights of each of its voters. The interested reader may wish to peruse [1] and [4] as a supplement to our introduction of weighted voting. Now that we have been acquainted with the basic concepts of weighted voting systems, let us introduce more mathematical definitions: Definition 1.1. A coalition A is a collection of j voters from a system with n voters such that every member of A votes the same way on a motion. The coalition A has the following properties: • 0 ≤ j ≤ n • wA = w1 + w2 + w3 + · · ·+ wj, where wA is the weight of coalition A The grand coalition is precisely the coalition containing all voters in the system. In other words, the grand coalition occurs when j = n. Similarly, the empty coalition contains no voters and occurs when j = 0. Definition 1.2. A weighted voting system (q : wn, wn−1, ..., w2, w1) is defined by the following characteristics: • There are n voters, to whom we give the names v1, v2, v3, ..., vn. • There is a weight wi ≥ 0 corresponding to each voter vi. • There is a quota q such that a motion will pass if, given the coalition A containing all voters voting in favor of the motion, we have wA ≥ q. In most literature, voters are referred to by their subscripts. In other words, we would simply call voter vn by the abbreviated name, voter n. As a convention, we enumerate our n voters by increasing weight. That is, voter 1 has the least weight, voter 2 has the second least weight,..., and voter n has the greatest weight. We do not require that the weights be strictly increasing. So, we have: 0 ≤ w1 ≤ w2 ≤ w3 ≤ · · · ≤ wn Contrastingly, we list the voters comprising a coalition by decreasing weight. Consequently, the coalition containing voters 1, 3, and 4 would be written {4, 3, 1} = {431} because RHIT Undergrad. Math. J., Vol. 14, No. 1 Page 159 voter 4 has the greatest weight, voter 3 has the second greatest weight, and voter 1 has the least weight. Similarly, the grand coalition containing all n voters would be written {n, n − 1, n − 2, ..., 2, 1}. We should note that this ordering is contrary to the majority of voting literature, which tends to list the weights of the voters in a coalition in increasing order. The convention of ordering weights in this new way came from a paper by Mason and Parsley [3]. The advantage to this ordering is that it makes it much easier to determine the rank of a coalition in a certain partially ordered set M(n), which is central to the study of weighted voting theory and to this paper. Essentially, weighted voting is predicated on the idea that not all voters are equal. Familiar real world applications of weighted voting range from shareholder meetings to the Electoral College of the United States. In shareholder meetings, some shareholders will have more weight than others by virtue of owning more stock in the company. For example, a shareholder’s weight might be equal to the number of shares he owns. Similarly, some states have more weight than others in the Electoral College because they have a larger population. Let us examine an example of weighted voting in a real-world scenario. Example 1.1. Alice, Bob, and Charlie are the sole shareholders of a small construction company. The distribution of shares is illustrated in the following table: Shareholder Number of Shares Alice 49 Bob 46 Charlie 5 The company has recently fallen on hard times because of the decline in the housing market. Alice proposes expanding the business to other markets, reasoning that this will increase their revenue. Bob opposes the idea, explaining that this proposition will also lead to higher expenditures. Given the share distribution shown above, how can Alice get her motion to pass? We will set the quota q at 51 shares. Because we can pick a quota (namely, q = 51) and assign weights to each voter i (simply take wi = the number of shares held by i), we have a weighted voting system. Using Definition 1.2, we notate this weighted voting system as (51 : 49, 46, 5). If Alice is the only one to vote for the motion, she will form a coalition with weight w = 49. But 49 < q, so Alice’s motion would fail. On the other hand, if Charlie votes with Alice on the motion, they will form a coalition with weight w = 49 + 5 = 54. Since 54 > q, Alice’s motion would pass. Finally, if Alice is able to convince Bob to support her motion, they will form a coalition with weight w = 49 + 46 = 95. Since 95 > q, Alice’s motion would pass. So, we have found that Alice cannot singlehandedly force her motion to pass; she needs the help of either Bob or Charlie. Of course, Alice could also win by teaming up with both Bob and Charlie. However, this observation is trivial. Once Alice forms a coalition with one of the other two shareholders, they already have enough weight between themselves to force her motion to pass. Adding the third shareholder to the coalition would have no effect on the outcome. This idea is important and will come up again when we discuss the notion of minimal winning coalitions. Page 160 RHIT Undergrad. Math. J., Vol. 14, No. 1 Example 1.1 is meant to be a representative example of weighted voting systems. So, we should note that while it was convenient for the weights of our voters to sum to 100, it was not necessary. Furthermore, we did not have to choose q = 51 as our quota; it could have just as easily been taken to be q = 30 or even q = 100. However, to avoid the situation in which two opposing coalitions are both winning, we will disallow quotas set at less than or equal 50 percent of the combined weight of all voters. The reader may also notice that the weights of each voter were different in Example 1.1. This was also not necessary. It is perfectly valid to have a weighted voting system in which the weights of certain (or even all) voters are equal. Consider the following example, which is a variation of Example 1.1: Example 1.2. Alice, Bob, and Charlie are the sole shareholders of the construction company. Their shares are distributed according to the following table. Shareholder Number of Shares Alice 33 Bob 33 Charlie 33 Alice again makes her proposition, and is opposed by Bob as in Example 1.1. In this scenario, how can Alice get her motion to pass? The quota q is set at 50 shares. Analyzing the system as we did before, we find that Alice needs the help of either Bob or Charlie to get her motion to pass. The reader may notice that this result is identical to the result from Example 1.1. This similarity will be even more apparent once the notion of winning coalitions is introduced. 2 Winning Coalitions Before we proceed any further, we will give a formal definition for an idea that we have been taking for granted, winning coalitions. Our intuitive notion of a winning coalition has served us well up to this point, but it will be useful to have an explicit definition. In a natural extension of this idea, we will also define losing coalitions. Definition 2.1. Given a weighted voting system (q : wn, wn−1, ..., w2, w1), define the coalition A as a collection of all voters who vote the same way on some motion. • A is a winning coalition if and only if wA ≥ q • A is a losing coalition if and only if wA < q By Definition 1.1, the weight of any given coalition A must be unique. In other words, no one coalition can have two weights at once. Therefore, the weight of coalition A must satisfy either wA ≥ q or wA < q, but not both. This tells us that a coalition cannot be both RHIT Undergrad. Math. J., Vol. 14, No. 1 Page 161 winning and losing; however, it must be one of the two. To help us see this more explicitly, we will find all of the winning and losing coalitions from Example 1.1. Recall that the quota was set at q = 51. Coalition Weight of Coalition Winning/Losing {∅} 0 Losing {Alice} 49 Losing {Bob} 46 Losing {Charlie} 5 Losing {Alice, Bob} 49 + 46 = 95 Winning {Alice, Charlie} 49 + 5 = 54 Winning {Bob, Charlie} 46 + 5 = 51 Winning {Alice, Bob, Charlie} 49 + 46 + 5 = 100 Winning Next, we will compute all of the winning and losing coali