Groups And Semigroups Generated By Automata, 2011 University of Nebraska-Lincoln

#### Groups And Semigroups Generated By Automata, David Mccune

*Dissertations, Theses, and Student Research Papers in Mathematics*

In this dissertation we classify the metabelian groups arising from a restricted class of invertible synchronous automata over a binary alphabet. We give faithful, self-similar actions of Heisenberg groups and upper triangular matrix groups. We introduce a new class of semigroups given by a restricted class of asynchronous automata. We call these semigroups ``expanding automaton semigroups''. We show that this class strictly contains the class of automaton semigroups, and we show that the class of asynchronous automaton semigroups strictly contains the class of expanding automaton semigroups. We demonstrate that undecidability arises in the actions of expanding automaton semigroups and semigroups ...

Hilbert-Samuel And Hilbert-Kunz Functions Of Zero-Dimensional Ideals, 2011 University of Nebraska-Lincoln

#### Hilbert-Samuel And Hilbert-Kunz Functions Of Zero-Dimensional Ideals, Lori A. Mcdonnell

*Dissertations, Theses, and Student Research Papers in Mathematics*

The Hilbert-Samuel function measures the length of powers of a zero-dimensional ideal in a local ring. Samuel showed that over a local ring these lengths agree with a polynomial, called the Hilbert-Samuel polynomial, for sufficiently large powers of the ideal. We examine the coefficients of this polynomial in the case the ideal is generated by a system of parameters, focusing much of our attention on the second Hilbert coefficient. We also consider the Hilbert-Kunz function, which measures the length of Frobenius powers of an ideal in a ring of positive characteristic. In particular, we examine a conjecture of Watanabe and ...

Markov Bases For Noncommutative Harmonic Analysis Of Partially Ranked Data, 2011 Harvey Mudd College

#### Markov Bases For Noncommutative Harmonic Analysis Of Partially Ranked Data, Ann Johnston

*HMC Senior Theses*

Given the result $v_0$ of a survey and a nested collection of summary statistics that could be used to describe that result, it is natural to ask which of these summary statistics best describe $v_0$. In 1998 Diaconis and Sturmfels presented an approach for determining the conditional significance of a higher order statistic, after sampling a space conditioned on the value of a lower order statistic. Their approach involves the computation of a Markov basis, followed by the use of a Markov process with stationary hypergeometric distribution to generate a sample.This technique for data analysis has become an accepted ...

Homology Of Artinian Modules Over Commutative Noetherian Rings, 2011 University of Nebraska-Lincoln

#### Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer

*Dissertations, Theses, and Student Research Papers in Mathematics*

This work is primarily concerned with the study of artinian modules over commutative noetherian rings.

We start by showing that many of the properties of noetherian modules that make homological methods work seamlessly have analogous properties for artinian modules. We prove many of these properties using Matlis duality and a recent characterization of Matlis reflexive modules. Since Matlis reflexive modules are extensions of noetherian and artinian modules many of the properties that hold for artinian and noetherian modules naturally follow for Matlis reflexive modules and more generally for mini-max modules.

In the last chapter we prove that if the Betti ...

A Survey Of Modern Mathematical Cryptology, 2011 The University of Tennessee

#### A Survey Of Modern Mathematical Cryptology, Kenneth Jacobs

*University of Tennessee Honors Thesis Projects*

No abstract provided.

On The Homology Of Automorphism Groups Of Free Groups., 2011 University of Tennessee, Knoxville

#### On The Homology Of Automorphism Groups Of Free Groups., Jonathan Nathan Gray

*Doctoral Dissertations*

Following the work of Conant and Vogtmann on determining the homology of the group of outer automorphisms of a free group, a new nontrivial class in the rational homology of Outer space is established for the free group of rank eight. The methods started in [8] are heavily exploited and used to create a new graph complex called the space of good chord diagrams. This complex carries with it significant computational advantages in determining possible nontrivial homology classes.

Next, we create a basepointed version of the Lie operad and explore some of it proper- ties. In particular, we prove a ...

Characteristic Polynomial Of Arrangements And Multiarrangements, 2011 The University of Western Ontario

#### Characteristic Polynomial Of Arrangements And Multiarrangements, Mehdi Garrousian

*Electronic Thesis and Dissertation Repository*

This thesis is on algebraic and algebraic geometry aspects of complex hyperplane arrangements and multiarrangements. We start by examining the basic properties of the logarithmic modules of all orders such as their freeness, the cdga structure, the local properties and close the first chapter with a multiarrangement version of a theorem due to M. Mustata and H. Schenck.

In the next chapter, we obtain long exact sequences of the logarithmic modules of an arrangement and its deletion-restriction under the tame conditions. We observe how the tame conditions transfer between an arrangement and its deletion-restriction.

In chapter 3, we use some ...

Annihilators Of Local Cohomology Modules, 2011 University of Nebraska-Lincoln

#### Annihilators Of Local Cohomology Modules, Laura Lynch

*Dissertations, Theses, and Student Research Papers in Mathematics*

In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider the annihilators of local cohomology modules. We examine these annihilators at various cohomological degrees, in particular at the cohomological dimension and at the height or the grade of the defining ideal. We also investigate the dimension of these annihilators at various degrees and we refine our results by specializing to particular types of rings, for example, Cohen Macaulay rings, unique factorization domains, and rings of small dimension.

Adviser: Thomas Marley

Descent Systems, Eulerian Polynomials And Toric Varieties, 2011 The University of Western Ontario

#### Descent Systems, Eulerian Polynomials And Toric Varieties, Letitia Mihaela Golubitsky

*Electronic Thesis and Dissertation Repository*

It is well-known that the Eulerian polynomials, which count permutations in S_n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the Sn -orbit for a generic weight in the weight lattice of Sn . Therefore the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra. In this thesis we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a non-generic weight, namely a weight fixed by only the simple reflections ...

Formalizing Categorical And Algebraic Constructions In Operator Theory, 2011 University of Nebraska-Lincoln

#### Formalizing Categorical And Algebraic Constructions In Operator Theory, William Benjamin Grilliette

*Dissertations, Theses, and Student Research Papers in Mathematics*

In this work, I offer an alternative presentation theory for C*-algebras with applicability to various other normed structures. Specifically, the set of generators is equipped with a nonnegative-valued function which ensures existence of a C*-algebra for the presentation. This modification allows clear definitions of a "relation" for generators of a C*-algebra and utilization of classical algebraic tools, such as Tietze transformations.

Much More Than Symbolics: The Early History Of Algebra And Its Significance For Introductory Algebra Education, 2011 Dordt College

#### Much More Than Symbolics: The Early History Of Algebra And Its Significance For Introductory Algebra Education, Calvin Jongsma

*Faculty Work Comprehensive List*

No abstract provided.

Distribution Of Prime Numbers,Twin Primes And Goldbach Conjecture, 2011 SelectedWorks

#### Distribution Of Prime Numbers,Twin Primes And Goldbach Conjecture, Subhajit Kumar Ganguly

*Subhajit Kumar Ganguly*

The following paper deals with the distribution of prime numbers, the twin prime numbers and the Goldbach conjecture. Starting from the simple assertion that prime numbers are never even, a rule for the distribution of primes is arrived at. Following the same approach, the twin prime conjecture and the Goldbach conjecture are found to be true.

Classical Kloosterman Sums: Representation Theory, Magic Squares, And Ramanujan Multigraphs, 2011 South Dakota School of Mines and Technology

#### Classical Kloosterman Sums: Representation Theory, Magic Squares, And Ramanujan Multigraphs, Patrick S. Fleming, Stephan Ramon Garcia, Gizem Karaali

*Pomona Faculty Publications and Research*

We consider a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of “magical” combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.

Review: Massey Products On Cycles Of Projective Lines And Trigonometric Solutions Of The Yang-Baxter Equations, 2011 Pomona College

#### Review: Massey Products On Cycles Of Projective Lines And Trigonometric Solutions Of The Yang-Baxter Equations, Gizem Karaali

*Pomona Faculty Publications and Research*

No abstract provided.

Spatial Isomorphisms Of Algebras Of Truncated Toeplitz Operators, 2011 Pomona College

#### Spatial Isomorphisms Of Algebras Of Truncated Toeplitz Operators, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen

*Pomona Faculty Publications and Research*

We examine when two maximal abelian algebras in the truncated Toeplitz operators are spatially isomorphic. This builds upon recent work of N. Sedlock, who obtained a complete description of the maximal algebras of truncated Toeplitz operators.

Filtering Irreducible Clifford Supermodules, 2011 Bard College

#### Filtering Irreducible Clifford Supermodules, Julia C. Bennett

*Senior Projects Spring 2011*

A Clifford algebra is an associative algebra that generalizes the sequence R, C, H, etc. Filtrations are increasing chains of subspaces that respect the structure of the object they are filtering. In this paper, we filter ideals in Clifford algebras. These filtrations must also satisfy a “Clifford condition”, making them compatible with the algebra structure. We define a notion of equivalence between these filtered ideals and proceed to analyze the space of equivalence classes. We focus our attention on a specific class of filtrations, which we call principal filtrations. Principal filtrations are described by a single element in complex projective ...

Ken Kunen: Algebraist, 2011 University of Denver

#### Ken Kunen: Algebraist, Michael Kinyon

*Mathematics Preprint Series*

Ken Kunen is justifiably best known for his work in set theory and topology. What I would guess many of his friends and students in those areas do not know is that Ken also did important work in algebra, especially in quasigroup and loop theory. In fact, I think it is not an exaggeration to say that his work in loop theory, both alone and in collaboration, revolutionized the field. In this paper, I would like to describe some of his accomplishments to nonspecialists. My point of view is personal, of course, and so I will give the most attention ...

Discrete Quantum Processes, 2011 University of Denver

#### Discrete Quantum Processes, S. Gudder

*Mathematics Preprint Series*

A discrete quantum process is defined as a sequence of local states ρt , t = 0, 1, 2, . . ., satisfying certain conditions on an L2 Hilbert space H. If ρ = lim ρt exists, then ρ is called a global state for the system. In important cases, the global state does not exist and we must then work with the local states. In a natural way, the local states generate a sequence of quantum measures which in turn define a single quantum measure µ on the algebra of cylinder sets C. We consider the problem of extending µ to other physically relevant sets ...

Predictors Of Student Outcomes In Developmental Math At A Public Community And Technical College, 2011 Shawnee State University

#### Predictors Of Student Outcomes In Developmental Math At A Public Community And Technical College, Linda Darlene Hunt

*Theses, Dissertations and Capstones*

With the wide range of abilities of community college students, proper course placement is crucial. Therefore, having better predictors of success can help improve placement of students for their achievement. This study analyzed student predictors, instructor predictors, and classroom predictors in relation to student final exam score and student final grade in Elementary Algebra and Intermediate Algebra classes. Student predictors included gender, ACT math score, SAT math score, community college enrollment, math pretest score, and ASC grade. Instructor predictors included gender, employment status, Mozart music use, and ALEKS software use. Classroom predictors included time of day, number of class meetings ...

On The Betti Number Of Differential Modules, 2011 University of Nebraska-Lincoln

#### On The Betti Number Of Differential Modules, Justin Devries

*Dissertations, Theses, and Student Research Papers in Mathematics*

Let *R = k[x _{1}, ..., x_{n}]* with

*k*a field. A multi-graded differential

*R*-module is a multi-graded

*R*-module

*D*with an endomorphism

*d*such that

*d*= 0. This dissertation establishes a lower bound on the rank of such a differential module when the underlying

^{2}*R*-module is free. We define the Betti number of a differential module and use it to show that when the homology ker

*d*/im

*d*of

*D*is non-zero and finite dimensional over

*k*then there is an inequality rank

*≥ 2*

_{R}D^{n}. This relates to a problem of Buchsbaum ...