Mathematics Commons^™

A General Setting For Functions Of Fueter Variables: Differentiability, Rational Functions, Fock Module And Related Topics, Daniel Alpay, Ismael L. Paiva, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

We develop some aspects of the theory of hyperholomorphic functions whose values are taken in a Banach algebra over a field—assumed to be the real or the complex numbers—and which contains the field. Notably, we consider Fueter expansions, Gleason’s problem, the theory of hyperholomorphic rational functions, modules of Fueter series, and related problems. Such a framework includes many familiar algebras as particular cases. The quaternions, the split quaternions, the Clifford algebras, the ternary algebra, and the Grassmann algebra are a few examples of them.

Art, Artfulness, Or Artifice?: A Review Of The Art Of Statistics: How To Learn From Data, By David Spiegelhalter, Jason Makansi

Numeracy

David Spiegelhalter. 2019. The Art of Statistics: How to Learn From Data. (London: The Penguin Group). 444 pp. ISBN 978-1541618510

The author successfully eases the reader away from the rigor of statistical methods and calculations and into the realm of statistical thinking. Despite an engaging style and attention-grabbing examples, the reader of The Art of Statistics will need more than a casual grounding in statistics to get what Spiegelhalter, I believe, intends from his book. It should be viewed as a companion to a more rigorous textbook on statistical methods but not necessarily a book that makes statistics any …

Locally Recoverable Codes From Planar Graphs, Kathryn Haymaker, Justin O'Pella

Kanbar College Faculty Papers

In this paper we apply Kadhe and Calderbank’s definition of LRCs from convex polyhedra and planar graphs [4] to analyze the codes resulting from 3-connected regular and almost regular planar graphs. The resulting edge codes are locally recoverable with availability two. We prove that the minimum distance of planar graph LRCs is equal to the girth of the graph, and we also establish a new bound on the rate of planar graph edge codes. Constructions of regular and almost regular planar graphs are given, and their associated code parameters are determined. In certain cases, the code families meet the rate …

On Possibility Of Binary Companion Of The Sun: A Serendipity Finding And Comparison With Uvs Model Of Solar System, Victor Christianto, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

While we completely understood that a binary dwarf companion of the Sun has not been accepted by majority of astronomers, allow us to present some new arguments, along with our own serendipitous encounter with such a binary companion of the Sun. We hope that the present note will be found useful for further investigations, in relation to Planet Nine and such a dwarf star companion of the Sun (sometimes dubbed as Nemesis). Nonetheless, this article is not an exhaustive review of such dwarf companion star theories.

Finite Element Analysis Of An Arbitrary Lagrangian–Eulerian Method For Stokes/Parabolic Moving Interface Problem With Jump Coefficients, Rihui Lan, Michael J. Ramirez, Pengtao Sun

Mathematical Sciences Faculty Research

In this paper, a type of arbitrary Lagrangian–Eulerian (ALE) finite element method in the monolithic frame is developed for a linearized fluid–structure interaction (FSI) problem — an unsteady Stokes/parabolic interface problem with jump coefficients and moving interface, where, the corresponding mixed finite element approximation in both semi- and fully discrete scheme are developed and analyzed based upon one type of ALE formulation and a novel H1- projection technique associated with a moving interface problem, and the stability and optimal convergence properties in the energy norm are obtained for both discretizations to approximate the solution of a transient Stokes/parabolic interface problem …

𝑘-Plane Constant Curvature Conditions, Maxine E. Calle

Rose-Hulman Undergraduate Mathematics Journal

This research generalizes the two invariants known as constant sectional curvature (csc) and constant vector curvature (cvc). We use k-plane scalar curvature to investigate the higher-dimensional analogues of these curvature conditions in Riemannian spaces of arbitrary finite dimension. Many of our results coincide with the known features of the classical k=2 case. We show that a space with constant k-plane scalar curvature has a uniquely determined tensor and that a tensor can be recovered from its k-plane scalar curvature measurements. Through two example spaces with canonical tensors, we demonstrate a method for determining constant k-plane …

Isoperimetric Problems On The Line With Density |𝑥|ᵖ, Juiyu Huang, Xinkai Qian, Yiheng Pan, Mulei Xu, Lu Yang, Junfei Zhou

Rose-Hulman Undergraduate Mathematics Journal

On the line with density |x|^p, we prove that the best single bubble is an interval with endpoint at the origin and that the best double bubble is two adjacent intervals that meet at the origin.

The Isoperimetric Inequality: Proofs By Convex And Differential Geometry, Penelope Gehring

Rose-Hulman Undergraduate Mathematics Journal

The Isoperimetric Inequality has many different proofs using methods from diverse mathematical fields. In the paper, two methods to prove this inequality will be shown and compared. First the 2-dimensional case will be proven by tools of elementary differential geometry and Fourier analysis. Afterwards the theory of convex geometry will briefly be introduced and will be used to prove the Brunn--Minkowski-Inequality. Using this inequality, the Isoperimetric Inquality in n dimensions will be shown.

On Consecutive Triples Of Powerful Numbers, Edward Beckon

Rose-Hulman Undergraduate Mathematics Journal

A powerful number is a positive integer such that every prime that appears in its prime factorization appears there at least twice. Erdős, Mollin and Walsh conjectured that three consecutive powerful numbers do not exist. This paper shows that if they do exist, the smallest of the three numbers must have remainder 7, 27, or 35 when divided by 36.

Consecutive Prime And Highly Total Prime Labeling In Graphs, Robert Scholle

Rose-Hulman Undergraduate Mathematics Journal

This paper examines the graph-theoretical concepts of consecutive prime labeling and highly total prime labeling. These are variations on prime labeling, introduced by Tout, Dabboucy, and Howalla in 1982. Consecutive prime labeling is defined here for the first time. Consecutive prime labeling requires that the labels of vertices in a graph be relatively prime to the labels of all adjacent vertices as well as all incident edges. We show that all paths, cycles, stars, and complete graphs have a consecutive prime labeling and conjecture that all simple connected graphs have a consecutive prime labeling.

This paper also expands on work …

Combinatorial Identities On Multinomial Coefficients And Graph Theory, Seungho Lee

Rose-Hulman Undergraduate Mathematics Journal

We study combinatorial identities on multinomial coefficients. In particular, we present several new ways to count the connected labeled graphs using multinomial coefficients.

Graham's Pebbling Conjecture Holds For The Product Of A Graph And A Sufficiently Large Complete Graph, Nopparat Pleanmani

Theory & Applications of Graphs

For connected graphs G and H, Graham conjectured that π(G □ H) ≤ π(G) π(H) where π(G), π (H), and π(G □ H) are the pebbling numbers of G, H, and the Cartesian product G □ H, respectively. In this paper, we show that the inequality holds when H is a complete graph of sufficiently large order in terms of graph parameters of G.

Ziplines And Stuntwork, Kelly W. Remijan

Teacher Resources

This activity involves an engineering activity which connects the work of stuntmen/stuntwomen working with ziplines to the concept of linear functions. Students create a physical model replicating a given situation and then model the zipline algebraically by writing the equation of the zipline.

Identifying Group Contributions In Nba Lineups With Spectral Analysis, Stephen Devlin, David Uminsky

Mathematics

We address the question of how to quantify the contributions of groups of players to team success. Our approach is based on spectral analysis, a technique from algebraic signal processing, which has several appealing features. First, our analysis decomposes the team success signal into components that are naturally understood as the contributions of player groups of a given size: individuals, pairs, triples, fours, and full five-player lineups. Secondly, the decomposition is orthogonal so that contributions of a player group can be thought of as pure: Contributions attributed to a group of three, for example, have been separated from the lower-order …

Level Algebras And S-Lecture Hall Polytopes, Mccabe Olsen, Florian Kohl

Faculty Publications - Mathematics

No abstract provided.

Single-Seed Cascades On Clustered Networks. Network Science, John Mcsweeney

Faculty Publications - Mathematics

No abstract provided.

Gehring Inequalities On Time Scales, Martin Bohner, S. H. Saker

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we first prove a new dynamic inequality based on an application of the time scales version of a Hardy-type inequality. Second, by employing the obtained inequality, we prove several Gehring-type inequalities on time scales. As an application of our Gehring-type inequalities, we present some interpolation and higher integrability theorems on time scales. The results as special cases, when the time scale is equal to the set of all real numbers, contain some known results, and when the time scale is equal to the set of all integers, the results are essentially new.

A Game-Theoretic Model Of Monkeypox To Assess Vaccination Strategies, Sri Vibhaav Bankuru, Samuel Kossol, William Hou, Parsa Mahmoudi, Jan Rychtář, Dewey Taylor

Mathematics and Applied Mathematics Publications

Monkeypox (MPX) is a zoonotic disease similar to smallpox. Its fatality rate is about 11% and it is endemic to the Central and West African countries. In this paper, we analyze a compartmental model of MPX dynamics. Our goal is to see whether MPX can be controlled and eradicated by voluntary vaccinations. We show that there are three equilibria—disease free, fully endemic and previously neglected semi-endemic (with disease existing only among humans). The existence of semi-endemic equilibrium has severe implications should the MPX virus mutate to increased viral fitness in humans. We find that MPX is controllable and can be …

Luminescence Emission In A Nanocrystal Doped By A Transition Metal Impurity, George Chappell Jr.

Theses, Dissertations and Capstones

In this thesis we consider the structure of magnetic ion centers with 3d-electrons in quantum dots under the effects of Coulomb and exchange interaction between the 3d-electrons of the impurity centers and the confined electrons (or holes) existing inside the nanocrystals. In particular, we are interested how this interaction changes the photoluminescence properties of those materials. We will make use of representation theory and the symmetry of the crystal structure to find the orthonormal wave functions that make up the wave functions of the outer, 3d-electrons inside our dot. The Coulomb and exchange interaction …

Markov Chain Epidemic Models And Parameter Estimation, Oluwatobiloba Ige

Theses, Dissertations and Capstones

Over the years, various parts of the world have experienced disease outbreaks. Mathematical models are used to describe these outbreaks. We study the transmission of disease in simple cases of disease outbreaks by using compartmental models with Markov chains. First, we explore the formulation of compartmental SIS (Susceptible-Infectious-Susceptible) and SIR (Susceptible-Infectious-Recovered) disease models. These models are the basic building blocks of other compartmental disease models. Second, we build SIS and SIR disease models using both discrete and continuous time Markov chains. In discrete time models, transmission occurs at fixed time steps, and in continuous time models, transmission may occur at …

Nested Adaptation Of Mcmc Algorithms, Dao Nguyen, Perry De Valpine, Yves Atchade, Daniel Turek, Nicholas Michaud, Christopher Paciorek

Faculty and Student Publications

© 2020 International Society for Bayesian Analysis Markov chain Monte Carlo (MCMC) methods are ubiquitous tools for simulation-based inference in many fields but designing and identifying good MCMC samplers is still an open question. This paper introduces a novel MCMC algorithm, namely, Nested Adaptation MCMC. For sampling variables or blocks of variables, we use two levels of adaptation where the inner adaptation optimizes the MCMC performance within each sampler, while the outer adaptation explores the space of valid kernels to find the optimal samplers. We provide a theoretical foundation for our approach. To show the generality and usefulness of the …

Representations Of Cuntz Algebras Associated To Random Walks, Nicholas Christoffersen

Honors Undergraduate Theses

In the present thesis, we investigate representations of Cuntz algebras coming from dilations of row co-isometries. First, we give some general results about such representations. Next, we show that by labeling a random walk, a row co-isometry appears naturally. We give an explicit form for representations that come from such random walks. Then, we give some conditions relating to the reducibility of these representations, exploring how properties of a random walk relate to the Cuntz algebra representation that comes from it

Derivation Of The (Closed-Form) Particular Solution Of The Poisson’S Equation In 3d Using Oscillatory Radial Basis Function, Anup R. Lamichhane, Steven Manns

Undergraduate Mathematics Day: Past Content

Partial differential equations (PDEs) are useful for describing a wide variety of natural phenomena, but analytical solutions of these PDEs can often be difficult to obtain. As a result, many numerical approaches have been developed. Some of these numerical approaches are based on the particular solutions. Derivation of these particular solutions are challenging. This work is about how the Laplace operator can be written in a more convenient form when it is applied to radial basis functions and then use this form to derive the (closed-form) particular solution of the Poisson’s equation in 3D with the oscillatory radial function in …

New Estimates For The Number Of Integer Polynomials With Given Discriminants, Natalia Budarina, Vasilii Bernik, Hugh O'Donnell

Articles

In this paper, we propose a new method of upper bounds for the number of integer polynomials of the fourth degree with a given discriminant. By direct calculation similar results were established by H. Davenport and D. Kaliada for polynomials of second and third degrees.

Discrete Evolutionary Population Models: A New Approach, K. Mokni, Saber Elaydi, M. Ch-Chaoui, A. Eladdadi

Mathematics Faculty Research

In this paper, we apply a new approach to a special class of discrete time evolution models and establish a solid mathematical foundation to analyse them. We propose new single and multi-species evolutionary competition models using the evolutionary game theory that require a more advanced mathematical theory to handle effectively. A key feature of this new approach is to consider the discrete models as non-autonomous difference equations. Using the powerful tools and results developed in our recent work [E. D'Aniello and S. Elaydi, The structure of ω-limit sets of asymptotically non-autonomous discrete dynamical systems, Discr. Contin. Dyn. Series B. …

A Discrete Analogue For The Poincaré-Hopf Theorem, Savana Ammons

HMC Senior Theses

In this thesis, we develop a discrete analogue to the Poincaré–Hopf Theorem. We define the notion of a vector field on a graph, and establish an index theory for such a field. Specifically, we create well-defined indices for the nodes and “cells" formed by a planar graph. Then, we show that the sum of these indices remains constant for certain types of planar graphs, regardless of the discrete vector fields they have.

Math Department 2019-2020 Newsletter, Mathematics Department

Mathematics Newsletter

Welcome to our annual Mathematics Department Newsletter. This was an unusual year for our department as the coronavirus cut short our time on campus this year. Hence we were unable to have our usual events such as the Annual Mathematics Lecture, the Pi Mu Epsilon Induction and Awards Night, and there was no in-person commencement ceremony. But despite that, many great things happened in our department.

Geometry Across The Curriculum, Corey Dunn

Q2S Enhancing Pedagogy

This project is designed for a multicalculus class already familiar with computing the arc length of a parameterized curve in space. The activity asks the student to first recall basic facts about arc length, and then introduces the notion of measuring lengths of vectors differently, depending on where their initial point is. This is a foundational concept in metric differential geometry, and, this activity attempts to motivate this generalization of computing lengths of vectors through this arc length activity. The activity concludes with a short discussion of basic concepts of Lorentzian geometry, including the idea that lightlike vectors have length …

Geogebra Activities: Tracing Points, Jeremy Aikin, Corey Dunn, Jeffrey Meyer, Rolland Trapp

Q2S Enhancing Pedagogy

In this activity, we will learn how to use GeoGebra (www.geogebra.org) to trace the movement of points, which depend on the movement of other objects. The paths of these points determine curves and we will provide algebraic descriptions of these curves.

The Gentle Man Who Taught Infinity, Sheshagiri K.M. Rao

Teacher India

Sheshagiri KM Rao describes the non-conventional teaching methods of his high school mathematics teacher Channa who helped students to discover the beauty in mathematics.