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.