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 ...

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.

Twisted Virtual Biracks, 2011 Claremont McKenna College

#### Twisted Virtual Biracks, Jessica Ceniceros

*CMC Senior Theses*

This thesis will take a look at a branch of topology called knot theory. We will first look at what started the study of this field, classical knot theory. Knot invariants such as the Bracket polynomial and the Jones polynomial will be introduced and studied. We will then explore racks and biracks along with the axioms obtained from the Reidemeister moves. We will then move on to generalize classical knot theory to what is now known as virtual knot theory which was first introduced by Louis Kauffman. Finally, we take a look at a newer aspect of knot theory, twisted ...

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.

The Mathematical Landscape, 2011 Claremont McKenna College

#### The Mathematical Landscape, Antonio Collazo

*CMC Senior Theses*

The intent of this paper is to present the reader will enough information to spark a curiosity in to the subject. By no means is the following a complete formulation of any of the topics covered. I want to give the reader a tour of the mathematical landscape. There are plenty of further details to explore in each section, I have just touched the tip the iceberg. The work is basically in four sections: Numbers, Geometry, Functions, Sets and Logic, which are the basic building blocks of Math. The first sections are a exposition into the mathematical objects and their ...

A Class Of Gaussian Processes With Fractional Spectral Measures, 2011 Chapman University

#### A Class Of Gaussian Processes With Fractional Spectral Measures, Daniel Alpay, Palle Jorgensen, David Levanony

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

We study a family of stationary increment Gaussian processes, indexed by time. These processes are determined by certain measures σ (generalized spectral measures), and our focus here is on the case when the measure σ is a singular measure. We characterize the processes arising from when σ is in one of the classes of affine self-similar measures. Our analysis makes use of Kondratiev-white noise spaces. With the use of a priori estimates and the Wick calculus, we extend and sharpen (see Theorem 7.1) earlier computations of Ito stochastic integration developed for the special case of stationary increment processes having ...

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 ...

The Unimodality Of Pure O-Sequences Of Type Three In Three Variables, 2011 Sacred Heart University

#### The Unimodality Of Pure O-Sequences Of Type Three In Three Variables, Bernadette Boyle

*Mathematics Faculty Publications*

We will give a positive answer for the unimodality of the Hilbert functions in the smallest open case, that of Artinian level monomial algebras of type three in three variables.

Left Centralizers On Rings That Are Not Semiprime, 2011 Iowa State University

#### Left Centralizers On Rings That Are Not Semiprime, Irvin R. Hentzel, M.S. Tammam El-Sayiad

*Mathematics Publications*

A (left) centralizer for an associative ring *R* is an additive map satisfying *T(xy)* = *T(x)y* for all *x*, *y* in *R*. A (left) Jordan centralizer for an associative ring *R* is an additive map satisfying *T*(*xy*+*yx*) = *T*(*x*)*y* + *T*(*y*)*x* for all *x*, *y* in *R*. We characterize rings with a Jordan centralizer *T*. Such rings have a *T* invariant ideal *I*, *T* is a centralizer on *R/I*, and *I* is the union of an ascending chain of nilpotent ideals. Our work requires 2-torsion free. This result has applications to (right) centralizers ...

Eventually Nonnegative Matrices And Their Sign Patterns, 2011 Xavier University

#### Eventually Nonnegative Matrices And Their Sign Patterns, Minerva Catral, Craig Erickson, Leslie Hogben, D. D. Olesky, P. Van Den Driessche

*Mathematics Conference Papers, Posters and Presentations*

No abstract provided.

Constructions Of Potentially Eventually Positive Sign Patterns With Reducible Positive Part, 2011 Iowa State University

#### Constructions Of Potentially Eventually Positive Sign Patterns With Reducible Positive Part, Marie Archer, Minerva Catral, Craig Erickson, Rana Haber, Leslie Hogben, Xavier Martinez-Rivera, Antonio Ochoa

*Mathematics Publications*

Potentially eventually positive (PEP) sign patterns were introduced by Berman et al. (Electron. J. Linear Algebra 19 (2010), 108–120), where it was noted that a matrix is PEP if its positive part is primitive, and an example was given of a 3×3 PEP sign pattern with reducible positive part. We extend these results by constructing n×n PEP sign patterns with reducible positive part, for every n≥3.

The Positive Real Lemma And Construction Of All Realizations Of Generalized Positive Rational Functions, 2011 Chapman University

#### The Positive Real Lemma And Construction Of All Realizations Of Generalized Positive Rational Functions, Daniel Alpay, Izchak Lewkowicz

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

We here extend the well known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. All state space realizations are partitioned into subsets, each is identified with a set of matrices satisfying the same Lyapunov inclusion. Thus, each subset forms a convex invertible cone, cic in short, and is in fact is replica of all realizations of positive functions of the same dimensions. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a ...

Inverse Limits With Set Valued Functions, 2011 University of Richmond

#### Inverse Limits With Set Valued Functions, Van C. Nall

*Math and Computer Science Faculty Publications*

We begin to answer the question of which continua can be homeomorphic to an inverse limit with a single upper semi-continuous bonding map from [O, 1) to 2^{(O,l)}. Several continua including (0, 1) x (0, 1) and all compact manifolds with dimension greater than one cannot be homeomorphic to such an inverse limit. It is also shown that if the upper semi-continuous bonding maps have only zero dimensional point values, then the dimension of the inverse limit does not exceed the dimension of the factor spaces.

Idempotents In Plenary Train Algebras, 2010 Universidad de Chile

#### Idempotents In Plenary Train Algebras, Antonio Behn, Irvin R. Hentzel

*Mathematics Publications*

In this paper we study plenary train algebras of arbitrary rank. We show that for most parameter choices of the train identity, the additional identity (x^2 -w(x)x)^2 =0 is satisfied. We also find sufficient conditions for *A* to have idempotents.

Descending Central Series Of Free Pro-P-Groups, 2010 University of Western Ontario

#### 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 ...

Information-Preserving Structures: A General Framework For Quantum Zero-Error Information, 2010 Perimeter Institute for Theoretical Physics

#### Information-Preserving Structures: A General Framework For Quantum Zero-Error Information, Robin Blume-Kohout, Hui Khoon Ng, David Poulin, Lorenza Viola

*Open Dartmouth: Faculty Open Access Articles*

Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system’s ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using *information-preserving structures*, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected ...

The Cohomology Of Modules Over A Complete Intersection Ring, 2010 University of Nebraska-Lincoln

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

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

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.