Algebra Commons^™

Descending Central Series Of Free Pro-P-Groups, German A. Combariza

Electronic Thesis and Dissertation Repository

In this thesis, we study the first three cohomology groups of the quotients of the descending central series of a free pro-p-group. We analyse the Lyndon-Hochschild- Serre spectral sequence up to degree three and develop what we believe is a new technique to compute the third cohomology group. Using Fox-Calculus we express the cocycles of a finite p-group G with coefficients on a certain module M as the kernel of a matrix composed by the derivatives of the relations of a minimal presentation for G. We also show a relation between free groups and finite fields, this is a new …

On Conjectures Concerning Nonassociate Factorizations, Jason A Laska

Doctoral Dissertations

We consider and solve some open conjectures on the asymptotic behavior of the number of different numbers of the nonassociate factorizations of prescribed minimal length for specific finite factorization domains. The asymptotic behavior will be classified for Cohen-Kaplansky domains in Chapter 1 and for domains of the form R=K+XF[X] for finite fields K and F in Chapter 2. A corollary of the main result in Chapter 3 will determine the asymptotic behavior for Krull domains with finite divisor class group.

Elasticity Of Krull Domains With Infinite Divisor Class Group, Benjamin Ryan Lynch

Doctoral Dissertations

The elasticity of a Krull domain R is equivalent to the elasticity of the block monoid B(G,S), where G is the divisor class group of R and S is the set of elements of G containing a height-one prime ideal of R. Therefore the elasticity of R can by studied using the divisor class group. In this dissertation, we will study infinite divisor class groups to determine the elasticity of the associated Krull domain. The results will focus on the divisor class groups Z, Z(p infinity), Q, and general infinite groups. For the groups Z and Z(p infinity), it has …

Assessing The Impact Of A Computer-Based College Algebra Course, Ningjun Ye

Dissertations

USM piloted the Math Zone in Spring 2007, a computer-based program in teaching MAT 101and MAT 099 in order to improve student performance. This research determined the effect of the re-design of MAT 101 on student achievements in comparison to a traditional approach to the same course. Meanwhile, the study investigated possible effects of the Math Zone program on students’ attitude toward studying mathematics.

This study shows that there was no statistically significant difference on MAT101 final exam scores between the Math Zone students and the Classroom students in Fall 2007, Spring 2008 and Fall 2008. At the same time, …

Analyzing Fractals, Kara Mesznik

Honors Capstone Projects - All

For my capstone project, I analyzed fractals. A fractal is a picture that is composed of smaller images of the larger picture. Each smaller picture is self- similar, meaning that each of these smaller pictures is actually the larger image just contracted in size through the use of the Contraction Mapping Theorem and shifted using linear and affine transformations.

Fractals live in something called a metric space. A metric space, denoted (X, d), is a space along with a distance formula used to measure the distance between elements in the space. When producing fractals …

Fractions Of Numerical Semigroups, Harold Justin Smith

Doctoral Dissertations

Let S and T be numerical semigroups and let k be a positive integer. We say that S is the quotient of T by k if an integer x belongs to S if and only if kx belongs to T. Given any integer k larger than 1 (resp., larger than 2), every numerical semigroup S is the quotient T/k of infinitely many symmetric (resp., pseudo-symmetric) numerical semigroups T by k. Related examples, probabilistic results, and applications to ring theory are shown.

Given an arbitrary positive integer k, it is not true in general that every numerical semigroup S is the …

On The Irreducibility Of The Cauchy-Mirimanoff Polynomials, Brian C. Irick

Doctoral Dissertations

The Cauchy-Mirimanoff Polynomials are a class of polynomials that naturally arise in various classical studies of Fermat's Last Theorem. Originally conjectured to be irreducible over 100 years ago, the irreducibility of the Cauchy-Mirimanoff polynomials is still an open conjecture.

This dissertation takes a new approach to the study of the Cauchy-Mirimanoff Polynomials. The reciprocal transform of a self-reciprocal polynomial is defined, and the reciprocal transforms of the Cauchy-Mirimanoff Polynomials are found and studied. Particular attention is given to the Cauchy-Mirimanoff Polynomials with index three times a power of a prime, and it is shown that the Cauchy-Mirimanoff Polynomials of index …