Physical Sciences and Mathematics Commons^™

Albert Forms, Quaternions, Schubert Varieties & Embeddability, Jasmin Omanovic

Electronic Thesis and Dissertation Repository

The origin of embedding problems can be understood as an effort to find some minimal datum which describes certain algebraic or geometric objects. In the algebraic theory of quadratic forms, Pfister forms are studied for a litany of powerful properties and representations which make them particularly interesting to study in terms of embeddability. A generalization of these properties is captured by the study of central simple algebras carrying involutions, where we may characterize the involution by the existence of particular elements in the algebra. Extending this idea even further, embeddings are just flags in the Grassmannian, meaning that their study …

Groups Satisfying The Converse To Lagrange's Theorem, Jonah N. Henry

MSU Graduate Theses

Lagrange’s theorem, which is taught early on in group theory courses, states that the order of a subgroup must divide the order of the group which contains it. In this thesis, we consider the converse to this statement. A group satisfying the converse to Lagrange’s theorem is called a CLT group. We begin with results that help show that a group is CLT, and explore basic CLT groups with examples. We then give the conditions to guarantee either CLT is satisfied or a non-CLT group exists for more advanced cases. Additionally, we show that CLT groups are properly contained between …

Supporting The Algebra I Curriculum With An Introduction To Computational Thinking Course, Michelle M. Laskowski

LSU Master's Theses

The Louisiana Workforce Commission predicts a 33.6% increase in computer science and mathematical occupations by 2022 and the Bureau of Labor Statistics foresees a 16% increase in computer scientists from 2018-2028. Despite these opportunities for job and financial security, the number of Louisiana students enrolled in a nationally accredited computing course is less than 1%, compared to national leaders California and Texas which have 3% and 3.8% of students respectively. Furthermore, the international assessments of mathematical literacy, PISA and TIMMS, both report American students continue to fall further behind their international peers in mathematics achievement.

This thesis rejects these statistics …

Alpha Capture Reaction Rates For Nucleosynthesis Within An Ab Initio Framework, Alison Constance Dreyfuss

LSU Doctoral Dissertations

Clustering in nuclear systems has broad impacts on all phases of stellar burning, and plays a significant role in our understanding of nucleosynthesis, or how and where nuclei are produced in the universe. The role of alpha particles in particular is extremely important for nuclear astrophysics: ⁴He was one of the earliest elements produced in the Big Bang, it is one of the most abundant elements in the universe, and helium burning -- in particular, the triple-alpha process -- is one of the most important ``engines'' in stars. To better understand nucleosynthesis and stellar burning, then, it is important …

Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, Matthew Bates

Doctoral Dissertations

Let k be the finite field with q elements, let F be the field of Laurent series in the variable 1/t with coefficients in k, and let A be the polynomial ring in the variable t with coefficients in k. Let SLn(F) be the ring of nxn-matrices with entries in F, and determinant 1. Given a polynomial g in A, let Gamma(g) subset SLn(F) be the full congruence subgroup of level g. In this thesis we examine the action of Gamma(g) on the Bruhat-Tits building Xn associated to SLn(F) for n equals 2 and n equals 3. Our first main …

Zeta Functions Of Classical Groups And Class Two Nilpotent Groups, Fikreab Solomon Admasu

Dissertations, Theses, and Capstone Projects

This thesis is concerned with zeta functions and generating series associated with two families of groups that are intimately connected with each other: classical groups and class two nilpotent groups. Indeed, the zeta functions of classical groups count some special subgroups in class two nilpotent groups.

In the first chapter, we provide new expressions for the zeta functions of symplectic groups and even orthogonal groups in terms of the cotype zeta function of the integer lattice. In his paper on universal $p$-adic zeta functions, J. Igusa computed explicit formulae for the zeta functions of classical algebraic groups. These zeta functions …

Algebraic Topics In The Classroom – Gauss And Beyond, Lisa Krance

Masters Essays

No abstract provided.

Quantum Cluster Algebras At Roots Of Unity, Poisson-Lie Groups, And Discriminants, Kurt Malcolm Trampel Iii

LSU Doctoral Dissertations

This dissertation studies quantum algebras at roots of unity in regards to cluster structure and Poisson structure. Moreover, quantum cluster algebras at roots of unity are rigorously defined. The discriminants of these algebras are described, in terms of frozen cluster variables for quantum cluster algebras and Poisson primes for specializations of quantum algebras. The discriminant is a useful invariant for representation theoretic and algebraic study, whose laborious computation deters direct evaluation. The discriminants of quantum Schubert cells at roots of unity will be computed from the two distinct approaches. These methods can be applied to many other quantum algebras.

Graded Character Rings, Mackey Functors And Tambara Functors, Beatrice Isabelle Chetard

Electronic Thesis and Dissertation Repository

Let $G$ be a finite group. The ring $R_\KK(G)$ of virtual characters of $G$ over the field $\KK$ is a $\lambda$-ring; as such, it is equipped with the so-called $\Gamma$-filtration, first defined by Grothendieck. In the first half of this thesis, we explore the properties of the associated graded ring $R^*_\KK(G)$, and present a set of tools to compute it through detailed examples. In particular, we use the functoriality of $R^*_\KK(-)$, and the topological properties of the $\Gamma$-filtration, to explicitly determine the graded character ring over the complex numbers of every group of order at most $8$, as well as …

Dehn Functions Of Bestvina-Brady Groups, Yu-Chan Chang

LSU Doctoral Dissertations

In this dissertation, we prove that if the flag complex on a finite simplicial graph is a 2-dimensional triangulated disk, then the Dehn function of the associated Bestvina--Brady group depends on the maximal dimension of the simplices in the interior of the flag complex. We also give some examples where the flag complex on a finite simplicial graph is not 2-dimensional, and we establish a lower bound for the Dehn function of the associated Bestvina--Brady group.

Analogues Between Leibniz's Harmonic Triangle And Pascal's Arithmetic Triangle, Lacey Taylor James

Electronic Theses, Projects, and Dissertations

This paper will discuss the analogues between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle by utilizing mathematical proving techniques like partial sums, committees, telescoping, mathematical induction and applying George Polya's perspective. The topics presented in this paper will show that Pascal's triangle and Leibniz's triangle both have hockey stick type patterns, patterns of sums within shapes, and have the natural numbers, triangular numbers, tetrahedral numbers, and pentatope numbers hidden within. In addition, this paper will show how Pascal's Arithmetic Triangle can be used to construct Leibniz's Harmonic Triangle and show how both triangles relate to combinatorics and arithmetic through the …

Category Theory And Universal Property, Niuniu Zhang

Honors Theses

Category theory unifies and formalizes the mathematical structure and concepts in a way that various areas of interest can be connected. For example, many have learned about the sets and its functions, the vector spaces and its linear transformation, and the group theories and its group homomorphism. Not to mention the similarity of structure in topological spaces, as the continuous function is its mapping. In sum, category theory represents the abstractions of other mathematical concepts. Hence, one could use category theory as a new language to define and simplify the existing mathematical concepts as the universal properties. The goal of …

On The Intersection Number Of Finite Groups, Humberto Bautista Serrano

Math Theses

Let G be a finite, nontrivial group. In a paper in 1994, Cohn defined the covering number of a finite group as the minimum number of nontrivial proper subgroups whose union is equal to the whole group. This concept has received considerable attention lately, mainly due to the importance of recent discoveries. In this thesis we study a dual concept to the covering number. We define the intersection number of a finite group as the minimum number of maximal subgroups whose intersection is equal to the Frattini subgroup. Similarly we define the inconjugate intersection number of a finite group as …

Weyl Groups And The Nil-Hecke Algebra, Arta Holaj

Mathematics and Statistics

We begin this paper with a short survey on finite reflection groups. First we establish what a reflection in Euclidean space is. Then we introduce a root system, which is then partitioned into two sets: one of positive roots and one with negative roots. Th is articulates our understanding of groups generated by simple reflections. Furthermore, we develop our insight to Weyl groups and crystallographic groups before exploring crystallographic root systems. The section section of this paper examines the twisted group algebra along with the Demazure element Xi and the Demazure-Lusztig element Ti. Lastly, the third section of this paper …

One Teacher's Transformation Of Practice Through The Development Of Covariational Thinking And Reasoning In Algebra : A Self-Study, Jacqueline Dauplaise

Theses, Dissertations and Culminating Projects

CCSSM (2010) describes quantitative reasoning as expertise that mathematics educators should seek to develop in their students. Researchers must then understand how to develop covariational reasoning. The problem is that researchers draw from students’ dialogue as the data for understanding quantitative relationships. As a result, the researcher can only conceive the students’ reasoning. The objective of using the self-study research methodology is to examine and improve existing teaching practices. To improve my practice, I reflected upon the implementation of my algebra curriculum through a hermeneutics cycle of my personal history and living educational theory. The critical friend provoked through dialogues …

On The Complexity Of Computing Galois Groups Of Differential Equations, Mengxiao Sun

Dissertations, Theses, and Capstone Projects

The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions.

Hrushovski first proposed an algorithm for computing the differential …

Analysis Of A Group Of Automorphisms Of A Free Group As A Platform For Conjugacy-Based Group Cryptography, Pavel Shostak

Dissertations, Theses, and Capstone Projects

Let F be a finitely generated free group and Aut(F) its group of automorphisms.

In this monograph we discuss potential uses of Aut(F) in group-based cryptography.

Our main focus is on using Aut(F) as a platform group for the Anshel-Anshel-Goldfeld protocol, Ko-Lee protocol, and other protocols based on different versions of the conjugacy search problem or decomposition problem, such as Shpilrain-Ushakov protocol.

We attack the Anshel-Anshel-Goldfeld and Ko-Lee protocols by adapting the existing types of the length-based attack to the specifics of Aut(F). We also present our own version of the length-based attack that significantly increases the attack' success …

Taking Notes: Generating Twelve-Tone Music With Mathematics, Nathan Molder

Electronic Theses and Dissertations

There has often been a connection between music and mathematics. The world of musical composition is full of combinations of orderings of different musical notes, each of which has different sound quality, length, and em phasis. One of the more intricate composition styles is twelve-tone music, where twelve unique notes (up to octave isomorphism) must be used before they can be repeated. In this thesis, we aim to show multiple ways in which mathematics can be used directly to compose twelve-tone musical scores.

A Differential Algebra Approach To Commuting Polynomial Vector Fields And To Parameter Identifiability In Ode Models, Peter Thompson

Dissertations, Theses, and Capstone Projects

In the first part, we study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field. One motivating factor is that we can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. We first show that a linear vector field admits a full complement of commuting vector fields. Then we study a type of planar vector field for which there exists an upper bound on the degree of a …

Patterns, Symmetries, And Mathematical Structures In The Arts, Sarah C. Deloach

Honors College Theses

Mathematics is a discipline of academia that can be found everywhere in the world around us. Mathematicians and scientists are not the only people who need to be proficient in numbers. Those involved in social sciences and even the arts can benefit from a background in math. In fact, connections between mathematics and various forms of art have been discovered since as early as the fourth century BC. In this thesis we will study such connections and related concepts in mathematics, dances, and music.

Does Teaching The History Of Mathematics In High School Aid In Student Understanding?, Anne Campbell

Undergraduate Honors Thesis Projects

This research will study the effect teaching the history of mathematics in a high school classroom has on student understanding. To accomplish this, lessons both including and excluding historical background on different topics were taught in an Honors Algebra 2 class in the high school setting. This research aims to engage student learning and investigation of topics that normally do not draw a lot of student focus and spark a new or revived interest in mathematics for students by broadening lessons to include material of which students would not otherwise be exposed. The lessons themselves aim to engage other current …

Enhanced Koszulity In Galois Cohomology, Marina Palaisti

Electronic Thesis and Dissertation Repository

Despite their central role in Galois theory, absolute Galois groups remain rather mysterious; and one of the main problems of modern Galois theory is to characterize which profinite groups are realizable as absolute Galois groups over a prescribed field. Obtaining detailed knowledge of Galois cohomology is an important step to answering this problem. In our work we study various forms of enhanced Koszulity for quadratic algebras. Each has its own importance, but the common ground is that they all imply Koszulity. Applying this to Galois cohomology, we prove that, in all known cases of finitely generated pro-$p$-groups, Galois cohomology is …

Polygonal Analogues To The Topological Tverberg And Van Kampen-Flores Theorems, Leah Leiner

Senior Projects Spring 2019

Tverberg’s theorem states that any set of (q-1)(d+1)+1 points in d-dimensional Euclidean space can be partitioned into q subsets whose convex hulls intersect. This is topologically equivalent to saying any continuous map from a (q-1)(d+1)-dimensional simplex to d-dimensional Euclidean space has q disjoint faces whose images intersect, given that q is a prime power. These continuous functions have a Fourier decomposition, which admits a Tverberg partition when all of the Fourier coefficients, except the constant coefficient, are zero. We have been working with continuous functions where all of the Fourier coefficients except the constant and one other coefficient are zero. …

Decidability For Residuated Lattices And Substructural Logics, Gavin St. John

Electronic Theses and Dissertations

We present a number of results related to the decidability and undecidability of various varieties of residuated lattices and their corresponding substructural logics. The context of this analysis is the extension of residuated lattices by various simple equations, dually, the extension of substructural logics by simple structural rules, with the aim of classifying simple equations by the decidability properties shared by their extensions. We also prove a number of relationships among simple extensions by showing the equational theory of their idempotent semiring reducts coincides with simple extensions of idempotent semirings. On the decidability front, we develop both semantical and syntactical …

Counting And Coloring Sudoku Graphs, Kyle Oddson

Mathematics and Statistics Dissertations, Theses, and Final Project Papers

A sudoku puzzle is most commonly a 9 × 9 grid of 3 × 3 boxes wherein the puzzle player writes the numbers 1 - 9 with no repetition in any row, column, or box. We generalize the notion of the n² × n² sudoku grid for all n ϵ Z ^≥2 and codify the empty sudoku board as a graph. In the main section of this paper we prove that sudoku boards and sudoku graphs exist for all such n we prove the equivalence of [3]'s construction using unions and products of graphs to the definition of …

Enhancing The Quandle Coloring Invariant For Knots And Links, Karina Elle Cho

HMC Senior Theses

Quandles, which are algebraic structures related to knots, can be used to color knot diagrams, and the number of these colorings is called the quandle coloring invariant. We strengthen the quandle coloring invariant by considering a graph structure on the space of quandle colorings of a knot, and we call our graph the quandle coloring quiver. This structure is a categorification of the quandle coloring invariant. Then, we strengthen the quiver by decorating it with Boltzmann weights. Explicit examples of links that show that our enhancements are proper are provided, as well as background information in quandle theory.

The Conditional Probability That An Elliptic Curve Has A Rational Subgroup Of Order 5 Or 7, Meagan Kenney

Senior Projects Spring 2019

Let E be an elliptic curve over the rationals. There are two different ways in which the set of rational points on E can be said to be divisible by a prime p. We will call one of these types of divisibility local and the other global. Global divisibility will imply local divisibility; however, the converse is not guaranteed. In this project we focus on the cases where p=5 and p=7 to determine the probability that E has global divisibility by p, given that E has local divisibility by p.

Factorization Lengths In Numerical Monoids, Maya Samantha Schwartz

Senior Projects Spring 2019

A numerical monoid M generated by the natural numbers n_1, ..., n_k is a subset of {0, 1, 2, ...} whose elements are non-negative linear combinations of the generators n_1, ..., n_k. The set of factorizations of an element in M is the set of all the different ways to write that element as a linear combination of the generators. The length of a factorization of an element is the sum of the coefficients of that factorization. Since an element in a monoid can be written in different ways in terms of the generators, its set of factorization lengths may …

Positivity Among P-Partition Generating Functions Of Partially Ordered Sets, Nate Lesnevich

Honors Theses

We find necessary and separate sufficient conditions for the difference between two labeled partially ordered set's (poset) partition generating functions to be positive in the fundamental basis. We define the notion of a jump sequence for a poset and show how different conditions on the jump sequences of two posets are necessary for those posets to have an order relation in the fundamental basis. Our sufficient conditions are of two types. First, we show how manipulating a poset's Hasse diagram produces a poset that is greater according to the fundamental basis. Secondly, we also provide tools to explain posets that …

Lattice Simplices: Sufficiently Complicated, Brian Davis

Theses and Dissertations--Mathematics

Simplices are the "simplest" examples of polytopes, and yet they exhibit much of the rich and subtle combinatorics and commutative algebra of their more general cousins. In this way they are sufficiently complicated --- insights gained from their study can inform broader research in Ehrhart theory and associated fields.

In this dissertation we consider two previously unstudied properties of lattice simplices; one algebraic and one combinatorial. The first is the Poincar\'e series of the associated semigroup algebra, which is substantially more complicated than the Hilbert series of that same algebra. The second is the partial ordering of the elements of …