# Mathematics

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Item Open Access An Introduction to Categories and Homological Algebra: A Tale of Four Functors(East Carolina University, July 2024) Ragland, Caedmon; Dr. Heather Ries; Dr. Chris Jantzen; Dr. Michael SpurrShow more We assume familiarity with group theory and discuss the technical background necessary to construct and understand the functors Hom, Ext, T or and tensor on the category of abelian groups, with special attention paid to Hom and Ext. In doing so, we hope to provide an introduction to the study of category theory and homological algebra understandable to an undergraduate with 1-2 semesters of abstract algebra.Show more Item Open Access First Order Definition of Rings Using Group of Units(East Carolina University, 2023-05-03) Kardos, Michael; MathematicsShow more We discuss the technical background and relevant research regarding the undecidability of $O_{\mathbb{Q}^{\text{ab}}}$. Given an algebraic extension $K/\mathbb{Q}$, we consider the subring defined by $$R_K=\{x\in O_K\,|\,\forall \varepsilon\in U_K\setminus\{1\}\,\exists\delta\in U_K:\delta-1\equiv x(\varepsilon-1)\bmod(\varepsilon-1)^2\}.$$ We later consider a similar construction over subrings of $\mathbb{Q}$ of characteristic $0$. In doing this, we hope to gain insight into the result of the construction of $R_K$ when $K=\mathbb{Q}^{\text{ab}}$.Show more Item Open Access Inferences Over Fields: A Preliminary Investigation into the Deductive Capabilities of Field-Theoretically Defined Logical Connectives(East Carolina University, 2023-05-03) Crumpler, Charles Wingate; MathematicsShow more In this paper, we will be concerned with developing an inferential structure over the field with four elements in characteristic $2$. We begin by discussing the historical context in which this research occurs. In subsequent sections, we will construct the field, called $\F_4$ and describing the algebraic structure over $\F_4$. We then define the connectives $\wedge$, $\vee$, and $\neg$ over $\F_4$ by extending their standard definition over $\F_2$. We define the basic syntax and semantics of $\F_4$. We show that $\wedge$ and $\vee$ are dual over $\F_4$ with respect to $\neg$ and that $\F_4$ is functionally complete over $\{ \ \wedge, \vee, \neg \ \} \cup \F_4$. We develop a notion of inferences over $\F_4$ by imbuing it with a partial order, defining validity and the material implication, and defining a proof. Upon completing this, we prove the Deduction, Soundness, and Completeness Theorems, thereby showing that inferences over $\F_4$ behaves in ways comparable to, but not equivalent to, those over a field of two values in characteristic $2$.Show more Item Open Access Boundary Quotient C*-Algebras of Semigroups(2022) Katsoulis, Elias G.; Kakariadis, Evgenios T.A.; Laca, Marcelo; Li, XinShow more Item Open Access Classification of Kuga Fiber Varieties(2022) Abdulali, SalmanShow more Item Open Access The Isomorphism Problem for Tensor Algebras of Multivariable Dynamical Systems(2022) Katsoulis, Elias G.; Ramsey, ChristopherShow more Item Open Access The Krein–Von Neumann Extension Revisited(2022) Fucci, Guglielmo; Gesztesy, Fritz; Littlejohn, Lance L.; Nichol, Roger; Nichols, Roger; Stanfill, JonathanShow more Item Open Access On Existential Definitions of C.E. Subsets of Rings of Functions of Characteristic 0(2022) Shlapentokh, Alexandra; Miller, RussellShow more Item Open Access C*-Envelopes for Operator Algebras with a Coaction and Co-Universal C*-Algebras for Product Systems(2022) Katsoulis, E.; Dor-on, A.; Kakariadis, E.T.A.; Laca, M.; Li, X.Show more Item Open Access Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms(2022) Pravica, David W.; Randriampiry, Njinasoa; Spurr, Michael J.Show more Item Open Access Mathematical Model of Subthalamic Nucleus Neuron: Characteristic Activity Patterns and Bifurcation Analysis(2021) Ahn, S.; Park, Choongseok; Rubchinsky, Leonid L.Show more Item Open Access Casimir Pistons with Generalized Boundary Conditions: a Step Forward(2021-02-17) Fucci, Guglielmo; Kirsten, Klaus; Mu˜noz-Casta˜neda, Jose M.Show more Item Open Access Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms(2022-07-07) Pravica, David W.; Randriampiry, Njinasoa; Spurr, Michael J.Show more For a wide class of solutions to multiplicatively advanced differential equations (MADEs), a comprehensive set of relations is established between their Fourier transforms and Jacobi theta functions. In demonstrating this set of relations, the current study forges a systematic connection between the theory of MADEs and that of special functions. In a large subset of the general case, we introduce a new family of Schwartz wavelet MADE solutions Wμ,λðtÞ for μ and λ rational with λ > 0. These Wμ,λðtÞ have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from λ, the connection of the Wμ,λðtÞ to the theory of wavelet frames is begun. For a second set of low parameter values derived from λ, the notion of a canonical extension is introduced. A number of examples are discussed. The study of convergence of the MADE solution to the solution of its analogous ODE is begun via an in depth analysis of a normalized example W−4/3,1/3ðtÞ/W−4/3,1/3ð0Þ. A useful set of generalized q-Wallis formulas are developed that play a key role in this study of convergence.Show more Item Open Access Theorems in Visual Art: Art and Math Teacher Collaboration toward Creative Leadership(2018) Song, Borim; Choi, JungminShow more An artist and a mathematician meet. The authors of this article, one an art education professor and the other a mathematics professor, collaborated to conduct research as a team, examining perceptions of K-12 art teachers regarding art and math integration. Because of changes in education such as the implementation of the Common Core State Standards and the Every Student Succeeds Act, some educational policymakers, administrators, and teachers have expressed interest in integrated curricula (Davis, Sumara, & Luce-Kapler, 2008; Franco & Unrath, 2014; Wexler, 2014). Recent research supports the positive impact of visual art learning on students’ test scores. For example, research outcomes from the Turnaround Arts Initiative indicated that the schools participating in the arts initiative demonstrated a 22.55% improvement in math proficiency (Turnaround: Arts Creating Success in Schools, 2016). In this Advisory, we want to share what we have learned from our partnership and exploration of visual art and math integration to help art teachers collaborate actively and efficiently with math teachers in their schools. Our project started with real-world problems. As parents of school-age children, we shared our concerns about changes in the math curriculum based on the Common Core State Standards and the uncertain status of art education in the public school system in our state. Conversations about these issues led us to devise an interdisciplinary research project about art and math integration. Our collaboration enabled us to work together and expand our knowledge and understanding, sometimes beyond our comfort zone, and to find new ways of practicing our disciplines.Show more Item Open Access IDENTIFYING SUBJECTIVE VALUE IN WOMEN’S COLLEGE GOLF RECRUITING REGARDLESS OF SOCIO-ECONOMIC CLASS(East Carolina University, 2018-05-03) Allred, Victoria; Carolan, Christopher (Christopher A.); MathematicsShow more College athletics have grown into a major industry and athletic departments are pushing coaches to recruit the top talent. To recruit golfers, college coaches depend on multiple ranking systems. These systems are bias towards more expensive, national tournaments over inexpensive, state tournaments. In response, players who come from a lower socio-economic class will have fewer financial aid opportunities than someone from a higher economic class. Analyzing junior girls’ golf statistics, has led to the creation of an objective methodology to compare golfers without regard to socio-economic imbalances.Show more Item Open Access Theoretical aspects and modelling of cellular decision making, cell killing and information-processing in photodynamic therapy of cancer(2013) Gkigkitzis, IoannisShow more BACKGROUND: The aim of this report is to provide a mathematical model of the mechanism for making binary fate decisions about cell death or survival, during and after Photodynamic Therapy (PDT) treatment, and to supply the logical design for this decision mechanism as an application of rate distortion theory to the biochemical processing of information by the physical system of a cell. METHODS: Based on system biology models of the molecular interactions involved in the PDT processes previously established, and regarding a cellular decision-making system as a noisy communication channel, we use rate distortion theory to design a time dependent Blahut-Arimoto algorithm where the input is a stimulus vector composed of the time dependent concentrations of three PDT related cell death signaling molecules and the output is a cell fate decision. The molecular concentrations are determined by a group of rate equations. The basic steps are: initialize the probability of the cell fate decision, compute the conditional probability distribution that minimizes the mutual information between input and output, compute the cell probability of cell fate decision that minimizes the mutual information and repeat the last two steps until the probabilities converge. Advance to the next discrete time point and repeat the process. RESULTS: Based on the model from communication theory described in this work, and assuming that the activation of the death signal processing occurs when any of the molecular stimulants increases higher than a predefined threshold (50% of the maximum concentrations), for 1800s of treatment, the cell undergoes necrosis within the first 30 minutes with probability range 90.0%-99.99% and in the case of repair/survival, it goes through apoptosis within 3-4 hours with probability range 90.00%-99.00%. Although, there is no experimental validation of the model at this moment, it reproduces some patterns of survival ratios of predicted experimental data. CONCLUSIONS: Analytical modeling based on cell death signaling molecules has been shown to be an independent and useful tool for prediction of cell surviving response to PDT. The model can be adjusted to provide important insights for cellular response to other treatments such as hyperthermia, and diseases such as neurodegeneration.Show more Item Open Access The Spectral Theorem for Self-Adjoint Operators(East Carolina University, 2016-04-25) Chilcoat, Kenneth; Ratcliff, Gail Dawn Loraine; MathematicsShow more The Spectral Theorem for Self-Adjoint Operators allows one to define what it means to evaluate a function on the operator for a large class of functions defined on the spectrum of the operator. This is done by developing a functional calculus that extends the intuitive notion of evaluating a polynomial on an operator. The Spectral Theorem is fundamentally important to operator theory and has applications in many fields, especially harmonic analysis on locally compact abelian groups. This thesis represents a merging of two traditional treatments of the Spectral Theorem and includes an extended example highlighting an important application in harmonic analysis.Show more Item Open Access Modeling Tsunami Waves Using Q-Advanced Waves in 2-D(East Carolina University, 2015-12-15) Cook, Cameron; Spurr, Michael J.; MathematicsShow more A two-dimensional numerical approximation for modeling tsunamis is developed. New q-advanced functions are used to model the forcing due to an earthquake. These results are used to model the Japanese tsunami of 2011, and compared to NOAA data generated by this earthquake and obtained on DART buoys at Wake Island.Show more Item Open Access Eigenvalues for Sums of Hermitian Matrices(East Carolina University, 2015) Taylor, James M.; Benson, Chal; MathematicsShow more In this thesis we explore how the eigenvalues of nxn Hermitian matrices A,B relate to the eigenvalues of their sum C=A+B. We mainly focus on inequalities bounding sums of r eigenvalues for C by sums of r eigenvalues for A with r eigenvalues for B, for some r less than n. We begin by using linear algebra to establish some classical results, including a result by R.C. Thompson that allows us to reformulate the eigenvalue problem in terms of nonempty intersections in the Grassmannian manifold of r-planes in complex n-dimensional space. In particular, every nonempty triple intersection of Schubert varieties in a Grassmannian yields an eigenvalue inequality. Such nonempty intersections correspond to nonzero cup products in the cohomology ring of the Grassmannian, and consequently, to nonzero Littlewood-Richardson coefficients. The Littlewood-Richardson rules provide us with an efficient method of detecting when these coefficients are nonzero, and hence of finding eigenvalue inequalities which necessarily hold for all nxn Hermitian matrices A,B,C=A+B. For the remainder of this thesis, we turn our attention to particular inequalities of the above form that Alfred Horn conjectured would completely determine the possible eigenvalues of A,B,C=A+B. Horn's conjecture, formulated in 1962, was resolved in the affirmative during the late 1990's in celebrated work of A. Knutson and T. Tao, building on results of A. Klyachko and others. We will develop the connection between Horn's inequalities and the earlier parts of this thesis. In particular, we will see that each Horn inequality corresponds to a nonzero cup product that lies in the top degree cohomology group of the Grassmannian. An alternate formulation of Horn's Theorem shows that indices yield a Horn inequality if and only if certain associated partitions occur as the eigenvalues for some rxr Hermitian matrices A, B, C=A+B. We will prove that when r=n-2 there are necessarily diagonal rxr matrices satisfying this condition.Show more Item Open Access Markov Chains, Random Walks, and Card Shuffling(East Carolina University, 2016) Outlaw, Nolan; Ratcliff, Gail Dawn Loraine; MathematicsShow more A common question in the study of random processes pertains to card shuffling. Whether or not a deck of cards is random can have huge implications on any game being played with those particular cards. This thesis explores the question of randomness by using techniques established through analysis of Markov chains, random walks, computer simulations, and some basic shuffling models. Ultimately, the aim is to explore the cutoff phenomenon, which asserts that at some point during the shuffling process there is a sharp decline in the shuffled deck's distance from random.Show more