Mathematics Commons^™

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].

Time Delay In The Kuramoto Model Of Coupled Oscillators, Stephen M.K. Yeung, S H. Strogatz

Mathematics

We generalize the Kuramoto model of coupled oscillators to allow time-delayed interactions. New phenomena include bistability between synchronized and incoherent states, and unsteady solutions with time-dependent order parameters. We derive exact formulas for the stability boundaries of the incoherent and synchronized states, as a function of the delay, in the special case where the oscillators are identical. The experimental implications of the model are discussed for populations of chirping crickets, where the finite speed of sound causes communication delays, and for physical systems such as coupled phase-locked loops or lasers.

Combinatorics Of Open Covers Vi: Selectors For Sequences Of Dense Sets, Marion Scheepers

Mathematics Faculty Publications and Presentations

We consider the following two selection principles for topological spaces:

[Principle 1:] { For each sequence of dense subsets, there is a sequence of points from the space, the n-th point coming from the n-th dense set, such that this set of points is dense in the space;

[Principle 2:]{ For each sequence of dense subsets, there is a sequence of finite sets, the n-th a subset of the n-th dense set, such that the union of these finite sets is dense in the space.

We show that for separable metric space X one of these principles holds for the …

Sextic Number Fields With Discriminant -J2A3B, John W. Jones, David P. Roberts

Mathematics Publications

. Complete lists of number fields, of given degree n and unramified outside a given finite set S of primes, are both of intrinsic interest and useful in some applications. For degrees n ≤ 5 and S = {∞, 2, 3}, the complete lists have appeared previously; there are in total 85 such fields. Here we give the complete list for n = 6 and S = {∞, 2, 3}, finding in particular exactly 398 such fields. We use a three-pronged approach to obtain this classification: an exhaustive computer search, sextic twinning, and class field theory. Also we completely identify …

Will Coaching Improve One's Math Score On The Scholastic Aptitude Test?, Judith B. Moore

The Corinthian

The Scholastic Aptitude Test (SAT) has long been a determining factor for admission into most of the nation's colleges and universities. Numerous studies have been conducted to question the reliability and predictive validity of SAT scores, yet they are still widely used. The widespread use of these scores in determining college admission is an indication that the SAT will be around for quite some time. For this reason and for personal reasons, I posed the following research question: "Will coaching improve one's math score on the Scholastic Aptitude Test?"

The Deconstruction Of Mathematics: A Criticism Of Reuben Hersh's What Is Mathematics, Really? And The Humanist Philosophy Of Mathematics, David J. Stucki

Mathematics Faculty Scholarship

Mathematics, as an academic discipline, has stood for many years as the last bastion against a growing tide of intellectual relativism that has become all but ubiquitous. More recently, however, efforts have been made to "humanize" mathematics by advocating a social-constructivist approach to the philosophy of mathematics, both in practice and education. This paper is intended to serve as a critical response to one advocate of this approach, Reuben Hersh (What Is Mathematics, Really?, 1997), and in the process a defense of Platonism.

A Lower Bound For The Cyclic Cutwidth Of The N-Cube, James Shigeo Namekata

Theses Digitization Project

No abstract provided.

A Sheaf Theoretic Approach To Consciousness, Goro Kato, Daniele C. Struppa

Mathematics

A new fundamental mathematical model of consciousness based on category theory is presented. The model is based on two philosophical-theological assumptions: a) the universe is a sea of consciousness, and b) time is multi-dimensional and non-linear.

Inequalities For Solutions Of Multipoint Boundary Value Problems, Paul W. Eloe, Johnny Henderson

Mathematics Faculty Publications

The concept of concavity is generalized to functions, y, satisfying nth-order differential inequalities. … An analogous inequality for a related Green’s function is also obtained. These inequalities are useful in applications of certain cone theoretic fixed-point theorems.

The Stable Manifold Theorem For Stochastic Differential Equations, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow

Articles and Preprints

We formulate and prove a local stable manifold theorem for stochastic differential equations (SDEs) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itô-type equations are treated. Starting with the existence of a stochastic flow for a SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich SDEs, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating SDE. The proof of the stable manifold theorem is based …

An Algorithm To Determine Non-Perfect Colorings That Arise From Plane Crystallographic Groups, Ma. Louise Antonette N. De Las Peñas

Mathematics Faculty Publications

This paper presents a computer algorithm that assists us in our research on non-perfect colorings of plane crystallographic patterns.

On The Number Of Radially Symmetric Solutions To Dirichlet Problems With Jumping Nonlinearities Of Superlinear Order, Alfonso Castro, Hendrik J. Kuiper

All HMC Faculty Publications and Research

This paper is concerned with the multiplicity of radially symmetric solutions u(x) to the Dirichlet problem

Δu+f(u)=h(x)+cφ(x)

on the unit ball Ω⊂R^N with boundary condition u=0 on ∂Ω. Here φ(x) is a positive function and f(u) is a function that is superlinear (but of subcritical growth) for large positive u, while for large negative u we have that f'(u)<μ, where μ is the smallest positive eigenvalue for Δψ+μψ=0 in Ω with ψ=0 on ∂Ω. It is shown that, given any integer k≥0, the value c may be chosen so large that there are 2k+1 solutions with k or less interior nodes. Existence of positive solutions is excluded for large enough values of c.

Knot Theory And Wild Knots, Cherie Annette Reardon

Theses Digitization Project

No abstract provided.

Can Mathematics Be Taught In A Christian Way?, Johan Deklerk

ACMS Journal 2004

Some people would argue that mathematics cannot be taught in a Christian way. Others would say it can be done, but that they are unsure how to do it, and therefore do not do it. Some methods mentioned from time to time will be briefly discussed in this address. The question will be asked whether these techniques bring one closer to a Christian perspective on mathematics. My view is that some of these methods are just not sufficient and that more should be done. More attention should be paid to the context in which a mathematical subject is imbedded. Attention …

Lewis Carroll: Author, Mathematician, And Christian, David L. Neuhouser

ACMS Journal 2004

Although a Christian, an author, and a mathematician, Charles Letwidge Dodgson (better known as Lewis Carroll) wrote very few works in which these three aspects of his person was present. The only examples of him merging these interests are in Sylvie and Bruno and Sylvie and Bruno Concluded. This paper will explore what motivated him to make these works and whether or not they were successful.

The Deconstruction Of Mathematics, David J. Stucki

ACMS Journal 2004

Mathematics, as an academic discipline, has stood for many years as the last bastion against a growing tide of intellectual relativism that has become all but ubiquitous. More recently, however, efforts have been made to "humanize" mathematics by advocating a social-constructivist approach to the philosophy of mathematics, both in practice and education. This paper is intended to serve as a critical response to one advocate of this approach, Reuben Hersh (What Is Mathematics, Really?, 1997), and in the process a defense of Platonism.

03. Preface Of Design And Analysis Of Experiments - 1st Edition, Angela Dean, Dan Voss, Danel Draguljic

Design and Analysis of Experiments

Preface of the first edition of Design and Analysis of Experiments.

Stability Of A Semilinear Cauchy Problem, Yi Liu, Yi Li, Yinbin Deng

Mathematics and Statistics Faculty Publications

A report of progress in linear and nonlinear partial differential equations, microlocal analysis, singular partial differential operators, spectral analysis and hyperfunction theory. The papers aretaken from a conference on partial differential equations and their applications, held in Wuhan.

Recurrence And Ergodicity Of Interacting Particle Systems, J. Theodore Cox, Achim Klenke

Mathematics - All Scholarship

Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of their ergodic invariant measures. The question arises whether a.s. the process eventually stays close to one of these ergodic states, or if it changes between the attainable ergodic states infinitely often ("recurrence"). Under the assumption that there exists a convergence--determining class of distributions that is (strongly) preserved under the dynamics, we show that the system is in fact recurrent in the above sense. We apply our method to several interacting particle systems, obtaining new or improved recurrence results. In addition, we answer …

Method For Identification Of Origins Of Replication And Genes Regulated By Dnaa In Bacteria, Olga G. Troyanskaya

Honors Theses

The study is focused on developing computer programs to identify origin of DNA replication based on analysis of total bacterial genomes, scoring regions for number of DnaA binding sites, AT content, DNA adenine methylase boxes, and integration host factors binding sites. The programs were tested on cyanobacterium Synechocystis, and several potential origins were identified. However, no one definite region could be located. Currently, software is being developed to analyze common motifs around the origins of all bacteria with known origins. Genes whose transcription could be regulated by DnaA were identified by searching for DnaA boxes preceding promoter regions.

The Topological Snake Lemma And Corona Algebras, Claude Schochet

Mathematics Faculty Research Publications

We establish versions of the Snake Lemma from homological algebra in the context of topological groups, Banach spaces, and operator algebras. We apply this tool to demonstrate that if ƒ : B → B′ is a quasi-unital C^*-map of separable C^*-algebras, so that it induces a map of Corona algebras ƒ̄ : QB → QB′, and if ƒ is mono, then the induced map ƒ̄ is also mono.

Cube Puzzles, Jeremiah Farrell

Scholarship and Professional Work - LAS

Jeremiah Farrell's contribution to "The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner"

Martin Gardner = Mint! Grand! Rare!, Jeremiah Farrell

Scholarship and Professional Work - LAS

Jeremiah Farrell's contirbution to "The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner"

Related works:

The puzzle from figure one is also available in Colloquy (Wordways 14,2) online at: http://digitalcommons.butler.edu/wordways/vol14/iss2/8/

Even Subgraphs Of A Graph, Hong-Jian Lai, Zhi-Hong Chen

Scholarship and Professional Work - LAS

No abstract provided.

Bounds On A Bug, Arthur T. Benjamin, Matthew T. Fluet '99

All HMC Faculty Publications and Research

In the game of Cootie, players race to construct a "cootie bug" by rolling a die to collect component parts. Each cootie bug is composed of a body, a head, two eyes, one nose, two antennae, and six legs. Players must first acquire the body of the bug by rolling a 1. Next, they must roll a 2 to add the head to the body. Once the body and head are both in place, the remaining body parts can be obtained in any order by rolling two 3s for the eyes, one 4 for the nose, two 5s for the …

The Bordalo Order On A Commutative Ring, Melvin Henriksen, Frank A. Smith

All HMC Faculty Publications and Research

If R is a commutative ring with identity and ≤ is defined by letting a ≤ b mean ab = a or a = b, then (R,≤) is a partially ordered ring. Necessary and sufficient conditions on R are given for (R,≤) to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings Z_n of integers mod n for n ≥ 2. In particular, if R is reduced, then (R,≤) is a lattice iff R is a weak Baer ring, and (R,≤) is a distributive lattice iff R …

An Inverse Function Theorem, Alfonso Castro, J. W. Neuberger

All HMC Faculty Publications and Research

In this note we present a local surjectivity result which is applicable to differential equations for which full boundary conditions may not be known. Our method uses continuous steepest descent and Sobolev gradients.

Invariant Measures For Set-Valued Dynamical Systems, Walter Miller, Ethan Akin

Mathematics and Statistics Faculty Research & Creative Works

A continuous map on a compact metric space, regarded as a dynamical system by iteration, admits invariant measures. For a closed relation on such a space, or, equivalently, an upper semicontinuous set-valued map, there are several concepts which extend this idea of invariance for a measure. We prove that four such are equivalent. In particular, such relation invariant measures arise as projections from shift invariant measures on the space of sample paths. There is a similarly close relationship between the ideas of chain recurrence for the set-valued system and for the shift on the sample path space. ©1999 American Mathematical …

A Monte Carlo Comparison Of Tests For Cointegration In Panel Data, Chihwa Kao

Center for Policy Research

This paper surveys recent developments and provides Monte Carlo comparison on various tests proposed for cointegration in panel data. In particular, tests for two panel models, varying intercepts and varying slopes, and varying intercepts and common slopes are presented from the literature with a total of seven tests being simulated. In all cases, results on empirical size and size-adjusted power are given.

On The Estimation And Inference Of A Cointegrated Regression In Panel Data, Chihwa Kao

Center for Policy Research

The main contribution of this paper is to add to the literature by suggesting a dynamic OLS (DOLS) estimator and providing a serious comparison of the finite sample properties of the OLS, fully modified OLS (FMOLS), and DOLS estimators in panel cointegrated regression models. Monte Carlo results illustrate the sampling behavior of the proposed estimators and show that (1) the OLS estimator has a non-negligible bias in finite samples, (2) the FMOLS estimator does not improve over the OLS estimator in general, and (3) the DOLS outperforms both the OLS and FMOLS estimators.