Mathematics Commons^™

Twisting Systems And Some Quantum P³S With Point Scheme A Rank-2 Quadric, Hung Viet Tran

Mathematics Dissertations - Archive

In 1996, J. J. Zhang introduced the concept of twisting a graded algebra by a twisting system, which generalizes the concept of twisting a graded algebra by an automorphism (the latter concept having been introduced in an article by M. Artin, J. Tate and M. Van den Bergh in 1991). Twisting using a twisting system is an equivalence relation and certain important algebraic properties of the original algebra are carried over to the twisted algebra. We call a twisting system nontrivial if it is not given by an automorphism. However, there are very few known examples of nontrivial twisting systems …

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.

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 …

Dimension Theory Of Conformal Iterated Function Systems, Sharon Sneha Spaulding

Honors Scholar Theses

This thesis is an expository investigation of the conformal iterated function system (CIFS) approach to fractals and their dimension theory. Conformal maps distort regions, subject to certain constraints, in a controlled way. Let $\mathcal{S} = (X, E, \{\phi_e\}_{e \in E})$ be an iterated function system where $X$ is a compact metric space, $E$ is a countable index set, and $\{\phi_e\}_{e \in E}$ is a family of injective and uniformly contracting maps. If the family of maps $\{\phi_e\}_{e \in E}$ is also conformal and satisfies the open set condition, then the distortion properties of conformal maps can be extended to the …

Quantum Dimension Polynomials: A Networked-Numbers Game Approach, Nicholas Gaubatz

Honors College Theses

The Networked-Numbers Game--a mathematical "game'' played on a simple graph--is incredibly accessible and yet surprisingly rich in content. The Game is known to contain deep connections to the finite-dimensional simple Lie algebras over the complex numbers. On the other hand, Quantum Dimension Polynomials (QDPs)--enumerative expressions traditionally understood through root systems--corresponding to the above Lie algebras are complicated to derive and often inaccessible to undergraduates. In this thesis, the Networked-Numbers Game is defined and some known properties are presented. Next, the significance of the QDPs as a method to count combinatorially interesting structures is relayed. Ultimately, a novel closed-form expression of …

A Priori And A Posteriori Error Estimates For The Quad-Curl Eigenvalue Problem, Lixiu Wang, Qian Zhang, Jiguang Sun, Zhimin Zhang

Michigan Tech Publications, Part 1

In this paper, we consider a priori and a posteriori error estimates of the H(curl2)-conforming finite element when solving the quad-curl eigenvalue problem. An a priori estimate of eigenvalues with convergence order 2(s 1) is obtained if the corresponding eigenvector u a Hs 1(Ω) and-u a Hs(Ω). For the a posteriori estimate, by analyzing the associated source problem, we obtain lower and upper bounds for the errors of eigenvectors in the energy norm and upper bounds for the errors of eigenvalues. Numerical examples are presented for validation.

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.

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 …

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 …

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 …

Waring Rank And Apolarity Of Some Symmetric Polynomials, Max Brian Sullivan

Boise State University Theses and Dissertations

We examine lower bounds for the Waring rank for certain types of symmetric polynomials. The first are Schur polynomials, a symmetric polynomial indexed by integer partitions. We prove some results about the Waring rank of certain types of Schur polynomials, based on their integer partition. We also make some observations about the Waring rank in general for Schur polynomials, based on the shape of their Semistandard Young Tableaux. The second type of polynomials we refer to as a Power of a Fermat-type polynomial, or a PFT polynomial. This is a Fermat type (or power sum) polynomial over n variables with …

Independent Dominating Sets In Unicyclic Graphs, Rachel Lopez

Theses, Dissertations and Culminating Projects

Wilf found the maximum number of independent dominating sets of a tree using algebraic methods, while Sagan gave an elementary proof. In this thesis, we maximize the number of independent dominating sets of unicyclic graphs, giving a new proof of a result of Jou and Chang. In our proof, we are able to reduce the problem to finding independent dominating sets of single-legged caterpillar graphs. We also study the number of single-legged caterpillar graphs, both oriented and unoriented, which are related to the Fibonacci Sequence. Finally, this thesis also examines the domination ratio in unicyclic graphs. The domination ratio is …

Modeling Functions Into An Angular Displacement Of An Elastic Pendulum, Brenda Lee Garcia

Theses and Dissertations

In this thesis we study the relation between analytic signals and a variety of pendulum systems. The representation of a signal as a pair of time varying amplitude and phase has been well studied and often related to linear mass spring systems. The differential equations describing pendulum systems are nonlinear and we provide analytical and numerical results regarding interpretation about the amplitude and the phase of signals in different pendulum settings. We report an explicit solution of the Elastic Pendulum problem in the case of linear phase. We develop an experimental procedure to piece-wise approximate bounded functions on a partition …

Sangaku In Multiple Geometries: Examining Japanese Temple Geometry Beyond Euclid, Nathan Hartmann

Honors College Theses

When the country of Japan was closed from the rest of the world from 1603 until
1867 during the Edo period, the field of mathematics developed in a different way
from how it developed in the rest of the world. One way we see this development
is through the sangaku, the thousands of geometric problems hung in various Shinto and Buddhist temples throughout the country. Written on wooden tablets by people from numerous walks of life, all these problems hold true within Euclidean geometry. During the 1800s, while Japan was still closed, non-Euclidean geometries began to develop across the …

Visions Of Science: An Art Historical Exploration Of Medieval Scientific Manuscripts, Olivia Brock

Undergraduate Honors Capstone Projects

The late 16th century saw a new movement in the world of science to push scientific ideas and practice out of academia and into the hands of the layman. No longer were scholars the sole proprietors of science –everyday laborers, craftsman, and artists now had practical scientific principles at their fingertips that they could incorporate into their professions. This new spread of science was facilitated in several ways, including the publication of books incorporating detailed, explanatory images, new utilitarian instruments, and public lectures. Though science was disseminated through a variety of means, I have been particularly interested in the ways …

Positive Solutions To Semilinear Elliptic Equations With Logistic-Type Nonlinearities And Harvesting In Exterior Domains, Eric Jameson

UNLV Theses, Dissertations, Professional Papers, and Capstones

Existing results provide the existence of positive solutions to a class of semilinear elliptic PDEs with logistic-type nonlinearities and harvesting terms both in RN and in bounded domains U ⊂ RN with N ≥ 3, when the carrying capacity of the environment is not constant. We consider these same equations in the exterior domain Ω, defined as the complement of the closed unit ball in RN , N ≥ 3, now with a Dirichlet boundary condition. We first show that the existing techniques forsolving these equations in the whole space RN can be applied to the exterior domain with some …

Sinusoidal Projection For 360° Image Compression And Triangular Discrete Cosine Transform Impact In The Jpeg Pipeline, Iker Vazquez Lopez

Boise State University Theses and Dissertations

The equirectangular projection is commonly used to store and transmit 360' images. However, using the equirectangular projection to store and transmit 360° images is not efficient due to its natural topographic redundancy. To generate the 360° image, captured pixels that form a spherical point cloud in the 3D space are projected onto a 2D plane using the equirectangular projection; generating redundant pixels in the process. These extra pixels in the image add extra memory requirements that have low impact in the final image quality. This dissertation presents results of research into the compression of 360° spherical imagery. It examines and …

The Matrix Sortability Problem, Seth Cleaver

Boise State University Theses and Dissertations

Sorting is such a fundamental component of achieving efficiency that a significant body of mathematics is dedicated to the investigation of sorting. Any modern textbook on algorithms contains chapters on sorting.

One approach to arranging a disorganized list of items into an organized list is to successively identify two blocks of contiguous items, and swap the two blocks. In a fundamental paper D.A. Christie showed that a special version of block swapping, in recent times called context directed swapping and abbreviated cds, is the most efficient among block swapping strategies to achieve an organized list of items. The cds …