Other Mathematics Commons^™

Math And Democracy, Kimberly A. Roth, Erika L. Ward

Journal of Humanistic Mathematics

Math and Democracy is a math class containing topics such as voting theory, weighted voting, apportionment, and gerrymandering. It was first designed by Erika Ward for math master’s students, mostly educators, but then adapted separately by both Erika Ward and Kim Roth for a general audience of undergraduates. The course contains materials that can be explored in mathematics classes from those for non-majors through graduate students. As such, it serves students from all majors and allows for discussion of fairness, racial justice, and politics while exploring mathematics that non-major students might not otherwise encounter. This article serves as a guide …

Dna Self-Assembly Of Trapezohedral Graphs, Hytham Abdelkarim

Electronic Theses, Projects, and Dissertations

Self-assembly is the process of a collection of components combining to form an organized structure without external direction. DNA self-assembly uses multi-armed DNA molecules as the component building blocks. It is desirable to minimize the material used and to minimize genetic waste in the assembly process. We will be using graph theory as a tool to find optimal solutions to problems in DNA self-assembly. The goal of this research is to develop a method or algorithm that will produce optimal tile sets which will self-assemble into a target DNA complex. We will minimize the number of tile and bond-edge types …

An Extension Of The Complex–Real (C–R) Calculus To The Bicomplex Setting, With Applications, Daniel Alpay, Kamal Diki, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we extend notions of complex ℂ−ℝ-calculus to the bicomplex setting and compare the bicomplex polyanalytic function theory to the classical complex case. Applications of this theory include two bicomplex least mean square algorithms, which extend classical real and complex least mean square algorithms.

Polynomial Density Of Compact Smooth Surfaces, Luke P. Broemeling

Electronic Thesis and Dissertation Repository

We show that any smooth closed surface has polynomial density 3 and that any connected compact smooth surface with boundary has polynomial density 2.

Multiplication Operators By White Noise Delta Functions And Associated Differential Equations, Luigi Accardi, Un Cig Ji, Kimiaki Saitô

Journal of Stochastic Analysis

No abstract provided.

Random Variables With Overlapping Number And Weyl Algebras Ii, Ruma Dutta, Gabriela Popa, Aurel Stan

Journal of Stochastic Analysis

No abstract provided.

An Explicit Construction Of Sheaves In Context, Tyler A. Bryson

Dissertations, Theses, and Capstone Projects

This document details the body of theory necessary to explicitly construct sheaves of sets on a site together with the development of supporting material necessary to connect sheaf theory with the wider mathematical contexts in which it is applied. Of particular interest is a novel presentation of the plus construction suitable for direct application to a site without first passing to the generated grothendieck topology.

Operators Induced By Certain Hypercomplex Systems, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we consider a family {H_t}_t∈R of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations {(C², π_t)}_t∈R of the hypercomplex system {H_t}_t∈R, and study the realizations π_t(h) of hypercomplex numbers h ∈ H_t, as (2 × 2)-matrices acting on C², for an arbitrarily fixed scale t ∈ R. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.

A Hörmander–Fock Space, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In a recent paper we used a basic decomposition property of polyanalytic functions of order 2 in one complex variable to characterize solutions of the classical ∂-problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander-Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by η= (η_n)_n≥0 leading to a special entire function E(z) that is used to express the kernel function of the Hörmander-Fock space. We …

Modelling Illiquid Stocks Using Quantum Stochastic Calculus: Asymptotic Methods, Will Hicks

Journal of Stochastic Analysis

No abstract provided.

Modelling Nuclear Weapon Effects In Wargaming Using Monte Carlo Simulations, Tyler Guetzke, Alexander Withenbury, Zachary Dugger

West Point Research Papers

The United States Army’s interpretation of nuclear weapon effects needs change and modernization. Wargaming exercises are commonplace in today’s military, however, despite the growing threat of non-strategic nuclear weapons (NSNW), little has been done to inform battlefield commanders on their true effects. Our research seeks to develop a tool for commanders to easily interpret quantifiable effects of a NSNW. Utilizing Monte Carlo simulation, we are developing a new methodology to analyze NSNW effects. Our model allows a commander to calculate the expected unit strength following a NSNW strike which will aid in their operational decision making ability. The Monte Carlo …

Adaptive And Topological Deep Learning With Applications To Neuroscience, Edward Mitchell

Doctoral Dissertations

Deep Learning and neuroscience have developed a two way relationship with each informing the other. Neural networks, the main tools at the heart of Deep Learning, were originally inspired by connectivity in the brain and have now proven to be critical to state-of-the-art computational neuroscience methods. This dissertation explores this relationship, first, by developing an adaptive sampling method for a neural network-based partial different equation solver and then by developing a topological deep learning framework for neural spike decoding. We demonstrate that our adaptive scheme is convergent and more accurate than DGM -- as long as the residual mirrors the …

Knot Equivalence, Jacob Trubey

Electronic Theses, Projects, and Dissertations

A knot is a closed curve in R3. Alternatively, we say that a knot is an embedding f : S1 → R3 of a circle into R3. Analogously, one can think of a knot as a segment of string in a three-dimensional space that has been knotted together in some way, with the ends of the string then joined together to form a knotted loop. A link is a collection of knots that have been linked together.

An important question in the mathematical study of knot theory is that of how we can tell when two knots are, or are …

Symmetric Functions Algebras (Sfa) Ii: Induced Matrices, Philip Feinsilver

Journal of Stochastic Analysis

No abstract provided.

The Theatre Of Math: The Stage As A Tool For Abstract Math Education, Blaine Hudak

Honors Projects

So often in the education of Mathematics does instruction solely consist of the lecture. While this can be an effective method of communicating the ideas of math, it leaves much to be desired in gathering the interest and intrigue of those who have not dedicated their lives to the study of the subject. Theatre, by contrast, is a tool that has been used in the past as means of teaching complicated and difficult to understand moral and emotional subjects. While Theatre and Mathematics have been used in combination many times in the past, it is most often done in the …

Defining Characteristics That Lead To Cost-Efficient Veteran Nba Free Agent Signings, David Mccain

Honors Projects in Mathematics

Throughout the history of the NBA, decisions regarding the signing of free agents have been riddled with complexity. Franchises are tasked with finding out what players will serve as optimal free agent signings prior to seeing them perform within the framework of their team. This study hypothesizes that the adequacy of an NBA free agent signing can be modeled and predicted through the implementation of a machine learning model. The model will learn the necessary information using training and testing data sets that include various player biometrics, game statistics, and financial information. The application of this machine learning model will …

Optimal Control Problems For Stochastic Processes With Absorbing Regime, Yaacov Kopeliovich

Journal of Stochastic Analysis

No abstract provided.

Sl(2,Z) Representations And 2-Semiregular Modular Categories, Samuel Nathan Wilson

LSU Doctoral Dissertations

We address the open question of which representations of the modular group SL(2,Z) can be realized by a modular category. In order to investigate this problem, we introduce the concept of a symmetrizable representation of SL(2,Z) and show that this property is necessary for the representation to be realized. We then prove that all congruence representations of SL(2,Z) are symmetrizable. The proof involves constructing a symmetric basis, which greatly aids in further calculation. We apply this result to the reconstruction of modular category data from representations, as well as to the classification of semiregular categories, which are defined via an …

Discovering Dune: Essays On Frank Herbert’S Epic Saga., Edited By Dominic J. Nardi And N. Trevor Brierly, G. Connor Salter

Mythlore: A Journal of J.R.R. Tolkien, C.S. Lewis, Charles Williams, and Mythopoeic Literature

G. Connor Salter reviews Discovering Dune: Essays on Frank Herbert’s Epic Saga, edited by Dominic J. Nardi and N. Trevor Brierly, considering its new contributions to studies of Frank Herbert's work. Essays included fit into four categories (Politics and Power, History and Religion, Biology and Ecology, and Philosophy, Choice and Ethics) and range from Herbert's use of ecology in Dune to how game theory may help explain certain characters' apparent ability to see the future. Discovering Dune also includes an appendix which contains the only up-to-date bibliography of Herbert's work (primary and secondary sources).

Hörmander’S L2 -Method, ∂-Problem And Polyanalytic Function Theory In One Complex Variable, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we consider the classical ∂-problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of polyanalytic functions in order to construct particular solutions of this problem and obtain new Hörmander type estimates using suitable powers of the Cauchy-Riemann operator. We also compute particular solutions of the ∂-problem for specific polyanalytic data such as the Itô complex Hermite polynomials and polyanalytic Fock kernels.