Mathematics Commons^™

The Galois Correspondence For Branched Covering Spaces And Its Relationship To Hecke Algebras, Matthew Ong

Mathematical Sciences Technical Reports (MSTR)

There is a very beautiful correspondence between branched covers of the Riemann sphere P¹ and subgroups of the fundamental group π₁(P¹ − {branch points}), exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F). This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois group to the case of non- Galois covers S/P¹. Specifically, we show that the isomorphism type of a Hecke algebra C[H\G/H] is completely determined by the decomposition of …

Fixed Point And Two-Cycles Of The Discrete Logarithm, Joshua Holden

Mathematical Sciences Technical Reports (MSTR)

We explore some questions related to one of Brizolis: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? We extend this question to ask about not only fixed points but also two-cycles. Campbell and Pomerance have not only answered the fixed point question for sufficiently large p but have also rigorously estimated the number of such pairs given certain conditions on g and h. We attempt to give heuristics for similar estimates given other conditions on g and h and also in the case …

Characterizing A Defect In A One-Dimensional Bar, Cynthia Gangi, Sameer Shah

Mathematical Sciences Technical Reports (MSTR)

We examine the inverse problem of locating and describing an internal point defect in a one­ dimensional rod W by controlling the heat inputs and measuring the subsequent temperatures at the boundary of W. We use a variation of the forward heat equation to model heat flow through W, then propose algorithms for locating an internal defect and quantifying the effect the defect has on the heat flow. We implement these algorithms, analyze the stability of the procedures, and provide several computational examples.

Tilings Of Low-Genus Surfaces By Quadrilaterals, John Gregoire, Isabel Averil

Mathematical Sciences Technical Reports (MSTR)

In contribution to the classification of all tilings of low-genus surfaces, the kaleidoscopic and non-kaleidoscopic tilings by quadrilaterals are given up to genus 12. As part of their classification, the algebraic structure of the conformal tiling groups and the geometric structure of the tiles are specified. In addition, several infinite classes of tilings and tiling groups are presented.

A Restricted Partition Function Modulo 3, Naomi Utgof

Mathematical Sciences Technical Reports (MSTR)

The ordinary partition function p(n) counts the number of representations of a positive integer n as the sum of positive integers. We denote by p₃(n) the number of partitions of n with no parts divisible by 3: We demonstrate congruence relations for arithmetic sequences qⁿ+(2q²-2)/24 where q is a prime other than 3 congruent to 3 (mod 4): We also prove a result when q = 5 and make a conjecture about a generalization .

Applications Of Graph Theory To Separability, Stephen Young

Mathematical Sciences Technical Reports (MSTR)

Let S be a surface with a triangular tiling T. Let R be a reflection a side of one of the triangles; so that R is an orientation reversing isometry of the surface. Define M = {s in S |S : Rs = s}. We then say that the surface S separates along the reflection R if S-R has two components. This paper considers the applications of graph theoretic methods to determining whether a reflection is separating or not and compares the algorithmic efficiency of these methods to the current known methods.

Separability Of Tilings, Nicholas Baeth, Jason Deblois, Lisa Powell

Mathematical Sciences Technical Reports (MSTR)

A tiling by triangles of an orientable surfaces is called kaleidoscopic if the local reflection in any edge of a triangle extends to a global isometry of the surface. Given such a global reflection the fixed point subset of the reflection consists of embedded circles (ovals) whose union is called the mirror of the reflection. The reflection is called separating if removal of the mirror disconnects the surface into two components. We consider surfaces such that the orientation preserving subgroup of the tiling group generated by the reflection is cyclic or abelian. A complete classification of those surfaces with separating …

Triangular Surface Tiling Groups For Low Genus, Sean A. Broughton, Robert M. Dirks, Maria Sloughter, C. Ryan Vinroot

Mathematical Sciences Technical Reports (MSTR)

Consider a surface, S, with a kaleidoscopic tiling by non-obtuse triangles (tiles), i.e., each local reflection in a side of a triangle extends to an isometry of the surface, preserving the tiling. The tiling is geodesic if the side of each triangle extends to a closed geodesic on the surface consisting of edges of tiles. The reflection group G*, generated by these reflections, is called the tiling group of the surface. This paper classifies, up to isometry, all geodesic, kaleidoscopic tilings by triangles, of hyperbolic surfaces of genus up to 13. As a part of this classification the tiling groups …

Lengths Of Systoles On Tileable Hyperbolic Surfaces, Kevin Woods

Mathematical Sciences Technical Reports (MSTR)

The same triangle may tile geometrically distinct surfaces of the same genus, and these tilings may determine isomorphic tiling groups. We determine if there are geometric differences in the surfaces that can be found using group theoretic methods. Specifically, we determine if the systole, the shortest closed geodesic on a surface, can distinguish a certain families of tilings. For example, there are three tilings of surfaces of genus 14 by the hyperbolic triangle with angles π/2 , π/3 , and π/7 whose tiling groups are all PSL₂(13). These tilings can be distinguished by the lengths of their systoles.

Ramanujan-Like Congreuences Of The Distinct Partition Function, Ian Blumenfeld, Christi Carlstead, Mimi Cukier, Wesley Terway

Mathematical Sciences Technical Reports (MSTR)

In his work with the partition function, Ramanujan observed several congruences of the form p(An + B) = 0 (mod m). We adapt this form to several congruences of the distinct partition function, p2(n). We show that one can determine all ordered pairs of integers (A;B) for which p₂(An + B)=0 (mod 2) and show families of congruences modulo 4. Finally, we offer a proof of a congruence modulo 5 satisfied by the distinct partition function.

Classification Of Cwatsets Through Order 23, Ben Goodwin, Dennis Lin

Mathematical Sciences Technical Reports (MSTR)

A cwatset of order n can be represented by a transitive subgroup of S_n. Previous work has shown that each conjugacy class of rep­resentation groups corresponds to an isomorphism class of cwatsets. We present a technique for determining whether a particular transitive subgroup of Sn can appear as the representation group for a cwatset of order n. Using this method, we provide a full classification of cwatset isomorphism classes through order 23.

Quest For Tilings On Riemann Surfaces Of Genus Six And Seven, Robert Dirks, Maria Sloughter

Mathematical Sciences Technical Reports (MSTR)

The problem of kaleidoscopically tiling a surface by congruent triangles is equivalent to finding groups generated in certain ways. In order to admit a tiling, a group must have a specific set of generators as well as an involutary automorphism, T, that acts to reverse the orientation of the tiles. The purpose of this paper is to explore group theoretic and computational methods for determining the existence of symmetry groups and tiling groups, as well as to classify the symmetry and tiling groups on hyperbolic Riemann surfaces of genus 6 and 7.

Lengths Of Geodesics On Klein’S Quartic Curve, Ryan Derby-Talbot

Mathematical Sciences Technical Reports (MSTR)

A well-known and much studied Riemann surface is Klein’s quartic curve. This surface is interesting since it is the smallest complex curve with maximal symmetry. In addition to this high degree of symmetry, Klein’s quartic curve can be tiled by triangles,giving rise to a tiling group generated by reflections. Using the tiling group and the universal cover of the tiling group we are able to compile a list of the lengths of the short,simple,closed geodesics on this surface. In particular,w e are able to determine whether the geodesic loops generated by the tiling are the systoles,i.e.,the shortest closed geodesics.

Singular Solutions To A Nonlinear Elliptic Boundary Value Problem Originating From Corrosion Modeling, Kurt M. Bryan, Michael Vogelius

Mathematical Sciences Technical Reports (MSTR)

We consider a nonlinear elliptic boundary value problem on a planar domain. The exponential type nonlinearity in the boundary condition is one that frequently appears in the modeling of electrochemical systems. For the case of a disk we construct a family of exact solutions that exhibit limiting logarithmic singularities at certain points on the boundary. Based on these solutions we develop two criteria that we believe predict the possible locations of the boundary singularities on quite general domains.

Cwatset Isomorphism And Its Consequences, Carolyn M. Girod, Matthew Lipinski, Joseph R. Mileti, Jennifer R. Paulhus

Mathematical Sciences Technical Reports (MSTR)

We explore the consequences of cwatset isomorphism (there are a finite number of non-isomorphic cwatsets of each order) and consider parallels between the theory of groups and the theory of cwatsets (cwatsets of prime order are cyclic but direct sums of isomorphic cwatsets aren't necessarily isomorphic).

Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

Let p be a reflection on a closed Riemann Surface S, i.e., an anti-conformal involutary isometry of S with a non-empty fixed point subset. Let Sp denote the fixed point subset of p, which is also called the mirror of p. If S −Sp has two components, then p is called separating and we say that S splits at the mirror Sp. Otherwise p is called non-separating. We assume that the system of mirrors, Sq, as q varies over all reflections in the isometry group Aut*(S) defines a tiling of the surface, consisting of triangles. In turn, the tiling determines …

Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith

Mathematical Sciences Technical Reports (MSTR)

We consider triangle-quadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of 1,2, and 3 parameter families as well as a finite number of exceptional cases.

Tilings Which Split A Mirror, Jim Belk

Mathematical Sciences Technical Reports (MSTR)

We consider the mirror of a reflection which consists of its subset of fixed points. We investigate a number of conditions on the tiling that guarantee that the surface splits at a mirror.

Automorphic Subsets Of The N-Dimensional Cube Are Translations Of Cwatsets, Matthew Lepinski

Mathematical Sciences Technical Reports (MSTR)

It is known that automorphic subsets are generalizations of cwatsets. In this paper we show that an automorphic subset is the translation of some cwatset, and therefore that each automorphic subset is internally isomorphic to a cwatset.

Elementary Inversion Of The Laplace Transform, Kurt M. Bryan

Mathematical Sciences Technical Reports (MSTR)

This paper provides an elementary derivation of a very simple "closed-form"

inversion formula for the Laplace Transform.

Constructing Kaleidscopic Tiling Polygons In The Hyperbolic Plane, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

We have all seen many of the beautiful patterns obtained by tiling the hyperbolic plane H by repeated reflection in the sides of a "kaleidoscopic" polygon. Though there are such patterns on the sphere and the euclidean plane, these positively curved and fiat geometries lack the richness we see in the hyperbolic plane. Many of these patterns have been popularized by the beautiful art of M.C. Escher. For a list of references and a more complete discussion on the construction of artistic tilings see [6].

Symmetry And Tiling Groups For Genus 4 And 5, C. Ryan Vinroot

Mathematical Sciences Technical Reports (MSTR)

All symmetry groups for surfaces of genus 2 and 3 are known. In this paper, we classify symmetry groups and tiling groups with three branch points for surfaces of genus 4 and 5. Also, a class of symmetry groups that are not tiling groups is presented, as well as a class of odd order non-abelian tiling groups.

Quadrilaterals Subdivided By Triangles In The Hyperbolic Plane, Dawn M. Haney, Lori T. Mckeough

Mathematical Sciences Technical Reports (MSTR)

In this paper, we consider triangle-quadrilateral pairs in the hyperbolic plane which “kaleidoscopically” tile the plane simultaneously. These tilings are called divisible tilings or subdivided tilings. We restrict our attention to the simplest case of divisible tilings, satisfying the corner condition, in which a single triangle occurs at each vertexof the quadrilateral. All possible such divisible tilings are catalogued as well as determining the minimal genus surface on which the divisible tiling exists. The tiling groups of these surfaces are also determined.

Maximally Disjoint Solutions Of The Set Covering Problem, David J. Rader, Peter L. Hammer

Mathematical Sciences Technical Reports (MSTR)

This paper is concerned with finding two solutions of a set covering problem that have a minimum number of variables in common. We show that this problem is NP­ complete, even in the case where we are only interested in completely disjoint solutions. We describe three heuristic methods based on the standard greedy algorithm for set covering problems. Two of these algorithms find the solutions sequentially, while the third finds them simultaneously. A local search method for reducing the overlap of the two given solutions is then described. This method involves the solution of a reduced set covering problem. Finally, …

Oval Intersections In Tilings On Surfaces, Dennis A. Schmidt

Mathematical Sciences Technical Reports (MSTR)

A tiling is a covering by polygons, without gaps or overlapping, of a compact, orientable surface. We are particularly interested in tilings by triangles that generate a large symmetry group of the surface. An oval of the tiling is a simple, closed curve that is a union of edges of the tiling. We investigate the number of points of intersection of two ovals. We have found that the number of intersections is bounded when the subgroup of orientation preserving symmetries is abelian. However, there is no upper bound on the number of intersections in the non-abelian case.

Counting Ovals On A Symmetric Riemann Surface, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

Let S be a compact Riemann surface without boundary. A symmetry of S is an anti-conformal, involutary automorphism. Its fixed point set is a disjoint union of circles, each of which is called an oval. A method is presented for counting the ovals of a symmetry when S admits a large group G of automorphisms. The method involves only calculations in G, based on the geometric description of S/G, and the knowledge of the action of the symmetry on G.

The Link Between Scrambling Numbers And Derangements, Barry Balof, Eric Farmer, Jamie Kawabata

Mathematical Sciences Technical Reports (MSTR)

The group equation abcdef = dabecf can be reduced to the equation xcde = dxec. In general, we are interested in how many variables are needed to represent group equations in which the right side is a permutation of the variables on the left side. Scrambling numbers capture this information about a permutation. In this paper we present several facts about scrambling numbers, and expose a striking relationship between permutations that cannot be reduced and derangements.

Generalized Conjugacy Classes, Pramod N. Achar

Mathematical Sciences Technical Reports (MSTR)

Generalized conjugation is the action of a group on its underlying set given by (g,x) -> p(g)xg^-1, where p is some fixed endomorphism of G. Here we study combinatorial properties of the sizes of the orbits of the preceding action. In particular, we reduce the problem to a simpler case if p has nontrivial kernel, or if it is an inner automorphism, and we give a construction that allows a partial analysis in the general case.

A Stronger Triangle Inequality, Herb Bailey

Mathematical Sciences Technical Reports (MSTR)

The triangle inequality is basic for many results in real and complex analysis. The geometric form states that the sum of any two sides of a triangle is greater than the third. This was included as Proposition XX in the first book of Euclid's Elements. Many geometric triangle inequalities involving sides, angles, altitudes, inscribed circles and circumscribed circles have been found. Hundreds of these inequalities are summarized in [l] and [2]. A nice geometric proof of the triangle inequality is given in [3].

Cwatsets: Weights, Cardinalities, And Generalizations, Richard Mohr

Mathematical Sciences Technical Reports (MSTR)

This report provides an upper bound on the average weight of an element in a cwatset and discusses the ratio of the cardinality of a cwatset to the cardinality of the group containing the cwatset. The concept of a generalized cwatset is also introduced.