Physical Sciences and Mathematics Commons^™

The Cohomology Of Modules Over A Complete Intersection Ring, Jesse Burke

Department of Mathematics: Dissertations, Theses, and Student Research

We investigate the cohomology of modules over commutative complete intersection rings. The first main result is that if M is an arbitrary module over a complete intersection ring R, and if one even self-extension module of M vanishes then M has finite projective dimension. The second main result gives a new proof of the fact that the support variety of a Cohen-Macaulay module whose completion is indecomposable is projectively connected.

Applications Of Linear Programming To Coding Theory, Nathan Axvig

Department of Mathematics: Dissertations, Theses, and Student Research

Maximum-likelihood decoding is often the optimal decoding rule one can use, but it is very costly to implement in a general setting. Much effort has therefore been dedicated to find efficient decoding algorithms that either achieve or approximate the error-correcting performance of the maximum-likelihood decoder. This dissertation examines two approaches to this problem.

In 2003 Feldman and his collaborators defined the linear programming decoder, which operates by solving a linear programming relaxation of the maximum-likelihood decoding problem. As with many modern decoding algorithms, is possible for the linear programming decoder to output vectors that do not correspond to codewords; such …

Vanishing Of Ext And Tor Over Complete Intersections, Olgur Celikbas

Department of Mathematics: Dissertations, Theses, and Student Research

Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of …

Bass’ Nk Groups And Cd H-Fibrant Hochschild Homology, G. Cortiñas, C. Haesemeyer, Mark E. Walker, C. Weibel

Department of Mathematics: Faculty Publications

The K-theory of a polynomial ring R[t ] contains the K-theory of R as a summand. For R commutative and containing Q, we describe K∗(R[t ])/K∗(R) in terms of Hochschild homology and the cohomology of Kähler differentials for the cdh topology.

We use this to address Bass’ question, whether K_n(R) = K_n(R[t ]) implies K_n(R) = K_n(R[t₁, t₂]). The answer to this question is affirmative when R is essentially of …

Generalized Fourier-Feynman Transforms, Convolution Products, And First Variations On Function Space, Seung Jun Chang, Jae Gil Choi, David Skough

Department of Mathematics: Faculty Publications

In this paper we examine the various relationships that exist among the first variation, the convolution product and the Fourier-Feynman transform for functionals of the form F(x) = f((α₁, x), . . . , (α_n, x)) with x in a very general function space C_a,b[0,T].

Properties Of The Generalized Laplace Transform And Transport Partial Dynamic Equation On Time Scales, Chris R. Ahrendt

Department of Mathematics: Dissertations, Theses, and Student Research

In this dissertation, we first focus on the generalized Laplace transform on time scales. We prove several properties of the generalized exponential function which will allow us to explore some of the fundamental properties of the Laplace transform. We then give a description of the region in the complex plane for which the improper integral in the definition of the Laplace transform converges, and how this region is affected by the time scale in question. Conditions under which the Laplace transform of a power series can be computed term-by-term are given. We develop a formula for the Laplace transform for …