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Stability And Reconstruction For An Inverse Problem For The Heat Equation, Kurt Bryan, Lester Caudill

Department of Math & Statistics Faculty Publications

We examine the inverse problem of determining the shape of some unknown portion of the boundary of a region Ω from measurements of the Cauchy data for solutions to the heat equation on Ω. By suitably linearizing the inverse problem we obtain uniqueness and continuous dependence results. We propose an algorithm for recovering estimates of the unknown portion of the surface and use the insight gained from a detailed analysis of the inverse problem to regularize the inversion. Several computational examples are presented.

Finding Cyclic Redundancy Check Polynomials For Multilevel Systems, James A. Davis, Miranda Mowbray, Simon Crouch

Department of Math & Statistics Faculty Publications

This letter describes a technique for finding cyclic redundancy check polynomials for systems for transmission over symmetric channels which encode information in multiple voltage levels, so that the resulting redundancy check gives good error protection and is efficient to implement. The codes which we construct have a Hamming distance of 3 or 4. We discuss a way to reduce burst error in parallel transmissions and some tricks for efficient implementation of the shift register for these polynomials. We illustrate our techniques by discussing a particular example where the number of levels is 9, but they are applicable in general.

Octary Codewords With Power Envelopes Of 3∗2M, Katherine M. Nieswand, Kara N. Wagner

Department of Math & Statistics Technical Report Series

This paper examines codewords of length 2^m in Z₈ with envelope power maxima of 3 ∗ 2^m. Using the general form for Golay pairs as a base, a general form is derived for the set of coset leaders that generate these codewords. From this general form it will be proven that there exists at least one element in the coset that achieves a power of 3 ∗ 2^m for each m-even and m-odd case.

New Semiregular Divisible Difference Sets, James A. Davis

Department of Math & Statistics Faculty Publications

We modify and generalize the construction by McFarland (1973) in two different ways to construct new semiregular divisible difference sets (DDSs) with λ₁≠0. The parameters of the DDS fall into a family of parameters found in Jungnickel (1982), where his construction is for divisible designs. The final section uses the idea of a K-matrix to find DDSs with a nonelementary abelian forbidden subgroup.

The Dirichlet Problem And Its Physical Motivations, Andrew E. Pitts

Honors Theses

In this work, we explore the basics of harmonic function theory and its relationship to problems in the theory of heat diffusion. In particular, we will focus on the classical Dirichlet problem.

Problems In Harmonic Function Theory, Ronald A. Walker

Honors Theses

Harmonic Function Theory is a field of differential mathematics that has both many theoretical constructs and physical connections, as well as its store of classical problems.

One such problem is the Dirichlet Problem. While the proof of the existence of a solution is well-founded on basic theory, and general methods for polynomial solutions have been well studied, much ground is still yet to be overturned. In this paper we focus on the examination, properties and computation methods and limitations, of solutions for rational boundary functions.

Another area that we shall study is the properties and generalizations of the zero sets …

The Set Of Hemispheres Containing A Closed Curve On The Sphere, Mary Kate Boggiano, Mark Desantis

Department of Math & Statistics Technical Report Series

Suppose you get in your car and take a drive on the sphere of radius R, so that when you return to your starting point the odometer indicates you've traveled less than 2πR. Does your path, γ, have to lie in some hemisphere?

This question was presented to us by Dr. Robert Foote of Wabash College. Previous authors chose two points, A and B, on γ such that these points divided γ into two arcs of equal length. Then they took the midpoint of the great circle arc joining A and B to be the North Pole and showed that …

New Families Of Semi-Regular Relative Difference Sets, James A. Davis, Jonathan Jedwab, Miranda Mowbray

Department of Math & Statistics Faculty Publications

We give two constructions for semi-regular relative difference sets (RDSs) in groups whose order is not a prime power, where the order u of the forbidden subgroup is greater than 2. No such RDSs were previously known. We use examples from the first construction to produce semi-regular RDSs in groups whose order can contain more than two distinct prime factors. For u greater than 2 these are the first such RDSs, and for u = 2 we obtain new examples.

Pseudocontinuations And The Backward Shift, William T. Ross, Alexandru Aleman, Stefan Richter

Department of Math & Statistics Faculty Publications

In this paper, we will examine the backward shift operator Lf = (f −f(0))/z on certain Banach spaces of analytic functions on the open unit disk D. In particular, for a (closed) subspace M for which LM Ϲ M, we wish to determine the spectrum, the point spectrum, and the approximate point spectrum of L│M. In order to do this, we will use the concept of “pseudocontinuation" of functions across the unit circle T.

We will first discuss the backward shift on a general Banach space of analytic functions and then for the weighted …

Hadamard Difference Sets In Nonabelian 2-Groups With High Exponent, James A. Davis, Joel E. Iiams

Department of Math & Statistics Faculty Publications

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Hadamard difference sets. In the abelian case, a group of order 2^{2t + 2} has a difference set if and only if the exponent of the group is less than or equal to 2^{t + 2}. In a previous work (R. A. Liebler and K. W. Smith, in “Coding Theory, Design Theory, Group Theory: Proc. of the Marshall Hall Conf.,” Wiley, New York, 1992), the authors constructed a difference set in a nonabelian group of order …

[Introduction To] Schaum's Outlines Fundamentals Of Computing With C++, John R. Hubbard

Bookshelf

This book is intended to be used primarily for self study, preferably in conjunction with a regular course in the fundamentals of computer science using the new ANSI/ISO Standard C++. The book covers topics from the fundamental units of the 1991 A.C.M. computing curricula.