Completeness Of Interacting Particles, 2011 Dublin Institute of Technology

#### Completeness Of Interacting Particles, Pavel Abramski

*Doctoral*

This thesis concerns the completeness of scattering states of n _-interacting particles in one dimension. Only the repulsive case is treated, where thereare no bound states and the spectrum is entirely absolutely continuous, so the scattering Hilbert space is the whole of L2(Rn). The thesis consists of 4 chapters: The first chapter describes the model, the scattering states as given by the Bethe Ansatz, and the main completeness problem. The second chapter contains the proof of the completeness relation in the case of two particles: n = 2. This case had in fact already been treated by B. Smit (1997 ...

The Eigenvalues Of A Tridiagonal Matrix In Biogeography, 2011 BMI Research

#### The Eigenvalues Of A Tridiagonal Matrix In Biogeography, Boris Igelnik, Daniel J. Simon

*Electrical Engineering & Computer Science Faculty Publications*

We derive the eigenvalues of a tridiagonal matrix with a special structure. A conjecture about the eigenvalues was presented in a previous paper, and here we prove the conjecture. The matrix structure that we consider has applications in biogeography theory.

The Square Discrete Exponentiation Map, 2011 DePaul University

#### The Square Discrete Exponentiation Map, A Wood

*Mathematical Sciences Technical Reports (MSTR)*

We will examine the square discrete exponentiation map and its properties. The square discrete exponentiation map is a variation on a commonly seen problem in cryptographic algorithms. This paper focuses on understanding the underlying structure of the functional graphs generated by this map. Specifically, this paper focuses on explaining the in-degree of graphs of safe primes, which are primes of the form p = 2q + 1, where q is also prime.

Base-Free Formulas In The Lattice-Theoretic Study Of Compacta, 2011 Marquette University

#### Base-Free Formulas In The Lattice-Theoretic Study Of Compacta, Paul Bankston

*Mathematics, Statistics and Computer Science Faculty Research and Publications*

The languages of finitary and infinitary logic over the alphabet of bounded lattices have proven to be of considerable use in the study of compacta. Significant among the sentences of these languages are the ones that are *base free*, those whose truth is unchanged when we move among the lattice bases of a compactum. In this paper we define syntactically the *expansive* sentences, and show each of them to be base free. We also show that many well-known properties of compacta may be expressed using expansive sentences; and that any property so expressible is closed under inverse limits and co-existential ...

The Derived Category And The Singularity Category, 2011 University of Pennsylvania

#### The Derived Category And The Singularity Category, Mehmet U. Isik

*Publicly Accessible Penn Dissertations*

We prove an equivalence between the derived category of a variety and the equivari- ant/graded singularity category of a corresponding singular variety. The equivalence also holds at the dg level.

Algebraic Solutions To Overdefined Systems With Applications To Cryptanalysis, 2011 Rose-Hulman Institute of Technology

#### Algebraic Solutions To Overdefined Systems With Applications To Cryptanalysis, Eric Crockett

*Mathematical Sciences Technical Reports (MSTR)*

Cryptographic algorithms are based on a wide variety of difficult problems in mathematics. One of these problems is finding a solution to a system of multivariate quadratic equations (MQ). A generalization of this problem is to find a solution to a system of higher order non-linear equations. Both of these problems are NP-hard over any field. Many cryptosystems such as AES, Serpent, Toyocrypt, and others can be reduced to some form of the MQ problem. In this paper we analyze the relinearization and XL algorithms for solving overdetermined systems of non-linear equations, as well as two variations of the XL ...

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 ...

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 ...

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 ...

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.

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 ...

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 ...