Digital Commons Network^™

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.

Extremal Absorbing Sets In Low-Density Parity-Check Codes, Emily Mcmillon, Allison Beemer, Christine A. Kelley

Department of Mathematics: Faculty Publications

Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of b for a given a for which an (a, b)-absorbing set may exist. We identify certain cases of extremal …

Regularity Theory Of Quasilinear Ellipitic And Parabolic Equations In The Heisenberg Group, Luca Capogna, Giovanna Citti, Xiao Zhong

Mathematics Sciences: Faculty Publications

This note provides a succinct survey of the existing literature concerning the H¨older regularity for the gradient of weak solutions of PDEs of the form modeled on the p-Laplacian in a domain Ω in the Heisenberg group Hⁿ, with 1 ≤ p < ∞, and of its parabolic counterpart. We present some open problems and outline some of the difficulties they present.

Classification Of Seven-Dimensional Solvable Lie Algebras With A5,2 And A5,3 Nilradicals, Robert Dolan

Honors Projects

This paper provides a classification of seven-dimensional indecomposable solvable Lie algebras over the reals for which the nilradical is isomorphic to A_5,2 and A_5,3. We follow a technique that was first introduced by Mubarakzyanov.

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).

Using Visual Imagery To Develop Multiplication Fact Strategies, Gina Kling

Dissertations

The learning of basic facts, or the sums and products of numbers 0–10 and their related differences and quotients, has always been a high priority for elementary school teachers. While memorization of basic facts has been a hallmark of elementary school, current recommendations focus on a more nuanced development of fluency with these facts. Fluency is characterized by the ability to demonstrate flexibility, accuracy, efficiency, and appropriate strategy use. Despite recommendations to focus on strategy use, there is insufficient information on instructional approaches that are effective for developing strategies, particularly for multiplication facts. Using visual imagery with dot patterns has …

Irregular Domination In Graphs, Caryn Mays

Dissertations

Domination in graphs has been a popular area of study due in large degree to its applications to modern society as well as the mathematical beauty of the topic. While this area evidently began with the work of Claude Berge in 1958 and Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of a survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then, a large number of variations of domination have surfaced and provided numerous applications to different areas of science and real-life problems. Among these variations are domination parameters …

The Reproducing Kernel Method For Solving Integro-Differential And Volterra Integral Equations, Khulood Gamal Qaid

Theses

Integro-differential equations are a class of mathematical equations that involve both derivatives and integrals. They have applications in a wide range of fields, including physics, engineering, finance, and biology such as the spread of diseases, population dynamics, and the behavior of financial markets. The study of these equations requires advanced mathematical techniques, including functional analysis, approximation methods, and numerical analysis. They are a rich area of research with many open questions and challenges.
In this thesis, we will develop and implement the reproducing kernel method to solve a class of integro-differential and Volterra integral equations. We discuss both cases when …

Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof, Zachary Warren

Senior Honors Theses

It is a well-known property of the integers, that given any nonzero a ∈ Z, where a is not a unit, we are able to write a as a unique product of prime numbers. This is because the Fundamental Theorem of Arithmetic (FTA) holds in the integers and guarantees (1) that such a factorization exists, and (2) that it is unique. As we look at other domains, however, specifically those of the form O(√D) = {a + b√D | a, b ∈ Z, D a negative, squarefree integer}, we find that …

Number Theoretic Arithmetic Functions And Dirichlet Series, Ivan V. Morozov

Publications and Research

In this study, we will study number theoretic functions and their associated Dirichlet series. This study lay the foundation for deep research that has applications in cryptography and theoretical studies. Our work will expand known results and venture into the complex plane.

Structure Of Extremal Unit Distance Graphs, Kaylee Weatherspoon

Senior Theses

This thesis begins with a selective overview of problems in geometric graph theory, a rapidly evolving subfield of discrete mathematics. We then narrow our focus to the study of unit-distance graphs, Euclidean coloring problems, rigidity theory and the interplay among these topics. After expounding on the limitations we face when attempting to characterize finite, separable edge-maximal unit-distance graphs, we engage an interesting Diophantine problem arising in this endeavor. Finally, we present a novel subclass of finite, separable edge-maximal unit distance graphs obtained as part of the author's undergraduate research experience.

An Exploration Of Computational Text Analysis Of Co-Design Discourse In A Research-Practice Partnership, Mei Tan, Victor R. Lee

Publications

In combination with contextualized human interpretation, computational text analysis offers a quantitative approach to interrogating the nature of participation and social positioning in discourse. Using meeting transcript data from the development of a co-design research-practice partnership, we examine the roles and forms of participation that contribute to an effective collaboration between a multileveled school system and researcher partners. We apply computational methods to explore the language of co-design and multi-stakeholder perspectives in support of educational improvement science efforts and our theoretical understanding of partnership roles. Results indicate participation patterns align with documented roles in co- design partnerships and highlight the …

Geometry And Coding: Introducing An Interactive And Integrated Mathematics-Computer Science Unit, Kimberly Beck, Jessica F. Shumway

Publications

As part of a collaborative project between Utah State University, the Cache County School District, and Stanford, instructional units were designed for fifth-grade students. These units integrated math concepts of geometrical shapes and computer science concepts of sequences, conditionals, and loops. One component of the unit was implemented in math classrooms by math teachers, and the other component was implemented in computer labs. This presentation will focus on the math unit as presented at the National Council of Teachers of Mathematics (NCTM-V).

Extreme Covering Systems, Primes Plus Squarefrees, And Lattice Points Close To A Helix, Jack Robert Dalton

Theses and Dissertations

This dissertation considers three different topics.

In the first part, we prove that if the least modulus of a distinct covering system is 4, its largest modulus is at least 60; also, if the least modulus is 3, the least common multiple of the moduli is at least 120; finally, if the least modulus is 4, the least common multiple of the moduli is at least 360. The constants 60, 120, and 360 are best possible, they cannot be replaced by larger constants. We also show that there do not exist distinct covering systems with all of the moduli in …

On The Algebraic And Geometric Multiplicity Of Zero As A Hypergraph Eigenvalue, Grant Ian Fickes

Theses and Dissertations

We consider the algebraic and geometric multiplicity of hypergraph eigenvalues, paying particular attention to nullities of hypertrees. Conjecture surrounds the relationship between these two multiplicites, but little work appears in the literature on this topic. Predominantly, we are interested in identifying the geometric structure of the nullvariety of certain hypergraph classes by listing the irreducible components and their accompanying dimensions. When applicable, we use this description to verify a conjecture relating the algebraic and geometric nullity. Furthermore, we refine these geometric structure tools to graph theoretic trees, defining a matroid on trees and relating skew zero forcing to the existence …

K-8 Preservice Teachers’ Statistical Thinking When Determining Best Measure Of Center, Ha Nguyen, Eryn M. Stehr Maher, Gregory Chamblee, Sharon Taylor

Mathematical Sciences: Faculty Publications

The purpose of this study was to determine K-8 preservice teacher (PST) candidates’ statistical thinking when selecting the best center representation for the given data. Forty-four PSTs enrolled in a Statistics and Probability for K-8 Teachers course in a university located in the southeastern region of the United States were asked to complete a 2007 National Assessment of Educational Progress test item. All 44 PSTs’ data were qualitatively analyzed for correctness and statistical thinking strategies used. Findings were that most PSTs either incorrectly selected the mean, rather than median, as the best measure of center for the given data or …

Ramanujan–Sato Series For 1/Π, Timothy Huber, Daniel Schultz, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

We compute Ramanujan–Sato series systematically in terms of Thompson series and their modular equations. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived.

P-Adic Cellular Neural Networks: Applications To Image Processing, B. A. Zambrano-Luna, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

The p-adic cellular neural networks (CNNs) are mathematical generalizations of the neural networks introduced by Chua and Yang in the 80s. In this work we present two new types of -adic CNNs that can perform computations with real data, and whose dynamics can be understood almost completely. The first type of networks are edge detectors for grayscale images. The stationary states of these networks are organized hierarchically in a lattice structure. The dynamics of any of these networks consists of transitions toward some minimal state in the lattice. The second type is a new class of reaction–diffusion networks. We investigate …

Counting Elliptic Curves With A Cyclic M-Isogeny Over Q, Grant S. Molnar

Dartmouth College Ph.D Dissertations

Using methods from analytic number theory, for m > 5 and for m = 4, we obtain asymptotics with power-saving error terms for counts of elliptic curves with a cyclic m-isogeny up to quadratic twist over the rational numbers. For m > 5, we then apply a Tauberian theorem to achieve asymptotics with power saving error for counts of elliptic curves with a cyclic m-isogeny up to isomorphism over the rational numbers.

A Support Theorem For A Wave Equation, Aysha Khaled Alshamsi

Theses

It is well known that the fundamental solution to the classical wave equation Δ�� (��, ��) − ∂_������(��,��) = 0 is supported on the light cone {(��, ��) ∈ ℝ^��× ℝ : ||��|| = |��|} if and only if the dimension �� is odd and ≥ 3. Because we are living in a 3-dimensional world we can hear each other clearly; One has a pure propagator without residual waves. In this thesis we consider the wave equation

2||��||Δ_����_��(��, ��) − ∂_������(��,��) = 0, (��, ��) ∈ ℝ^�� × ℝ …

Valuation Of Asian Options In A High Volatility Market With Jumps, Zeeshan Khalid

Theses

The evaluation of financial derivatives represents a central part of financial risk management. There are many types of derivatives among other path-dependent options. In this study, we aim at valuing Asian options. They are path dependent and have several benefits. For instance, their values are habitually lower than European options. Also, an Asian option on a commodity drops the risk value close to maturity. Though, the disadvantage is that they are in general difficult to value since the distribution of the payoff is usually unknown. It is agreed in the literature that a stochastic process with a jumps model for …

Enots Wolley Variations And Related Sequences, Nathan Myles Nichols

Master's Theses (2009 -)

The Enots Wolley sequence is a lexicographically earliest sequence (LES) that is closely related to the Yellowstone sequence. It is an open conjecture by N. J. Sloane that every number with at least two distinct prime factors appears as a term of the Enots Wolley sequence. In this thesis, this conjecture is proved for a variation of the Enots Wolley sequence that operates on the binary representation of a positive integer rather than the prime factorization. The methods used are then applied to prove some new properties of the prime factorization Enots Wolley sequence.

A Short Solution Of The Kissing Number Problem In Dimension Three, Alexey Glazyrin

School of Mathematical & Statistical Sciences Faculty Publications

We give a short solution of the kissing number problem in dimension three.

Dynamic Equations, Control Problems On Time Scales, And Chaotic Systems, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

The unification of integral and differential calculus with the calculus of finite differences has been rendered possible by providing a formal structure to study hybrid discrete-continuous dynamical systems besides offering applications in diverse fields that require simultaneous modeling of discrete and continuous data concerning dynamic equations on time scales. Therefore, the theory of time scales provides a unification between the calculus of the theory of difference equations with the theory of differential equations. In addition, it has become possible to examine diverse application problems more precisely by the use of dynamical systems on time scales whose calculus is made up …

Mth 50 Syllabus, Koby Kohulan

Open Educational Resources

No abstract provided.

Matrix Models Of 2d Critical Phenomena, Nathan Hayford

USF Tampa Graduate Theses and Dissertations

The 2D Ising model has played an important role in the theory of phase transitions, as one of only ahandful of exactly solvable models in statistical mechanics. The original model, introduced in the 1920s, has a rich mathematical structure. It thus came as a pleasant surprise when physicists studying matrix models of 2D gravity found that, coupled to quantum gravity, the planar Ising model still had an elegant solution. The methods used by V. Kazakov and his collaborators involved the method of orthogonal polynomials. However, these methods were formal, and no direct analytic derivation of the phase transition has been …

Recovering Generators Of Principal Ideals Using Subfield Structure And Applications To Cryptography, William Youmans

USF Tampa Graduate Theses and Dissertations

The principal ideal problem (PIP) is the problem of determining if a given ideal of a number field is principal, and if so, of finding a generator.Algorithms for resolving the PIP can be efficiently adapted to solve many hard problems in algebraic number theory, such as the computation of the class group, unit group, or $S$-unit group of a number field. The PIP is also connected to the search for approximate short vectors, known as the $\gamma$-Shortest Vector Problem ($\gamma$-SVP), in certain structured lattices called ideal lattices, which are prevalent in cryptography. We present an algorithm for resolving the PIP …

Semi-Analytic Approach To Solving Rosenau-Hyman And Korteweg-De Vries Equations Using Integral Transform, Adedapo Chis Loyinmi, Kabir Oluwatobi Idowu

Tanzania Journal of Science

In this research, we proposed the fusing of Elzaki transform and projected differential transform (PDTM) to obtain an analytical or approximate solution of the Rosenau-Hyman and Korteweg-de Vries equations which respectively govern pattern formation in liquid drops and model of waves on shallow water surfaces. The results obtained presented in tables and graphs showed better efficiency, accuracy, and convergence of the method to handle Rosenau-Hyman and Korteweg-de Vries equations when compared to other methods in the literature. Keywords:          Rosenau-Hyman Equation; Korteweg-de Vries equation; Elzaki Projected differential transform method; Semi-analytic approach

Simulation Of Mathematical Model Of Network Interference On Global System For Mobile Communication, Yidiat O. Aderinto, Yusuf Musa

Tanzania Journal of Science

Network interference is the incorporation of undesirable signals to desirable ones in an interconnected system. In this paper, the simulation of mathematical model of network interference with respect to global system of mobile communication is presented. The model was formulated and analyzed mathematically. Numerical simulation was carried out using real life data from Communication Towers Nigeria Limited, Northwest regional office Kaduna, Nigeria. The results obtained were very close to that of laboratory investigation. Keywords: Mathematical Model, Network, Interference, Congestion, Mobile communication, Stability

Data-Driven Learning Algorithm Via Densely-Defined Multiplication Operators And Occupation Kernels., John Kyei

USF Tampa Graduate Theses and Dissertations

Consider a nonautonomous nonlinear evolution $\dot{x}=f(x,t,\mu)$, where the vector $x(t) \in \mathbb{R}^n$ represents the state of the dynamical system at time $t$, $\mu$ contains system parameters, and $f(\cdot)$ represents a dynamic constraint. In most practical applications, the nonlinear dynamic constraint $f$ is unknown analytically. The problem of approximating $f$ directly from data measurements generated by the system is a main goal of this manuscript. In the postulates of the Nonlinear Autoregressive (NAR) framework, we show that the problem of approximating $f$ can be studied through symbols of densely defined multiplication operators over a Reproducing Kernel Hilbert Spaces (RKHS). In this …