Mathematics Commons^™

Curve Interpolation And Coding Theory, Darren B. Glass

Math Faculty Publications

Whether it is downloading files from the Internet, having conversations between cell phones, or sending information from a laptop to a printer, we often want to transmit data in situations where we need to worry about interference from other signals that may cause errors in the transmission. The branch of mathematics known as coding theory is dedicated to finding ways to tell when these are errors in transmission and, when possible, how to correct those errors. The goal of coding theory is to build as much redundancy as possible into a message without greatly increasing its length. [excerpt]

Review: Nontangential Limits In Pt(Μ)-Spaces And The Index Of Invariant Subgroups, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Review: Common Cyclic Vectors For Unitary Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar

All HMC Faculty Publications and Research

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

The Marshall Differential Analyzer: A Visual Interpretation Of Mathematics, Bonita A. Lawrence, Richard P. Merritt, Devon A. Tivener

Undergraduate Mathematics Day: Past Content

Mechanical integration is an idea dating back to the late 1800's discovered by James Thomson, brother of Lord Kelvin. This idea was then expanded to build a calculating machine, called a differential analyzer, by Vannevar Bush (M.I.T) in 1929. The Marshall University Differential Analyzer Team has followed in the footsteps of Dr. Bush and a gentleman named Dr. Arthur Porter, who was the first to build a differential analyzer in England when he was a student of Dr. Douglas Hartree. He built his machine of Meccano components, the British version of Erector Set. In the early days of Arthur Porter's …

Lifting Module Maps Between Different Noncommutative Domain Algebras, Jonathan Von Stroh

Electronic Theses and Dissertations

The classical Carathéodory interpolation problem is the following: let n be a natural number, a₀, a₁, . . . , a_N be complex numbers, and D the unit disk. When does there exist an analytic function F : D → C and complex numbers a_N+1, a_N+2, . . . such that F(z) = a₀ + a₁z + a₂z² + . . . + a_Nz^N + a_N+1z^N+1 + . . . and ||F||_∞ < 1? In 1967, Sarason used operator theory techniques to give an elegant solution to the Carathéodory interpolation problem. In 1968, Sz.-Nagy and Foias extended Sarason's approach into a commutant lifting theorem. Both the theorem and the technique of the proof have become standard tools in control theory. In particular, the commutant lifting theorem approach lends itself to a wide range of generalizations. This thesis concerns one such generalization.

Arias presented generalizations of the original …

Snort: A Combinatorial Game, Keiko Kakihara

Theses Digitization Project

This paper focuses on the game Snort, which is a combinatorial game on graphs. This paper will explore the characteristics of opposability through examples. More fully, we obtain some neccessary conditions for a graph to be opposable. Since an opposable graph guarantees a second player win, we examine graphs that result in a first player win.

Symmetric Generators Of Order 3, Stewart Contreras

Theses Digitization Project

The main purpose of this project is to construct finite homomorphic images of infinite semi-direct products.

Symmetric Generation, Dung Hoang Tri

Theses Digitization Project

In this thesis we construct finite homorphic images of infinite semi-direct products, 2*n : N, where 2*n is a free product of n copies the cyclic group of permutations on n letter.

An Investigation Of Kurosh's Theorem, Keith Anthony Earl

Theses Digitization Project

The purpose of this project will be an exposition of the Kurosh Theorem and the necessary and suffcient condition that A must be algebraic and satisfy a P.I. to be locally finite.

U-Singularity And T-Topos Theoretic Entropy, Goro C. Kato

Mathematics

We will give descriptions of u-singularities as we introduce the notion of t-topos theoretic entropies. The unifying methodology for a u-singularity is the universal mapping property of an inverse or direct limit. The qualitative, conceptual, and structural analyses of u-singularities are given in terms of inverse and direct limits of micro decompositions of a presheaf and coverings of an object in t-site in the theory of temporal topos.

Gronwall-Ouiang-Type Integral Inequalities On Time Scales, Ailian Liu, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

We present several Gronwall-OuIang-type integral inequalities on time scales. Firstly, an OuIang inequality on time scales is discussed. Then we extend the Gronwall-type inequalities to multiple integrals. Some special cases of our results contain continuous Gronwall-type inequalities and their discrete analogues. Several examples are included to illustrate our results at the end.

Weyl-Titchmarsh Theory For Hamiltonian Dynamic Systems, Shurong Sun, Shaozhu Chen, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale T , which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for T= ℝ and T= ℤ within one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i) M(λ) theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for …

Urcohomologies And Cohomologies Of N -Complexes, Naoya Hiramatsu, Goro Kato

Mathematics

For a general sequence of objects and morphisms, we construct two N-complexes. Then we can define cohomologies (i, k)-type of the N-complexes not only on a diagonal region but also in the triangular region. We obtain an invariant defined on a general sequence of objects and morphisms. For a short exact sequence of N-complexes, we get the associated long exact sequence generalizing the classical long exact sequence. Lastly, several properties of the vanishing cohomologies of N-complexes are given.

Conformal Classification Of (K, Μ)-Contact Manifolds, Ramesh Sharma, Luc Vrancken

Mathematics Faculty Publications

First we improve a result of Tanno that says "If a conformal vector field on a contact metric manifold M is a strictly infinitesimal contact transformation, then it is an infinitesimal automorphism of M" by waiving the "strictness" in the hypothesis. Next, we prove that a (k, μ)-contact manifold admitting a non-Killing conformal vector field is either Sasakian or has k = –n – 1, μ = 1 in dimension > 3; and Sasakian or flat in dimension 3. In particular, we show that (i) among all compact simply connected (k, μ)-contact manifolds of dimension > 3, only the unit sphere S2n+1 …

Mathematical Biology At An Undergraduate Liberal Arts College, Stephen C. Adolph, Lisette G. De Pillis

All HMC Faculty Publications and Research

Since 2002 we have offered an undergraduate major in Mathematical Biology at Harvey Mudd College. The major was developed and is administered jointly by the mathematics and biology faculty. In this paper we describe the major, courses, and faculty and student research and discuss some of the challenges and opportunities we have experienced.

Stability And Dynamics Of Self-Similarity In Evolution Equations, Andrew J. Bernoff, Thomas P. Witelski

All HMC Faculty Publications and Research

A methodology for studying the linear stability of self-similar solutions is discussed. These fundamental ideas are illustrated on three prototype problems: a simple ODE with finite-time blow-up, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, the use of dimensional analysis to derive scaling invariant similarity variables is discussed, as well as the role of symmetries in the context of stability of self-similar dynamics. The …

Two-Player Envy-Free Multi-Cake Division, John Cloutier '03, Kathryn L. Nyman, Francis E. Su

All HMC Faculty Publications and Research

We introduce a generalized cake-cutting problem in which we seek to divide multiple cakes so that two players may get their most-preferred piece selections: a choice of one piece from each cake, allowing for the possibility of linked preferences over the cakes. For two players, we show that disjoint envy-free piece selections may not exist for two cakes cut into two pieces each, and they may not exist for three cakes cut into three pieces each. However, there do exist such divisions for two cakes cut into three pieces each, and for three cakes cut into four pieces each. The …

Review: The Spectrum Of Some Compressions Of Unilateral Shifts, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Unitary Equivalence To A Complex Symmetric Matrix: Geometric Criteria, Levon Balayan '09, Stephan Ramon Garcia

Pomona Faculty Publications and Research

We develop several methods, based on the geometric relationship between the eigenspaces of a matrix and its adjoint, for determining whether a square matrix having distinct eigenvalues is unitarily equivalent to a complex symmetric matrix. Equivalently, we characterize those matrices having distinct eigenvalues which lie in the unitary orbit of the complex symmetric matrices.

Review: Classification Of Quasi-Trigonometric Solutions Of The Classical Yang-Baxter Equation, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: Quantization Of Hamiltonian-Type Lie Algebras, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

The Norm Of A Truncated Toeplitz Operator, Stephan Ramon Garcia, William T. Ross

Pomona Faculty Publications and Research

We prove several lower bounds for the norm of a truncated Toeplitz operator and obtain a curious relationship between the H² and H^∞ norms of functions in model spaces.

The Riesz Representation Theorem For Linear Functionals, Thomas Daniel Schellhous

Theses Digitization Project

This study will investigate the Riesz representation theorem for linear functionals in relation to locally compact Hausdorff spaces. Two other theorems that are commonly called "Riesz representation theorem" are the theorem for finite-dimensional inner product spaces and the theorem for Hilbert spaces [BN00], and studying these interesting topics helps us to not only gain a better understanding of how linear functionals interact with vector spaces over which they are defined, but also to see faint threads that hint at a deep connection between the various fields of modern mathematics.

EΠi + 1=0: The History & Development, Dawne Charters-Nelson

Undergraduate Review

I have on occasion run across the equation in books, articles and in conversation with other mathematicians. In each of these encounters the person alluded to a fascination with this equation which links the five most important constants in the whole of analysis:

• 0 = The additive identity
• 1 = The multiplicative identity
• π = The circular constant
• e = The base of the natural logarithms
• i = The imaginary unit

Being a novice mathematician, I wondered how all these fundamental constants could end up in one equation and what it meant. Along with this thought came the realization that …

Counting The Number Of Locally Convex Topologies On A Totally Ordered Finiate Set, Thomas Tyler Clark

Mahurin Honors College Capstone Experience/Thesis Projects

We look at locally convex topologies on a totally ordered finite set. We determine a method of finding an upper bound on the number of such topologies on an n element. We show how this problem is related to Pascal’s Triangle and the Fibonacci Numbers. We explain an algorithm for determining the number of locally convex topologies consisting of nested intervals.

Derivatives Of The Dedekind Zeta Function Attached To A Complex Quadratic Field Extention, Nathan Salazar

Mahurin Honors College Capstone Experience/Thesis Projects

The Riemann Zeta Function is a function of vital importance in the study of number theory and other branches of mathematics. This is primarily due to its intrinsic link with the prime numbers of the ring of integers. The value of the Riemann Zeta Function at 0 and the values of the first few derivatives at 0 have been determined by various mathematicians. Apostol obtained a closed expression for the nth derivative of the Riemann Zeta Function at 0 that generalized previously known results. For higher derivatives, his result is useful for numerical computations. The Dedekind Zeta Function is a …

On A Semigroup Variety Of György Pollák, Edmond W. H. Lee

Mathematics Faculty Articles

Let P be the variety of semigroups defined by the identity xyzx = x². By a result of György Pollák, every subvariety of P is finitely based. The present article is concerned with subvarieties of P and the lattice they constitute, where the main result is a characterization of finitely generated subvarieties of P. It is shown that a subvariety of P is finitely generated if and only if it contains finitely many subvarieties, and the identities defining these varieties are described. Specifically, it is decidable when a finite set of identities defines a finitely generated subvariety …

Right Focal Boundary Value Problems For Difference Equations, Johnny Henderson, Xueyan Liu, Jeffrey W. Lyons, Jeffrey T. Neugebauer

Mathematics Faculty Articles

An application is made of a new Avery et al. fixed point theorem of compression and expansion functional type in the spirit of the original fixed point work of Leggett and Williams, to obtain positive solutions of the second order right focal discrete boundary value problem. In the application of the fixed point theorem, neither the entire lower nor entire upper boundary is required to be mapped inward or outward. A nontrivial example is also Provided.

Using Correlation Coefficients To Estimate Slopes In Multiple Linear Regression, Rudy Gideon

Mathematical Sciences Faculty Publications

This short note takes correlation coefficients as the starting point to obtain inferential results in linear regression. Under certain conditions, the population correlation coefficient and the sampling correlation coefficient can be related via a Taylor series expansion to allow inference on the coefficients in simple and multiple regression. This general method includes nonparametric correlation coefficients and so gives a universal way to develop regression methods. This work is part of a correlation estimation system that uses correlation coefficients to perform estimation in many settings, for example, time series, nonlinear and generalized linear models, and individual distributions.