Physical Sciences and Mathematics Commons^™

Automorphisms Of A Generalized Quadrangle Of Order 6, Ryan Pesak

Undergraduate Honors Theses

In this thesis, we study the symmetries of the putative generalized quadrangle of order 6. Although it is unknown whether such a quadrangle Q can exist, we show that if it does, that Q cannot be transitive on either points or lines. We first cover the background necessary for studying this problem. Namely, the theory of groups and group actions, the theory of generalized quadrangles, and automorphisms of GQs. We then prove that a generalized quadrangle Q of order 6 cannot have a point- or line-transitive automorphism group, and we also prove that if a group G acts faithfully on …

Examining Factors Using Standard Subspaces And Antiunitary Representations, Paul Anderson

Undergraduate Honors Theses

In an effort to provide an axiomization of quantum mechanics, John von Neumann and Francis Joseph Murray developed many tools in the theory of operator algebras. One of the many objects developed during the course of their work was the von Neumann algebra, originally called a ring of operators. The purpose of this thesis is to give an overview of the classification of elementary objects, called factors, and explore connections with other mathematical objects, namely standard subspaces in Hilbert spaces and antiunitary representations. The main results presented here illustrate instances of these interconnections that are relevant in Algebraic Quantum Field …

Voting Rules And Properties, Zhuorong Mao

Undergraduate Honors Theses

This thesis composes of two chapters. Chapter one considers the higher order of Borda Rules (Bp) and the Perron Rule (P) as extensions of the classic Borda Rule. We study the properties of those vector-valued voting rules and compare them with Simple Majority Voting (SMV). Using simulation, we found that SMV can yield different results from B1, B2, and P even when it is transitive. We also give a new condition that forces SMV to be transitive, and then quantify the frequency of transitivity when it fails.

In chapter two, we study the `protocol paradox' of approval voting. In approval …

Modern Theory Of Copositive Matrices, Yuqiao Li

Undergraduate Honors Theses

Copositivity is a generalization of positive semidefiniteness. It has applications in theoretical economics, operations research, and statistics. An $n$-by-$n$ real, symmetric matrix $A$ is copositive (CoP) if $x^T Ax \ge 0$ for any nonnegative vector $x \ge 0.$ The set of all CoP matrices forms a convex cone. A CoP matrix is ordinary if it can be written as the sum of a positive semidefinite (PSD) matrix and a symmetric nonnegative (sN) matrix. When $n < 5,$ all CoP matrices are ordinary. However, recognizing whether a given CoP matrix is ordinary and determining an ordinary decomposition (PSD + sN) is still an unsolved problem. Here, we give an overview on modern theory of CoP matrices, talk about our progress on the ordinary recognition and decomposition problem, and emphasis the graph theory aspect of ordinary CoP matrices.

Enumerating Switching Isomorphism Classes Of Signed Graphs, Nathaniel Healy

Undergraduate Honors Theses

Let Γ be a simple connected graph, and let {+,−}^E(Γ) be the set of signatures of Γ. For σ a signature of Γ, we call the pair Σ = (Γ,σ) a signed graph of Γ. We may define switching functions ζ_X ∈ {+, −}^V (Γ) that negate the sign of every edge {u, v} incident with exactly one vertex in the fiber X = ζ^{−1}(−). The group Sw(Γ) of switching functions acts X on the set of signed graphs of Γ and induces an equivalence relation of switching classes in its orbits; there are 2^{|E(Γ)|−|V (Γ)|+1} such classes. More interestingly, …

Period Doubling Cascades From Data, Alexander Berliner

Undergraduate Honors Theses

Orbit diagrams of period doubling cascades represent systems going from periodicity to chaos. Here, we investigate whether a Gaussian process regression can be used to approximate a system from data and recover asymptotic dynamics in the orbit diagrams for period doubling cascades. To compare the orbits of a system to the approximation, we compute the Wasserstein metric between the point clouds of their obits for varying bifurcation parameter values. Visually comparing the period doubling cascades, we note that the exact bifurcation values may shift, which is confirmed in the plots of the Wasserstein distance. This has implications for studying dynamics …

The Enumeration Of Minimum Path Covers Of Trees, Merielyn Sher

Undergraduate Honors Theses

A path cover of a tree T is a collection of induced paths of T that are vertex disjoint and cover all the vertices of T. A minimum path cover (MPC) of T is a path cover with the minimum possible number of paths, and that minimum number is called the path cover number of T. A tree can have just one or several MPC's. Prior results have established equality between the path cover number of a tree T and the largest possible multiplicity of an eigenvalue that can occur in a symmetric matrix whose graph is that tree. We …

Determining Quantum Symmetry In Graphs Using Planar Algebras, Akshata Pisharody

Undergraduate Honors Theses

A graph has quantum symmetry if the algebra associated with its quantum automorphism group is non-commutative. We study what quantum symmetry means and outline one specific method for determining whether a graph has quantum symmetry, a method that involves studying planar algebras and manipulating planar tangles. Modifying a previously used method, we prove that the 5-cycle has no quantum symmetry by showing it has the generating property.

The Minimum Number Of Multiplicity 1 Eigenvalues Among Real Symmetric Matrices Whose Graph Is A Tree, Wenxuan Ding

Undergraduate Honors Theses

For a tree T, U(T) denotes the minimum number of eigenvalues of multiplicity 1 among all real symmetric matrices whose graph is T. It is known that U(T) >= 2. A tree is linear if all its vertices of degree at least 3 lie on a single induced path, and k-linear if there are k of these high degree vertices. If T′ is a linear tree resulting from the addition of 1 vertex to T, we show that |U(T′)−U(T)|

A Survey Of Methods To Determine Quantum Symmetry Of Graphs, Samantha Phillips

Undergraduate Honors Theses

We introduce the theory of quantum symmetry of a graph by starting with quantum permutation groups and classical automorphism groups. We study graphs with and without quantum symmetry to provide a comprehensive view of current techniques used to determine whether a graph has quantum symmetry. Methods provided include specific tools to show commutativity of generators of algebras of quantum automorphism groups of distance-transitive graphs; a theorem that describes why nontrivial, disjoint automorphisms in the automorphism group implies quantum symmetry; and a planar algebra approach to studying symmetry.

Controlling Infectious Disease: Prevention And Intervention Through Multiscale Models, Adrienna N. Bingham

Dissertations, Theses, and Masters Projects

Controlling infectious disease spread and preventing disease onset are ongoing challenges, especially in the presence of newly emerging diseases. While vaccines have successfully eradicated smallpox and reduced occurrence of many diseases, there still exists challenges such as fear of vaccination, the cost and difficulty of transporting vaccines, and the ability of attenuated viruses to evolve, leading to instances such as vaccine derived poliovirus. Antibiotic resistance due to mistreatment of antibiotics and quickly evolving bacteria contributes to the difficulty of eradicating diseases such as tuberculosis. Additionally, bacteria and fungi are able to produce an extracellular matrix in biofilms that protects them …

The Logarithmic Method And The Solution To The Tp2-Completion Problem, Shahla Nasserasr

Dissertations, Theses, and Masters Projects

A matrix is called TP2 if all 1-by-1 and 2-by-2 minors are positive. A partial matrix is one with some of its entries specified, while the remaining, unspecified, entries are free to be chosen. A TP2-completion, of a partial matrix T , is a choice of values for the unspecified entries of T so that the resulting matrix is TP2. The TP2-completion problem asks which partial matrices have a TP2-completion. A complete solution is given here. It is shown that the Bruhat partial order on permutations is the inverse of a certain natural partial order induced by TP2 matrices and …

Calculation Of Equilibrants For Semipositive Matrices, Zheng Tong

Dissertations, Theses, and Masters Projects

No abstract provided.

A Bayesian Network Approach To Feature Selection In Mass Spectrometry Data, Karl W. Kuschner

Dissertations, Theses, and Masters Projects

One of the key goals of current cancer research is the identification of biologic molecules that allow non-invasive detection of existing cancers or cancer precursors. One way to begin this process of biomarker discovery is by using time-of-flight mass spectroscopy to identify proteins or other molecules in tissue or serum that correlate to certain cancers. However, there are many difficulties associated with the output of such experiments. The distribution of protein abundances in a population is unknown, the mass spectroscopy measurements have high variability, and high correlations between variables cause problems with popular methods of data mining. to mitigate these …

Dynamic Adaptation To Cpu And Memory Load In Scientific Applications, Richard Tran Mills

Dissertations, Theses, and Masters Projects

As commodity computers and networking technologies have become faster and more affordable, fairly capable machines have become nearly ubiquitous while the effective "distance" between them has decreased as network connectivity and capacity has multiplied. There is considerable interest in developing means to readily access such vast amounts of computing power to solve scientific problems, but the complexity of these modern computing environments pose problems for conventional computer codes designed to run on a static, homogeneous set of resources. One source of problems is the heterogeneity that is naturally present in these settings. More problematic is the competition that arises between …

On Certain Sets Of Matrices: Euclidean Squared Distance Matrices, Ray-Nonsingular Matrices And Matrices Generated By Reflections, Thomas W. Milligan

Dissertations, Theses, and Masters Projects

In this dissertation, we study three different sets of matrices. First, we consider Euclidean distance squared matrices. Given n points in Euclidean space, we construct an n x n Euclidean squared distance matrix by assigning to each entry the square of the pairwise interpoint Euclidean distance. The study of distance matrices is useful in computational chemistry and structural molecular biology. The purpose of the first part of the thesis is to better understand this set of matrices and its different characterizations so that a number of open problems might be answered and known results improved. We look at geometrical properties …

Spontaneous Pulse Formation In Bistable Systems, George A. Andrews

Dissertations, Theses, and Masters Projects

This thesis considers localized spontaneous pulse formation in nonlinear, dissipative systems that are far from equilibrium and which exhibit bistability. It is shown that such pulses can form in systems that are dominated by the combined effects of: (1) a saturable amplifying or gain region, (2) a saturable absorbing or loss region, and (3) cavity effects. Analysis is based upon novel models for both an inertialess material in which the absorber responds instantaneously and inertial material in which there is temporal delay in the response. Additionally, we include the situation where the material does not fully relax between pulses, i.e. …

Simulation And Numerical Solution Of Stochastic Petri Nets With Discrete And Continuous Timing, Robert Linzey Jones Iii

Dissertations, Theses, and Masters Projects

We introduce a novel stochastic Petri net formalism where discrete and continuous phase-type firing delays can appear in the same model. By capturing deterministic and generally random behavior in discrete or continuous time, as appropriate, the formalism affords higher modeling fidelity and efficiencies to use in practice. We formally specify the underlying stochastic process as a general state space Markov chain and show that it is regenerative, thus amenable to renewal theory techniques to obtain steady-state solutions. We present two steady-state analysis methods depending on the class of problem: one using exact numerical techniques, the other using simulation. Although regenerative …

Algorithms For Operations On Probability Distributions In A Computer Algebra System, Diane Lynn Evans

Dissertations, Theses, and Masters Projects

In mathematics and statistics, the desire to eliminate mathematical tedium and facilitate exploration has lead to computer algebra systems. These computer algebra systems allow students and researchers to perform more of their work at a conceptual level. The design of generic algorithms for tedious computations allows modelers to push current modeling boundaries outward more quickly.;Probability theory, with its many theorems and symbolic manipulations of random variables is a discipline in which automation of certain processes is highly practical, functional, and efficient. There are many existing statistical software packages, such as SPSS, SAS, and S-Plus, that have numeric tools for statistical …

Totally Nonnegative Matrices, Shaun M. Fallat

Dissertations, Theses, and Masters Projects

An m-by-n matrix A is called totally nonnegative (resp. totally positive) if the determinant of every square submatrix (i.e., minor) of A is nonnegative (resp. positive). The class of totally nonnegative matrices has been studied considerably, and this class arises in a variety of applications such as differential equations, statistics, mathematical biology, approximation theory, integral equations and combinatorics. The main purpose of this thesis is to investigate several aspects of totally nonnegative matrices such as spectral problems, determinantal inequalities, factorizations and entry-wise products. It is well-known that the eigenvalues of a totally nonnegative matrix are nonnegative. However, there are many …

Discrete-Time Linear And Nonlinear Aerodynamic Impulse Responses For Efficient Cfd Analyses, Walter A. Silva

Dissertations, Theses, and Masters Projects

This dissertation discusses the mathematical existence and the numerical identification of linear and nonlinear aerodynamic impulse response functions. Differences between continuous-time and discrete-time system theories, which permit the identification and efficient use of these functions, will be detailed. Important input/output definitions and the concept of linear and nonlinear systems with memory will also be discussed. It will be shown that indicial (step or steady) responses (such as Wagner's function), forced harmonic responses (such as Theodorsen's function or those from doublet lattice theory), and responses to random inputs (such as gusts) can all be obtained from an aerodynamic impulse response function. …

Structured Eigenvectors, Interlacing, And Matrix Completions, Brenda K. Kroschel

Dissertations, Theses, and Masters Projects

This dissertation presents results from three areas of applicable matrix analysis: structured eigenvectors, interlacing, and matrix completion problems. Although these are distinct topics, the structured eigenvector results provide connections.;It is a straightforward matrix calculation that if {dollar}\lambda{dollar} is an eigenvalue of A, x an associated structured eigenvector and {dollar}\alpha{dollar} the set of positions in which x has nonzero entries, then {dollar}\lambda{dollar} is also an eigenvalue of the submatrix of A that lies in the rows and columns indexed by {dollar}\alpha{dollar}. We present a converse to this statement and apply the results to interlacing and to matrix completion problems. Several corollaries …

Completion Of Partial Operator Matrices, M.(Mihaly) Bakonyi

Dissertations, Theses, and Masters Projects

This work concerns completion problems for partial operator matrices. A partial matrix is an m-by-n array in which some entries are specified and the remaining are unspecified. We allow the entries to be operators acting between corresponding vector spaces (in general, bounded linear operators between Hilbert spaces). Graphs are associated with partial matrices. Chordal graphs and directed graphs with a perfect edge elimination scheme play a key role in our considerations. A specific choice for the unspecified entries is referred to as a completion of the partial matrix. The completion problems studied here involve properties such as: zero-blocks in certain …

A Structural Factoring Approach For Analyzing Probabilistic Networks, Kelly J. Hayhurst

Dissertations, Theses, and Masters Projects

No abstract provided.

Numerical Experiments With The Multi-Grid Method, Theodore Craig Poling

Dissertations, Theses, and Masters Projects

No abstract provided.

Steepest Descent Techniques For Operator Equations, William T. Suit

Dissertations, Theses, and Masters Projects

No abstract provided.

A Study Of Unique Factorization Domains, James D. Harris

Dissertations, Theses, and Masters Projects

No abstract provided.

Numerical Integration Of Systems With Large Frequency Ratios, James Thompson Howlett

Dissertations, Theses, and Masters Projects

No abstract provided.

On The Solution To Partial Differential Equations By Means Of Bergman's Integral Operator, George R. Young

Dissertations, Theses, and Masters Projects

No abstract provided.

Fourier Transforms In Euclidean K Space, Terry A. Straeter

Dissertations, Theses, and Masters Projects

No abstract provided.