Mathematics

The following dissertation investigates algebraic frames and their spaces of minimal prime elements with respect to the Hull-Kernel topology and Inverse topology. Much work by other authors has been done in obtaining internal characterizations in frame-theoretic terms for when these spaces satisfy certain topological properties, but most of what is done is under the auspices of the finite intersection property. In the first half of this dissertation, we shall add to the literature more characterizations in this context, and in the second half we will study general algebraic frames and investigate which, if any, of the known theorems generalize to algebraic frames not necessarily with the FIP.
Throughout this investigative journey, we have found that certain ideals and filters of algebraic frames play a pivotal role in determining internal characterizations of the algebraic frames for when interesting topological properties occur in its space of minimal prime elements. In this dissertation, we investigate completely prime filters and compactly generated filters on algebraic frames. We introduce a new concept of subcompact elements and subcompactly generated filters. One of our main results is that the inverse topology on the space of minimal prime elements is compact if and only if every maximal subcompactly generated filter is completely prime. Furthermore, when the space of minimal prime elements is compact, then each minimal prime has what we are calling the compact absoluteness property.

In this report we study the Aizawa field by first computing a Taylor series
expansion for the solution of an initial value problem. We then look for singularities
(equilibrium points) of the field and plot the set of solutions which lie in the linear
subspace spanned by the eigenvectors. Finally, we use the Parameterization Method
to compute one and two dimensional stable and unstable manifolds of equilibria for
the system.

Discussion begins with a modular method for determining cycle types of permutations in the Galois group of a given separable irreducible polynomial over Q. As the Galois group is a transitive permutation group on the n roots of its irreducible polynomial, a list of all transitive groups of degree n, together with the cycle type distributions of each group, allows a probablistic determination of the group in a process of elimination. In the case of prime degree, transitive groups are primitive and by the O'Nan-Scott theorem are of restricted form. Theory is presented in order to arrive at these results and others, so that in conjunction with the classification theorem on finite simple groups, it is possible (in principle) to list all primitive permutation groups of particular prime degree. The case of degree 17 is examined to obtain a list of the transitive permutation groups of degree seventeen, as well as the cycle type of distributions of the groups identified.

This thesis is based on the paper "Polynomial Structures in Order Statistics Distributions" by M. Denuit, Cl. Lefevre and Ph. Picard. We study the exact order distributions for i.i.d. random variables with arbitrary common law. The left tail distributions can be written as Abel-Gontcharoff polynomials and the right tail distributions can be expressed by Appell polynomials. The polynomial structure makes it easier to obtain closed forms and recursive methods for evaluating the distribution of frequently occurring statistics related to empirical distribution functions.

We study the decay in time of solutions of Schrodinger equations of the type du/du=idelta u+iV(t)u, establishing that for small potentials and initial data in L1 the solution u satisfies sup[u(x,t)](x element of R)<const * t^-n/2. In the process we also develop a number of results on operators of evolution; i.e., on the existence of solutions of the abstract initial value problem du/dt=A(t)u,u(0)=u0 where u0 element of X, X a Banach space, A(t) an operator in X for each t, and the
solution is an X-valued function u.

The relations between complete and $\sigma$-complete covers of a Boolean algebra are examined. The Dedekind completion of a Boolean algebra is shown to be a quotient of any complete cover. Atoms of a Boolean algebra correspond to atoms of the Dedekind completion hence the Dedekind completion of an atomic Boolean algebra is isomorphic to the power set of the set of all atoms. There exists a correspondence between complete (sigma-complete) homomorphisms and full (sigma-complete) ideals. The explicit form of the Dedekind completion is given for the Boolean algebra generated by all semiopen subintervals of [0,1) as the atomless, complete Boolean algebra of all regularly closed subsets of [0,1). A compatible topology for a Boolean algebra is a topology for which addition and multiplication are continuous. The properties concerning products, quotients, subspaces and uniform completions of topological Boolean algebras are examined. Compact algebras are isomorphic and homeomorphic with power sets, endowed with the product topology. Measure algebras endowed with the weak* topology are compatible if and only if the underlying measure is purely atomic. A new proof of Stone Representation Theorem for a field of sets is given, providing a tool for establishing relations between Stone representation spaces of algebras, covers, subalgebras and quotients.

This dissertation discusses umbral calculus and lattice path enumeration and then continues by explicitly enumerating weighted directed lattice paths staying above a boundary using finite operator calculus. In Part I we discuss the history and representative results of the two topics. We separate umbral calculus into two fields, classical umbral calculus and finite operator calculus, and attempt to correct their intertwined histories. We discuss the beginnings of lattice path enumeration and survey the types of lattice path enumeration problems and solution methods found in the literature. In Part II, we give necessary conditions of a step set or of its equivalent operator equation such that the path count functions coincide with Sheffer polynomials where the path counts are nonzero. We derive the polynomials from an expansion theorem that includes a polynomial basis and initial conditions. The polynomial basis is derived from a known basic sequence with a transfer formula and a linear operator equation based on the step set. The initial conditions are functionals on the polynomials designed to vanish when evaluated along the boundary line for all but finitely many values. We solve lattice path enumeration problems with four types of boundary conditions and various step sets. We work out general solutions for paths that stay in the first quadrant, paths that stay in the first quadrant and above a line with an integer slope, and paths that can reach the boundary with an additional privileged access step set. We count the number of paths, and in one example we count the paths refined by the number of times they contact the boundary. We explore step sets including a general three-element step set, weighted finite step sets, weighted infinite step sets, and step sets that include paths as steps called pathlets. We research if our methods still give explicit solutions as we complicate and expand the step sets. The example sections include fourteen explicitly worked out problems. Part II of the dissertation includes and extends the three papers on the subject by Humphreys and Niederhausen written between 2000 and 2004.

This dissertation has two chapters. In the first chapter we talk about the discrete logarithm problem, more specifically we concentrate on the Diffie-Hellman key exchange protocol. We survey the current state of security for the Diffie-Hellman key exchange protocol. We also motivate the reader to think about the Diffie-Hellman key exchange in terms of group automorphisms. In the second chapter we study two key exchange protocols similar to the Diffie-Hellman key exchange protocol using an abelian subgroup of the automorphism group of a non-abelian group. We also generalize group no. 92 of the Hall-Senior table, for arbitrary prime p and study the automorphism group of these generalized group. We show that for those groups, the group of central automorphisms is an abelian group. We use these central automorphisms for the key exchange we are studying. We also develop a signature scheme.

The problem of this experiment was to test and compare
the effects of an audio-tutorial method and a traditional
lecture method of instruction of Intermediate College
Algebra at Broward Community College, and to determine how
the commercially-prepared and teacher-made audio-tutorial
materials should be revised or altered for the improvement
of the instruction of Intermediate College Algebra.
The null hypotheses were based on the assumption
that if significant initial differences in intellectual
aptitude as measured by the Otis Mental Abilities Test,
Form Am, existed between the groups, these differences
would be adjusted with an Otis covariate. The Otis analysis revealed no significant differences,
at the .05 level, between groups in mental ability. The seven null hypotheses were tested, as a part of the overall
analysis of the study. The interpretation of analysis results
lead to the rejection at the .05 level of the first
hypothesis. (There is no significant difference in the
mathematics achievement of the audio-tutorial and traditionally
taught students.) The other hypotheses, two
through seven, were accepted.
As a result of the rejection of only the first
hypothesis and an investigation of the linear trends of the
pre- and post-test results it was concluded that the audiotutorial
instructional method was superior, under the defined
parameters, to the traditional method.
It was concluded from the results of the attitude
questionnaire that the audio-tutorial instructional approach
had a positive effect on the students' attitude toward the
course.

The purpose of this study was to examine the relationship between teacher retention and student achievement as measured by the Florida Comprehensive Achievement Test (FCAT) Math Developmental Scale Scores (DSS). This study examined the impact of teacher transience on high school student math scores over a three-year period and considered the effect of teacher years of experience in relation to transience and achievement. For the purposes of this study teachers were identified into the following four classifications: Stayers, Within District Movers, Cross District Movers, or Beginning teachers. The findings indicated that students of beginning teachers scored significantly lower on the ninth grade math test than students of teachers in the other three classifications. At the 10th grade level there was no significant difference among the teacher transience groups. Based upon the findings, the following conclusion resulted from the study. Since an analysis of the data indicated that teacher retention is likely to improve ninth grade student score gains on the FCAT Math assessment, it is recommended that High School administrators carefully review the teaching assignments of ninth grade math teachers, especially in this era of high stakes testing and accountability.