Digital Commons Network^™

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.

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.

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.

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 …

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 …