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Articles 1 - 8 of 8
Full-Text Articles in Physical Sciences and Mathematics
Hyperbolic Billiard Paths, Rebecca Lehman, Chad White
Hyperbolic Billiard Paths, Rebecca Lehman, Chad White
Mathematical Sciences Technical Reports (MSTR)
A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at the billiard path described by a closed geodesic on a single tile. When looking at billiard paths it is possible to ignore surfaces and restrict ourselves to the tiling of the hyperbolic plane. We classify the smallest billiard paths by wordlength and parity. We also demonstrate the existence of orientable paths and investigate conjectures about the billiard spectrum for the (2, 3, 7)-tiling.
Pigeon-Holing Monodromy Groups, Niles G. Johnson
Pigeon-Holing Monodromy Groups, Niles G. Johnson
Mathematical Sciences Technical Reports (MSTR)
A simple tiling on a sphere can be used to construct a tiling on a d-fold branched cover of the sphere. By lifting a so-called equatorial tiling on the sphere, the lifted tiling is locally kaleidoscopic, yielding an attractive tiling on the surface. This construction is via a correspondence between loops around vertices on the sphere and paths across tiles on the cover. The branched cover and lifted tiling give rise to an associated monodromy group in the symmetric group on d symbols. This monodromy group provides a beautiful connection between the cover and its base space. Our investigation …
The Galois Correspondence For Branched Covering Spaces And Its Relationship To Hecke Algebras, Matthew Ong
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
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
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
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
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
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.