Mathematics Commons^™

The Mean Sum Of Squared Linking Numbers Of Random Piecewise-Linear Embeddings Of $K_N$, Yasmin Aguillon, Xingyu Cheng, Spencer Eddins, Pedro Morales

Rose-Hulman Undergraduate Mathematics Journal

DNA and other polymer chains in confined spaces behave like closed loops. Arsuaga et al. \cite{AB} introduced the uniform random polygon model in order to better understand such loops in confined spaces using probabilistic and knot theoretical techniques, giving some classification on the mean squared linking number of such loops. Flapan and Kozai \cite{flapan2016linking} extended these techniques to find the mean sum of squared linking numbers for random linear embeddings of complete graphs $K_n$ and found it to have order $\Theta(n(n!))$. We further these ideas by inspecting random piecewise-linear embeddings of complete graphs and give introductory-level summaries of the ideas …

On The Spectrum Of Quaquaversal Operators, Josiah Sugarman

Dissertations, Theses, and Capstone Projects

In 1998 Charles Radin and John Conway introduced the Quaquaversal Tiling. A three dimensional hierarchical tiling with the property that the orientations of its tiles approach a uniform distribution faster than what is possible for hierarchical tilings in two dimensions. The distribution of orientations is controlled by the spectrum of a certain Hecke operator, which we refer to as the Quaquaversal Operator. For example, by showing that the largest eigenvalue has multiplicity equal to one, Charles Radin and John Conway showed that the orientations of this tiling approach a uniform distribution. In 2008, Bourgain and Gamburd showed that this operator …

Differential Calculus: From Practice To Theory, Eugene Boman, Robert Rogers

Milne Open Textbooks

Differential Calculus: From Practice to Theory covers all of the topics in a typical first course in differential calculus. Initially it focuses on using calculus as a problem solving tool (in conjunction with analytic geometry and trigonometry) by exploiting an informal understanding of differentials (infinitesimals). As much as possible large, interesting, and important historical problems (the motion of falling bodies and trajectories, the shape of hanging chains, the Witch of Agnesi) are used to develop key ideas. Only after skill with the computational tools of calculus has been developed is the question of rigor seriously broached. At that point, the …

Double Barrier Backward Doubly Stochastic Differential Equations, Tadashi Hayashi

Journal of Stochastic Analysis

No abstract provided.

Symmetric Functions Algebras (Sfa) Iii: Stochastic And Constant Row Sum Matrices, Philip Feinsilver

Journal of Stochastic Analysis

No abstract provided.

Signings Of Graphs And Sign-Symmetric Signed Graphs, Ahmad Asiri

Theses and Dissertations

In this dissertation, we investigate various aspects of signed graphs, with a particular focus on signings and sign-symmetric signed graphs. We begin by examining the complete graph on six vertices with one edge deleted ($K_6$\textbackslash e) and explore the different ways of signing this graph up to switching isomorphism. We determine the frustration index (number) of these signings and investigate the existence of sign-symmetric signed graphs. We then extend our study to the $K_6$\textbackslash 2e graph and the McGee graph with exactly two negative edges. We investigate the distinct ways of signing these graphs up to switching isomorphism and demonstrate …

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 …

Unilinear Residuated Lattices, Xiao Zhuang

Electronic Theses and Dissertations

We characterize all residuated lattices that have height equal to 3 and show that the variety they generate has continuum-many subvarieties. More generally, we study unilinear residuated lattices: their lattice is a union of disjoint incomparable chains, with bounds added. We give the characterization of all unilinear residuated lattices. By presenting the constructions and axiomatizations for different classes of unilinear residuated lattices, we conclude that the study of unilinear residuated lattices can be reduced to the study of the ⊤-unital ones. Using the classification of unilinear residuated lattices, the idempotent unilinear residuated lattices are studied and amalgamation property and strong …

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 …

A Dilation Theoretic Approach To Approximation By Inner Functions, Daniel Alpay, Tirthankar Bhattacharyya, Abhay Jindal, Poornendu Kumar

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using results from the theory of operators on a Hilbert space, we prove approximation results for matrix-valued holomorphic functions on the unit disc and the unit bidisc. The essential tools are the theory of unitary dilation of a contraction and the realization formula for functions in the unit ball of . We first prove a generalization of a result of Carathéodory. This generalization has many applications. A uniform approximation result for matrix-valued holomorphic functions which extend continuously to the unit circle is proved using the Potapov factorization. This generalizes a theorem due to Fisher. Approximation results are proved for matrix-valued …

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.

Platonism, De Re, And (Philosophy Of) Mathematical Practice, Marco Panza

MPP Published Research

The chapter advances a reformulation of the classical problem of the nature of mathematical objects (if any), here called “Plato’s problem,” in line with the program of a philosophy of mathematical practice. It then provides a sketch of a platonist solution, following the same perspective. This solution disregards as nonsensical the question of the existence of abstract, and specifically mathematical, objects, by rather focusing on the modalities of our access to them: objects (in general, both concrete and abstract) are regarded as individual contents that we have (or can have) a de re epistemic access to. The question of the …

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.

Engaging Students With High-Stakes Problems, Deepak Basyal

Mathematics and Statistics

Engaging students in meaningful mathematics problem-solving is the intention of many education stakeholders around the world. Research suggests that the implementation of high-stakes problems in mathematics teaching is one way to strengthen students’ conceptual understanding. Many carefully crafted open-ended problems constitute high-stakes problems, and proper use of such problems in teaching and learning not only encourages learners’ flexible thinking but also helps detect their misconceptions. However, what is less practiced and understood is: how exactly one should aim to implement such problems in a classroom setting. Teaching pre-service middle school teachers for a few years using high-stakes (mostly open-ended problems) …

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.

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 …

Invariants Of 3-Braid And 4-Braid Links, Mark Essa Sukaiti

Theses

In this study, we established a connection between the Chebyshev polynomial of the first kind and the Jones polynomial of generalized weaving knots of type W(3,n,m).
Through our analysis, we demonstrated that the coefficients of the Jones polynomial of weaving knots are essentially the Whitney numbers of Lucas lattices which allowed us to find an explicit formula for the Alexander polynomial of weaving knots of typeW(3,n).
In addition to confirming Fox’s trapezoidal conjecture, we also discussed the zeroes of the Alexander Polynomial of weaving knots of type W(3,n) as they relate to Hoste’s conjecture. In addition, …

Propuesta De Un Modelo De Enseñanza En Las Matemáticas Enfocado En La Solución De Problemas En El Nivel Secundario En Puerto Rico, Carlos J. Colon Rivera

Theses and Dissertations

El propósito del estudio fue proponer un modelo educativo enfocado en la solución de problemas matemáticos en el nivel secundario, y se realizó la revisión sistemática para evaluarlo. El marco teórico incluyó teorías heurísticas y modelos educativos. La metodología de seis fases que se empleó en este estudio, incluyendo la formulación de preguntas investigativas, búsqueda de literatura, selección de investigaciones, levantamiento de información, análisis y resumen de resultados y exposición y discusión de estos. Se siguieron guías para revisiones sistemáticas y criterios de inclusión y exclusión para evaluar la efectividad de modelos didácticos en la disciplina de matemáticas con énfasis …

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

Journal of Stochastic Analysis

No abstract provided.

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 …

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 …

Mth 125 - Modeling With Exponential Functions, Stivi Manoku

Open Educational Resources

The file includes a variety of problems that emphasize the importance of modeling exponential growth and/or radioactive decay. Through different exercises and problems, the assignment goal is to improve their comprehension of exponential functions and hone their problem-solving abilities.

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 …

Project Based Learning, Christopher Imm

Sabbatical Projects

Project-based learning (PBL) is a pedagogy which encourages students to be actively involved in their learning process through gaining information to solve real-world problems with projects. In areas of STEM education, PBL can supplement the traditional classroom lecture, where students have a predominantly passive role in the learning process. Through practical activities, interactive discussions, independent reflection and/or team cooperation, students achieve the desired learning outcomes.

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).