Physical Sciences and Mathematics Commons^™

Lasalle's Invariance Principle On Measure Chains, Anders Floor '00

Honors Projects

It was in 1892 that Lyapunov published his paper giving his "second method". The basic guiding principle was that we might be able to know something about the stability of the system from the form of the equations describing it. Specifically, the idea was that it would not be necessary to know the solutions of the equations involved. This is of course very useful since in most cases solutions are extremely difficult or even impossible to find. Lyapunov's insight was that if a function could be found with, among other properties, a negative rate of change along the solution of …

A Survey Of The Taniyama-Shimura Conjecture, Kady Schneiter

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Perhaps the most famous problem in all of mathematics is the theorem that states that the equation aⁿ + bⁿ = cⁿ has no non-trivial solutions for integers a, b, and c, and n ≥ 2. This theorem was proposed by a seventeenth century French mathematician named Pierre de Fermat. Though the theorem is easy to understand, the proof has been elusive. Over the past 350 years many mathematicians have attempted to prove Fermat's theorem. They have used a variety of methods and many have been successful in proving the theorem in specific cases. …

An Optimal-Order Error Estimate For An Ellam Scheme For Two-Dimensional Linear Advection-Diffusion Equations, Hong Wang

Faculty Publications

An Eulerian-Lagrangian localized adjoint method (ELLAM) is presented and an- alyzed for two-dimensional linear advection-diffusion partial differential equations (PDEs). An optimal-order error estimate in the L^2 norm and a superconvergence estimate in a discrete H^1 norm are derived. Numerical experiments are performed to verify the theoretical estimates.

A Census Of Rational Maps, Eva Brezin, Rosemary Byrne, Joshua Levy, Kevin M. Pilgrim, Kelly Plummer

Mathematics and Statistics Faculty Research & Creative Works

We discuss the general combinatorial, topological, algebraic, and dynamical issues underlying the enumeration of postcritically finite rational functions, regarded as holomorphic dynamical systems on the Riemann sphere. We present findings from our creation of a census of all degree two and three hyperbolic nonpolynomial maps with four or fewer postcritical points. Our data is tabulated in detail at. © 1999 American Mathematical Society.

Knot Factoring, Michael C. Sullivan

Articles and Preprints

A knot is just a closed loop in a three-dimensional space. Strangely enough, there is a notion of factoring knots and even a prime factoring theorem, due to Horst Schubert. We hope that our presentation of this classic result will be accessible to advanced undergraduates.

Math Induction, Lawrence Mark Lesser

Humanistic Mathematics Network Journal

No abstract provided.

What We Say, What Our Students Hear: A Case For Active Listening, Dorothy Buerk

Humanistic Mathematics Network Journal

No abstract provided.

The Word Problem And The Child, Kenneth J. Preskenis

Humanistic Mathematics Network Journal

No abstract provided.

Written To Me Upon Getting A B In Linear Algebra, Sandra Z. Keith

Humanistic Mathematics Network Journal

No abstract provided.

The Need For Interviews In The Mathematics Classroom, Emam Hoosain

Humanistic Mathematics Network Journal

No abstract provided.

A Glorious Constant, Ze'ev Barel

Humanistic Mathematics Network Journal

No abstract provided.

Using Environmental News To Help Teach Mathematics, Barry Schiller

Humanistic Mathematics Network Journal

No abstract provided.

Book Review: Women In Mathematics By Claudia Henrion, Natasha Keith, Sandra Z. Keith

Humanistic Mathematics Network Journal

No abstract provided.

A Tribute To Ramanujan, Mahesh Dube

Humanistic Mathematics Network Journal

No abstract provided.

Use Your Head: Mathematics As Therapy, Miriam Lipschutz-Yevick

Humanistic Mathematics Network Journal

No abstract provided.

Coherence In Theories Relating Mathematics And Language, Carl Winsløw

Humanistic Mathematics Network Journal

No abstract provided.

Operationalizing Interactive Learning Paradigms Through Cooperative Learning Activities 100% Of The Time In Math Classes, Ted Panitz

Humanistic Mathematics Network Journal

No abstract provided.

Mathematics And Sex, Yan Kow Cheong

Humanistic Mathematics Network Journal

No abstract provided.

Imaginary, Lawrence Mark Lesser

Humanistic Mathematics Network Journal

No abstract provided.

Counting On Continued Fractions, Arthur T. Benjamin, Francis E. Su, Jennifer J. Quinn

All HMC Faculty Publications and Research

No abstract provided in this article.

Cramer-Rao Bound And Optimal Amplitude Estimator Of Superimposed Sinusoidal Signals With Unknown Frequencies, Shaohui Jia

Doctoral Dissertations

This dissertation addresses optimally estimating the amplitudes of superimposed sinusoidal signals with unknown frequencies. The Cramer-Rao Bound of estimating the amplitudes in white Gaussian noise is given, and the maximum likelihood estimator of the amplitudes in this case is shown to be asymptotically efficient at high signal to noise ratio but finite sample size. Applying the theoretical results to signal resolutions, it is shown that the optimal resolution of multiple signals using a finite sample is given by the maximum likelihood estimator of the amplitudes of signals.

Explicit Resolutions Of Cubic Cusp Singularities, Helen G. Grundman

Mathematics Faculty Research and Scholarship

Resolutions of cusp singularities are crucial to many techniques in computational number theory, and therefore finding explicit resolutions of these singularities has been the focus of a great deal of research. This paper presents an implementation of a sequence of algorithms leading to explicit resolutions of cusp singularities arising from totally real cubic number fields. As an example, the implementation is used to compute values of partial seta functions associated to these cusps.

Computational Geometry Column 38, Joseph O'Rourke

Computer Science: Faculty Publications

Recent results on curve reconstruction are described.

Molecular Study Of Turbulence In Three Dimensional Cavity Flow, Donald Greenspan

Mathematics Technical Papers

Three dimensional molecular cavity problems are formulated and solved numerically. The fluid considered is water at 15°C. The turbulent flows generated are characterized by strong crosscurrents over the usual primary vortex direction.

The Fermat Point And The Steiner Problem, J. N. Boyd, P. N. Raychowdhury

Virginia Journal of Science

We revisit the convex coordinates of the Fermat point of a triangle. We have already computed these convex coordinates in a general setting. In this note, we obtain the coordinates in the context of the Steiner problem. Thereafter, we pursue calculations suggested by the problem.

An Object-Oriented Algorithmic Laboratory For Ordering Sparse Matrices, Gary Karl Kumfert

Computer Science Theses & Dissertations

We focus on two known NP-hard problems that have applications in sparse matrix computations: the envelope/wavefront reduction problem and the fill reduction problem. Envelope/wavefront reducing orderings have a wide range of applications including profile and frontal solvers, incomplete factorization preconditioning, graph reordering for cache performance, gene sequencing, and spatial databases. Fill reducing orderings are generally limited to—but an inextricable part of—sparse matrix factorization.

Our major contribution to this field is the design of new and improved heuristics for these NP-hard problems and their efficient implementation in a robust, cross-platform, object-oriented software package. In this body of research, we (1) examine …

Diffusion Problems In Wound Healing And A Scattering Approach To Immune System Interactions, Julia Suzanne Arnold

Mathematics & Statistics Theses & Dissertations

A theoretical model for the existence of a Critical Size Defect (CSD) in certain animals is the focus of the majority of this dissertation. Adam [1] recently developed a one-dimensional model of this phenomenon, and chapters I–V address the exist the CSD in a two-dimensional model and a three-dimensional model. The two dimensional (or 1-d circular) model is the more appropriate for a study of CSD's. In that model we assume a circular wound of uniform depth and develop a time-independent form of the diffusion equation relevant to the study of the CSD phenomenon. It transpires that the range of …

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.

Essays On First Best Implementable Incentive Problem., Manipushpak Mitra Dr.

Doctoral Theses

The theory of mechanism design originated in the mid 1930s with the work of Lange[27], Lerner (28] and Hayek [20] on market socialism'. Further regular was added to their ideas by Arrow and Hurwicz (1). Hurwicz extended them to the general problem of mechanism design. An important aspect of mechanism design is asymmetric information. Information asymmetry typically imposes constraints on the goals which can be attained. For example. in the classic pure public goods problem, mechanisms that achieve truthful revelation of private information are Pareto sub-optimal i.e. these mechanisms lead to a welfare loss (see Hurwicz [24]). When can mechanism …

Feature Evaluation, Classification And Rule Generation Using Fuzzy Sets And Neural Networks., Rajat Kumar De Dr.

Doctoral Theses

Pattern recognition and machine learning form a major area of research and develop- ment activity that encompasses the processing of pictorial and other non-numerical information obtained from the interaction between science, technology and society. A motivation for the spurt of activity in this field is the need for people to com- municate with the computing machines in their natural mode of communication. Another important motivation is that the scientists are also concerned with the idea of designing and making intelligent machines that can carry out certain tasks that we human beings do. The most salient outcome of these is the …