Mathematics Commons^™

Exponential Tensor Modules, Khoa Hoang Nguyen

Mathematics Dissertations - Archive

Representation theory of Lie algebra of a finite dimensional reductive Lie algebra g is a long-standing problem. The ultimate goal is to classify all representations of g. However. the only case only case when a complete classification is obtained is the case of g = sl(2). Hence, it is natural to study certain categories of representations of g for which some finiteness conditions on the action of certain elements of g is enforced. In this thesis, we introduce a class of representations T (g, V, S) of sl(n + 1) of mixed tensor type. By varying the polynomial g, the …

Quantized Enveloping Superalgebra Of Type P, Saber Murad Ahmed

Mathematics Dissertations - Archive

We introduce a new quantized enveloping superalgebra $\mathfrak{U}_q\mathfrak{p}_n$ attached to the Lie superalgebra $\mathfrak{p}_n$ of type P. The superalgebra $\mathfrak{U}_q\mathfrak{p}_n$ is a quantization of a Lie bisuperalgebra structure on $\mathfrak{p}_n$ and we study some of its basic properties. We determine representations of the superalgebra $\mathfrak{U}_q\mathfrak{p}_n$ and derive its Drinfeld-Jimbo relations. We prove the triangular decomposition of $\mathfrak{U}_q\mathfrak{p}_n$ and introduce some preliminary results concerning the highest weight representation of $\mathfrak{U}_q\mathfrak{p}_n$. We also introduce the periplectic q-Brauer algebra and prove that it is the centralizer of the $\mathfrak{U}_q\mathfrak{p}_n$-module structure on $\mathbb{C}(n|n)^{\otimes \ell}$. Finally, we propose a definition for a new periplectic q-Schur …

Strict Prime-Intersective Polynomials For A Fixed Prime Number, Rob Rexler Baello

Theses, Dissertations and Culminating Projects

In this thesis, we examine intersective polynomials, which are polynomials with integer coefficients that have a root modulo any positive integer greater than 1. For any prime number p, a p-intersective polynomial is a polynomial with integer coefficients which has a root in Zp. We define a special type of p-intersective polynomial called strict p-intersective polynomial that can be factored as the product of a p-intersective polynomial and an irreducible polynomial mod p. The main results include methods of construction of strict p-intersective polynomials for certain prime numbers p and enumeration of such polynomials of certain degrees.

Chapter 1 gives …

Slope Conjecture And Normal Surface Theory, Helene Swanepoel

Graduate Theses and Dissertations (2019 - present)

In this thesis we explore the relationship between quantum link invariants and the geometric and topological properties for a family of pretzel knots P(−2r, 2r + 1, 2r + s), r ≥ 1 and s ≥ 3, as predicted by the Slope conjecture. This conjecture asserts a connection between the degree of the colored Jones polynomial and boundary slopes of these knots. We introduce fundamentals of knot theory and prove that the minimum degree of the Jones polynomial for the family of pretzel knots P(−2r, 2r + 1, 2r + s), r ≥ 1 and s ≥ 3, is −6r …

On Uniqueness And Stability For The Boltzmann-Enskog Equation, Martin Friesen, Barbara Ruediger, Padmanabhan Subdar

Faculty Publications

The time-evolution of a moderately dense gas in a vacuum is described in classical mechanics by a particle density function obtained from the Boltzmann-Enskog equation. Based on a McKean-Vlasov equation with jumps, the associated stochastic process was recently constructed by modified Picard iterations with the mean-field interactions, and more generally, by a system of interacting particles. By the introduction of a shifted distance that exactly compensates for the free transport term that accrues in the spatially inhomogeneous setting, we prove in this work an inequality on the Wasserstein distance for any two measure-valued solutions to the Boltzmann-Enskog equation. As a …

Identifying How Home Advantage Manifests In Butler Basketball Using Sfa, Emily M. Shoemaker

Undergraduate Honors Thesis Collection

Butler University’s basketball team has been in the Big East Conference since 2013 and is known for having a distinct home court advantage named ‘Hinkle Magic’ by fans. It is of interest to identify how home court advantage affects a Big East team’s ability to play to its full potential for generating wins and what factors are most accurate in measuring this potential. This project will utilize the Stochastic Frontier Approach (SFA) model to identify a team’s efficiency when playing at home vs. away in order to identify when teams meet their potential and if game location has an effect …

Studying Extended Sets From Young Tableaux, Eric S. Nofziger

Undergraduate Honors Thesis Collection

Young tableaux are combinatorial objects related to the partitions of an integer that have various applications in representation theory. These tableaux are defined as a left-justified set of n boxes filled with the numbers 1 through n and organized in rows, with the length of each row corresponding to a summand in the partition. In recent work of Graham–Precup–Russell, an association has been made between a given row-strict tableau and three disjoint subsets I, J, and K, also called extended sets. In this project, we begin to classify which extended sets correlate to a valid row-strict or standard tableau. We …

Magic Squares Of Squares Of Order 5 Modulo A Prime Number, Imani L. Mosquera

Theses, Dissertations and Culminating Projects

In this paper, I examine magic squares of squares (MSS) of order 5 over Zp where p is a prime number. The first approach to the problem is to find how many distinct elements an MSS may have (called the degree of the MSS). In the next step, I study the relationship between the magic sum and the center entry of any MSS. In order to develop construction methods and configurations for magic squares of squares of order 5 with desired degrees, I study Pythagorean triples and sequences of consecutive quadratic residues modulo p. Properties of these sequences are provided …

A Network-Based Analysis Of Student Learning Trajectories And Andragogical Design, John Kerry O’Meara

Theses, Dissertations and Culminating Projects

The core goal of this thesis project is to formalize the complex system that exists naturally in a formal classroom environment. The three factors that are considered in this study are the roles of the student, the roles of the teacher, and the respective environments from which these members arise and how these act as determinants of curriculum development and design. At the curricular scale, educational practices should be treated as a complex system composed of various inherently connected concepts and exchanges of ideas and ways of knowing. This synthesis of previous work and ongoing research efforts employs a network …

The Independence Polynomial Of A Graph At −1, Kyle Robbins

Theses, Dissertations and Culminating Projects

No abstract provided.

John Horton Conway: The Man And His Knot Theory, Dillon Ketron

Electronic Theses and Dissertations

John Horton Conway was a British mathematician in the twentieth century. He made notable achievements in fields such as algebra, number theory, and knot theory. He was a renowned professor at Cambridge University and later Princeton. His contributions to algebra include his discovery of the Conway group, a group in twenty-four dimensions, and the Conway Constellation. He contributed to number theory with his development of the surreal numbers. His Game of Life earned him long-lasting fame. He contributed to knot theory with his developments of the Conway polynomial, Conway sphere, and Conway notation.

The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

Electronic Theses, Projects, and Dissertations

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i² = 3, j² = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL₂(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …

Lattice Reduction Algorithms, Juan Ortega

Electronic Theses, Projects, and Dissertations

The purpose of this thesis is to propose and analyze an algorithm that follows
similar steps of Guassian Lattice Reduction Algorithm in two-dimensions and applying
them to three-dimensions. We start off by discussing the importance of cryptography in
our day to day lives. Then we dive into some linear algebra and discuss specific topics that
will later help us in understanding lattice reduction algorithms. We discuss two lattice
problems: the shortest vector problem and the closest vector problem. Then we introduce
two types of lattice reduction algorithms: Guassian Lattice Reduction in two-dimensions
and the LLL Algortihm. We illustrate how both …

De Rham Cohomology, Homotopy Invariance And The Mayer-Vietoris Sequence, Stacey Elizabeth Cox

Electronic Theses, Projects, and Dissertations

This thesis will discuss the de Rham cohomology, homotopy invariance and the Mayer-Vietoris sequence. First the necessary information for this thesis is discussed such as differential p-forms, the exterior derivative as well as pull back of a map. The de Rham cohomology is defined explicitly, some properties of the de Rham cohomology will also be discussed. It will be shown that the de Rham cohomology is in fact a homotopy invariant as well as some examples using homotopy invariance are provided. Finally the Mayer-Vietoris sequence will be established, an example of using the Mayer-Vietoris sequence to compute the de …

An Exposition Of Elliptic Curve Cryptography, Travis Severns

Electronic Theses, Projects, and Dissertations

Protecting information that is being communicated between two parties over
unsecured channels is of huge importance in today’s world. The use of mathematical concepts to achieve high levels of security when communicating over these unsecured platforms is cryptography. The world of cryptography is always expanding and growing. In this paper, we set out to explore the use of elliptic curves in the cryptography of today, as well as the cryptography of the future.
We also offer our own original cryptosystem, CSDH. This system on its own
offers some moderate level of security. It shares many similarities to the post-quantum, SIDH …

Symmetric Generation, Ana Gonzalez

Electronic Theses, Projects, and Dissertations

We will examine progenitors. We start with progenitors of the form $m^{*n} : N$ where $m^{*n}$ is a free group and $N$ is a permutation group of degree $n$. But, $m^{*n} : N$ is a group of infinite order so we will factor by the necessary relations to get finite homomorphic images. These groups are constructed through the manual double coset enumeration method. We will examine how to construct progenitors for wreath products.

Error Terms For The Trapezoid, Midpoint, And Simpson's Rules, Jessica E. Coen

Electronic Theses, Projects, and Dissertations

When it is not possible to integrate a function we resort to Numerical Integration. For example the ubiquitous Normal curve tables are obtained using Numerical Integration. The antiderivative of the defining function for the normal curve involves the formula for antiderivative of e^{-x^2} which can't be expressed in the terms of basic functions.

Simpson's rule is studied in most Calculus books, and in all undergraduate Numerical Analysis books, but proofs are not provided. Hence if one is interested in a proof of Simpson's rule, either it can be found in advanced Numerical Analysis books as a special case …

The Decomposition Of The Space Of Algebraic Curvature Tensors, Katelyn Sage Risinger

Electronic Theses, Projects, and Dissertations

We decompose the space of algebraic curvature tensors (ACTs) on a finite dimensional, real inner product space under the action of the orthogonal group into three inequivalent and irreducible subspaces: the real numbers, the space of trace-free symmetric bilinear forms, and the space of Weyl tensors. First, we decompose the space of ACTs using two short exact sequences and a key result, Lemma 3.5, which allows us to express one vector space as the direct sum of the others. This gives us a decomposition of the space of ACTs as the direct sum of three subspaces, which at this point …

Fault-Free Tilings, T. Elise Dettling, Amelia Goetzinger

Mathematics Undergraduate Research

Given a p×q rectangular board (height p and width q), we may fill in the area with 2×1 tiles. We say the board is tileable if the board can be filled with non-overlapping tiles leaving no open space. Not all boards are tileable, for example a 5 × 5 board is not tileable because it has an odd area. A fault-free tiling exists on the board if every line through the board parallel to a side goes through a tile. Previous work on this subject has been completed by Ron Graham in “Fault-Free Tilings of Rectangles" and Emily Montelius in …

Finite Dimensional Approximation And Pin(2)-Equivariant Property For Rarita-Schwinger-Seiberg-Witten Equations, Minh Lam Nguyen

Graduate Theses and Dissertations

The Rarita-Schwinger operator Q was initially proposed in the 1941 paper by Rarita and Schwinger to study wave functions of particles of spin 3/2, and there is a vast amount of physics literature on its properties. Roughly speaking, 3/2−spinors are spinor-valued 1-forms that also happen to be in the kernel of the Clifford multiplication. Let X be a simply connected Riemannian spin 4−manifold. Associated to a fixed spin structure on X, we define a Seiberg-Witten-like system of non-linear PDEs using Q and the Hodge-Dirac operator d∗ + d+ after suitable gauge-fixing. The moduli space of solutions M contains (3/2-spinors, purely …