Mathematics Commons^™

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.

Matching Functions And Graphs At Multiple Levels Of Bloom’S Revised Taxonomy, Kris H. Green

Mathematical and Computing Sciences Faculty/Staff Publications

This paper illustrates the power of Bloom's revised taxonomy for teaching, learning and assessing [3] in aligning our curriculum expectations and our assessment tools in multivariable calculus. The particular assessment tool considered involves a common matching problem to evaluate students' abilities to think about functions from graphical and formulaic representations. Through this analysis we gain additional understanding of why students may have difficulty in performing well on certain activities.

Some New Classes Of Complex Symmetric Operators, Stephan Ramon Garcia, Warren R. Wogen

Pomona Faculty Publications and Research

We say that an operator $T \in B(H)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:H\to H$ so that $T = CT^*C$. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data $(\dim \ker T, \dim \ker T^*)$.

Review: Intertwining Symmetry Algebras Of Quantum Superintegrable Systems, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: The Semi-Dynamical Reflection Equation: Solutions And Structure Matrices, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

A Vision For Acms, James Bradley

ACMS Journal 2010-2011

This paper presents a vision for the mission of the ACMS. It considers some plausible connections between mathematics and theology as well as some big questions that Christian mathematicians might fruitfully explore. It examines the current cultural norms of the mathematics guild, why these are problematic for Christians, and some possible ways the ACMS might respond.

When Does A Category Built On A Lattice With A Monoidal Structure Have A Monoidal Structure?, Lawrence Stout

Scholarship

In a word, sometimes. And it gets harder if the structure on L is not commutative. In this paper we consider the question of what properties are needed on the lattice L equipped with an operation * for several different kinds of categories built using Sets and L to have monoidal and monoidal closed structures. This works best for the Goguen category Set(L) in which membership, but not equality, is made fuzzy and maps respect membership. Commutativity becomes critical if we make the equality fuzzy as well. This can be done several ways, so a progression of categories is considered. …

Traveling Wave Solutions For A Nonlocal Reaction-Diffusion Model Of Influenza A Drift, Joaquin Riviera, Yi Li

Mathematics and Statistics Faculty Publications

In this paper we discuss the existence of traveling wave solutions for a nonlocal reaction-diffusion model of Influenza A proposed in Lin et. al. (2003). The proof for the existence of the traveling wave takes advantage of the different time scales between the evolution of the disease and the progress of the disease in the population. Under this framework we are able to use the techniques from geometric singular perturbation theory to prove the existence of the traveling wave.

Quantum Geons And Noncommutative Spacetimes, A. P. Balachandran, A. Ibort, G. Marmo, M. Martone

Physics - All Scholarship

Physical considerations strongly indicate that spacetime at Planck scales is noncommutative. A popular model for such a spacetime is the Moyal plane. The Poincare group algebra acts on it with a Drinfel'd-twisted coproduct. But the latter is not appropriate for more complicated spacetimes such as those containing the Friedman-Sorkin (topological) geons. They have rich diffeomorphism groups and in particular mapping class groups, so that the statistics groups for N identical geons is strikingly different from the permutation group SN. We generalise the Drinfel'd twist to (essentially) generic groups including to finite and discrete ones and use it to modify the …

Mathematics In Motion: Linear Systems Of Differential Equations On The Differential Analyzer, Devon A. Tivener

Theses, Dissertations and Capstones

In this work, I will provide an introduction to the dierential analyzer, a machine designed to solve dierential equations through a process called mechanical integration. I will give a brief historical account of dierential analyzers of the past, and discuss the Marshall University Dierential Analyzer Project. The goal of this work is to provide an analysis of solutions of systems of dierential equations using a dierential analyzer. In particular, we are interested in the points at which these systems are in equilibrium and the behavior of solutions that start away from equilibrium. After giving a description of linear systems of …