Algebra Commons^™

Q-Polymatroids And Their Application To Rank-Metric Codes., Benjamin Jany

Theses and Dissertations--Mathematics

Matroid theory was first introduced to generalize the notion of linear independence. Since its introduction, the theory has found many applications in various areas of mathematics including coding theory. In recent years, q-matroids, the q-analogue of matroids, were reintroduced and found to be closely related to the theory of linear vector rank metric codes. This relation was then generalized to q-polymatroids and linear matrix rank metric codes. This dissertation aims at developing the theory of q-(poly)matroid and its relation to the theory of rank metric codes. In a first part, we recall and establish preliminary results for both q-polymatroids and …

Toric Bundles As Mori Dream Spaces, Courtney George

Theses and Dissertations--Mathematics

A projective, normal variety is called a Mori dream space when its Cox ring is finitely generated. These spaces are desirable to have, as they behave nicely under the Minimal Model Program, but no complete classification of them yet exists. Some early work identified that all toric varieties are examples of Mori dream spaces, as their Cox rings are polynomial rings. Therefore, a natural next step is to investigate projectivized toric vector bundles. These spaces still carry much of the combinatorial data as toric varieties, but have more variable behavior that means that they aren't as straightforward as Mori dream …

Understanding And Advancing College Students' Mathematical Reasoning Using Collaborative Argumentation, Rachel Kay Heili

Graduate Theses/Dissertations

This study explored students’ mathematical reasoning skills and offered supports to advance them through a collaborative argumentation framework in a college intermediate algebra class. The goals of this study were to make observations about student reasoning, identify specific actions to address those observations, and document student growth in reasoning as a result of those actions. An iterative analysis, mixed method study was conducted in which the researcher engaged students in responding to questions that required conceptual understandings using a collaborative argumentation framework as a tool to identify and code components of their responses—claim, evidence, and reasoning. After coding and analyzing …

On Covering Groups With Proper Subgroups, Collin B. Moore

Graduate Theses/Dissertations

In this paper, we explore groups that can be expressed as a union of proper subgroups. Using “covering number” to denote the minimal number of proper subgroups required to cover a group, we explore the nature of groups with covering numbers 3 and 4, while also finding covering numbers for p-groups, dihedral, and generalized dihedral groups.

Permutations, Representations, And Partition Algebras: A Random Walk Through Algebraic Statistics, Ian Shors

HMC Senior Theses

My thesis examines a class of functions on the symmetric group called permutation statistics using tools from representation theory. In 2014, Axel Hultman gave formulas for computing expected values of permutation statistics sampled via random walks. I present analogous formulas for computing variances of these statistics involving Kronecker coefficients – certain numbers that arise in the representation theory of the symmetric group. I also explore deep connections between the study of moments of permutation statistics and the representation theory of the partition algebras, a family of algebras introduced by Paul Martin in 1991. By harnessing these partition algebras, I derive …

The Multiset Partition Algebra: Diagram-Like Bases And Representations, Alexander N. Wilson

Dartmouth College Ph.D Dissertations

There is a classical connection between the representation theory of the symmetric group and the general linear group called Schur--Weyl Duality. Variations on this principle yield analogous connections between the symmetric group and other objects such as the partition algebra and more recently the multiset partition algebra. The partition algebra has a well-known basis indexed by graph-theoretic diagrams which allows the multiplication in the algebra to be understood visually as combinations of these diagrams. My thesis begins with a construction of an analogous basis for the multiset partition algebra. It continues with applications of this basis to constructing the irreducible …

Cayley Map Embeddings Of Complete Graphs With Even Order, Michael O'Connor

Honors Program Theses

German mathematician Claus Michael Ringel used voltage graphs to embed complete graphs onto orientable surfaces such that none of the graph's edges cross each other. Cayley maps do the same whilst being simpler to work with. The goal is to determine the efficiency of Cayley maps in embedding complete graphs onto orientable surfaces. This article focus on complete graphs of even order with an emphasis on graphs whose orders are congruent to 6 modulo 12 and 0 modulo 12. We establish 12 distinct classes that each have their own unique qualities. Through the generalization of a previous technique, we prove …

Another Way To Divide Two Complex Numbers: Identification Method, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this short note to design another method to divide two complex numbers and present a numerical example.

Math 301: Abstract Algebra I, Nicholas Vlamis

Open Educational Resources

No abstract provided.

Quasisymmetric Functions Distinguishing Trees, Jean-Christophe Aval, Karimatou Djenabou, Peter R. W. Mcnamara

Faculty Journal Articles

A famous conjecture of Stanley states that his chromatic symmetric function distinguishes trees. As a quasisymmetric analogue, we conjecture that the chromatic quasisymmetric function of Shareshian and Wachs and of Ellzey distinguishes directed trees. This latter conjecture would be implied by an affirmative answer to a question of Hasebe and Tsujie about the P-partition enumerator distinguishing posets whose Hasse diagrams are trees. They proved the case of rooted trees and our results include a generalization of their result.

The Mceliece Cryptosystem As A Solution To The Post-Quantum Cryptographic Problem, Isaac Hanna

Senior Honors Theses

The ability to communicate securely across the internet is owing to the security of the RSA cryptosystem, among others. This cryptosystem relies on the difficulty of integer factorization to provide secure communication. Peter Shor’s quantum integer factorization algorithm threatens to upend this. A special case of the hidden subgroup problem, the algorithm provides an exponential speedup in the integer factorization problem, destroying RSA’s security. Robert McEliece’s cryptosystem has been proposed as an alternative. Based upon binary Goppa codes instead of integer factorization, his cryptosystem uses code scrambling and error introduction to hinder decrypting a message without the private key. This …

Strong Homotopy Lie Algebras And Hypergraphs, Samuel J. Bevins, Marco Aldi

Undergraduate Research Posters

We study hypergraphs by attaching a nilpotent strong homotopy Lie algebra. We especially focus on hypergraph theoretic information that is encoded in the cohomology of the resulting strong homotopy Lie algebra.

The Lie Algebra Sl2(C) And Krawtchouk Polynomials, Nkosi Alexander

UNF Graduate Theses and Dissertations

The Lie algebra L = sl2(C) consists of the 2 × 2 complex matrices that have trace zero, together with the Lie bracket [y, z] = yz − zy. In this thesis we study a relationship between L and Krawtchouk polynomials. We consider a type of element in L said to be normalized semisimple. Let a, a^∗ be normalized semisimple elements that generate L. We show that a, a^∗ satisfy a pair of relations, called the Askey-Wilson relations. For a positive integer N, we consider an (N + 1)-dimensional irreducible L-module V consisting of the homogeneous polynomials in two variables …

Studies On Depth And Torsion In Tensor Products Of Modules, Uyen Huyen Thao Le

Graduate Theses, Dissertations, and Problem Reports (ETD)

This dissertation represents an in-depth exploration of two distinct yet interconnected research topics within commutative algebra: one centered around a conjecture of Huneke and R. Wiegand and the other concerns a depth inequality of Auslander. It consists of the following three papers as well as the author's work under the direction of Professor Olgur Celikbas:

• Remarks on a conjecture of Huneke and Wiegand and the vanishing of (co)homology, Journal of Mathematical Society of Japan Advance Publication. (joint work with Olgur Celikbas, Hiroki Matsui, and Arash Sadeghi).
• An extension of a depth inequality of Auslander, Taiwanese Journal of Mathematics, …