Mathematics Commons^™

Towards Reinforcement Learning Driven Mesh Adaptivity For Second Order Elliptic Problems, Augustine Twumasi

Open Access Theses & Dissertations

Adaptive mesh refinement techniques have become an indispensable tool in achieving accurate and efficiently computed solutions to problems which require impractically fine uniform meshes to obtain an accurate approximation. The adaptive algorithm involves a recursive application of SOLVE-ESTIMATE-MARK-REFINE steps where in particular the step `ESTIMATE' involves computing a posteriori error estimator based on only the numerical solution and the data of the problem. Over the years, several a posteriori error estimators have been developed and successfully applied but often times, the choice of estimator is ill suited for the problem at hand. In this research, we present two estimators namely …

A Novel Numerical Method For Solving Fractional Diffusion-Wave And Nonlinear Fredholm And Volterra Integral Equations With Zero Absolute Error, Mutaz Mohammad, Alexandre Trounev, Mohammed Alshbool

All Works

In this work, a new numerical method for the fractional diffusion-wave equation and nonlinear Fredholm and Volterra integro-differential equations is proposed. The method is based on Euler wavelet approximation and matrix inversion of an M × M collocation points. The proposed equations are presented based on Caputo fractional derivative where we reduce the resulting system to a system of algebraic equations by implementing the Gaussian quadrature discretization. The reduced system is generated via the truncated Euler wavelet expansion. Several examples with known exact solutions have been solved with zero absolute error. This method is also applied to the Fredholm and …

Tail-Measurable Functions And Their Corresponding Induced Classes, And Some Determinacy Conditions Involving 3-Player Games, Joshua K. Reagan

UNLV Theses, Dissertations, Professional Papers, and Capstones

In this dissertation, we have two main categories of results. The first is regarding certain point-classes, and the second is regarding 3-player games.

The point-classes of Baire Space, \mathcal{N}, in the Borel and Projective Hierarchies, as well as Hausdorff's Difference Hierarchy have been well studied, and there has been much research into further stratifying these hierarchies. One area of particular interest falls in between the point-classes \mathbf{\Pi}_\mathbf{1}^\mathbf{1} and \Delta\left(\omega^2-\mathbf{\Pi}_\mathbf{1}^\mathbf{1}\right). It is well known that the point-classes \beta-\mathbf{\Pi}_\mathbf{1}^\mathbf{1}, for \beta\in\omega^2, stratify this region of the projective hierarchy, with the point-class \bigcup_{\beta\in\omega^2}\beta-\mathbf{\Pi}_\mathbf{1}^\mathbf{1} still falling strictly below \Delta\left(\omega^2-\mathbf{\Pi}_\mathbf{1}^\mathbf{1}\right). Dr. Derrick DuBose developed multiple …

Multilateration Index., Chip Lynch

Electronic Theses and Dissertations

We present an alternative method for pre-processing and storing point data, particularly for Geospatial points, by storing multilateration distances to fixed points rather than coordinates such as Latitude and Longitude. We explore the use of this data to improve query performance for some distance related queries such as nearest neighbor and query-within-radius (i.e. “find all points in a set P within distance d of query point q”). Further, we discuss the problem of “Network Adequacy” common to medical and communications businesses, to analyze questions such as “are at least 90% of patients living within 50 miles of a covered emergency …

Node Generation For Rbf-Fd Methods By Qr Factorization, Tony Liu, Rodrigo B. Platte

Faculty Publications

Polyharmonic spline (PHS) radial basis functions (RBFs) have been used in conjunction with polynomials to create RBF finite-difference (RBF-FD) methods. In 2D, these methods are usually implemented with Cartesian nodes, hexagonal nodes, or most commonly, quasi-uniformly distributed nodes generated through fast algorithms. We explore novel strategies for computing the placement of sampling points for RBF-FD methods in both 1D and 2D while investigating the benefits of using these points. The optimality of sampling points is determined by a novel piecewise-defined Lebesgue constant. Points are then sampled by modifying a simple, robust, column-pivoting QR algorithm previously implemented to find sets of …

Matroids Determinable By Two Partial Representations, Aurora Calderon Dojaquez

Electronic Theses, Projects, and Dissertations

A matroid is a mathematical object that generalizes and connects notions of independence that arise in various branches of mathematics. Some matroids can be represented by a matrix whose entries are from some field; whereas, other matroids cannot be represented in this way. However, every matroid can be partially represented by a matrix over the field GF(2). In fact, for a given matroid, many different partial representations may exist, each providing a different collection of information about the matroid with which they are associated. Such a partial representation of a matroid usually does not uniquely determine the matroid on its …

The Color Number Of Cubic Graphs Having A Spanning Tree With A Bounded Number Of Leaves, Analen A. Malnegro, Gina A. Malacas, Kenta Ozeki

Theory & Applications of Graphs

The color number c(G) of a cubic graphG is the minimum cardinality of a color class of a proper 4-edge-coloring of G. It is well-known that every cubic graph G satisfies c(G) = 0 if G

Contributions To The Teaching And Learning Of Fluid Mechanics, Ashwin Vaidya

Department of Mathematics Faculty Scholarship and Creative Works

This issue showcases a compilation of papers on fluid mechanics (FM) education, covering different sub topics of the subject. The success of the first volume [1] prompted us to consider another follow-up special issue on the topic, which has also been very successful in garnering an impressive variety of submissions.

As a classical branch of science, the beauty and complexity of fluid dynamics cannot be overemphasized. This is an extremely well-studied subject which has now become a significant component of several major scientific disciplines ranging from aerospace engineering, astrophysics, atmospheric science (including climate modeling), biological and biomedical science …

Probability Distributions For Elliptic Curves In The Cgl Hash Function, Dhruv Bhatia, Kara Fagerstrom, Max Watson

Mathematical Sciences Technical Reports (MSTR)

Hash functions map data of arbitrary length to data of predetermined length. Good hash functions are hard to predict, making them useful in cryptography. We are interested in the elliptic curve CGL hash function, which maps a bitstring to an elliptic curve by traversing an inputdetermined path through an isogeny graph. The nodes of an isogeny graph are elliptic curves, and the edges are special maps betwixt elliptic curves called isogenies. Knowing which hash values are most likely informs us of potential security weaknesses in the hash function. We use stochastic matrices to compute the expected probability distributions of the …

Trilinear Smoothing Inequalities And A Variant Of The Triangular Hilbert Transform, Michael Christ, Polona Durcik, Joris Roos

Mathematics, Physics, and Computer Science Faculty Articles and Research

Lebesgue space inequalities are proved for a variant of the triangular Hilbert transform involving curvature. The analysis relies on a crucial trilinear smoothing inequality developed herein, and on bounds for an anisotropic variant of the twisted paraproduct.

The trilinear smoothing inequality also leads to Lebesgue space bounds for a corresponding maximal function and a quantitative nonlinear Roth-type theorem concerning patterns in the Euclidean plane.

Fourth Voices Conference On Teaching Stem With Song, Lawrence M. Lesser

Journal of Humanistic Mathematics

The fourth VOICES (Virtual Ongoing Interdisciplinary Collaborations on Educating with Song) conference will be held September 26, 2021, with its Sunday date especially targeting those whose teaching schedule precludes attendance on a weekday. This online-only conference explores the use of song to teach STEM content at the college or secondary school level and will include reviewed interactive sessions, video-posters, and discussions of interest to both practitioners and researchers.

An Interview With The (Other) Corona Graph, Bonnie Jacob, Jobby Jacob

Journal of Humanistic Mathematics

The following is an interview, conducted in April 2021, with a graph product known as the corona graph. Disclaimer: The views expressed herein are not ours. Actually, this entire interview is fiction, other than the mathematics, of course.

The Music Is All That Counts! A Poem-Collage Pair Created During The Pandemic, Sarah Glaz, Mark Sanders

Journal of Humanistic Mathematics

The poem-collage pair presented here is a work of collaboration between the mathematician and poet, Sarah Glaz, and the collage and ceramic artist, Mark Sanders. The piece is part of their larger joint poem-collage project involving the history of mathematics. Created at the height of the pandemic, the poem-collage pair reflects the reality of living during a plague and embodies the solace offered by long-distance collaboration and by the arts. Included as background is a brief discussion on the history and mathematics involved, information about Sarah’s poem and Mark’s collage, and a reflection on how the pandemic influenced both the …

Letters From Incompleteness, Jonah Howell

Journal of Humanistic Mathematics

In a series of letters to a faraway lover, an unnamed narrator attempts to make sense of his own internal struggles by fixating on the strange death of Kurt Gödel.

A Life Cut Short, Cacey L. Wells

Journal of Humanistic Mathematics

Connecting scholars and students with the human side of mathematics is critical. This poem about Évariste Galois provides a synopsis of his controversial life, while also providing details into his life that some may not be familiar with.

Asymptotic Dream, Oscar Gonzalez

Journal of Humanistic Mathematics

A love poem about breaching mathematical limits, inspired by the tragic beauty of calculus.

Train Algebra, Mary Soon Lee

Journal of Humanistic Mathematics

No abstract provided.

The Uncertainty Of Confidence, Michael J. Leach

Journal of Humanistic Mathematics

This is a free-verse poem about the estimation of population parameters in statistical models. The spacing of words is intended to reflect uncertainty.

Mathematical Poetry In The Time Of Covid

Journal of Humanistic Mathematics

The COVID-19 pandemic has affected mathematicians and teachers in many ways. In our January 2021 issue, we invited our community to submit essays and reflections, as well as fiction and poetry, based on their experiences during this globally unsettling (and still ongoing) event. This folder presents a select collection of poems submitted in response to that call, by Christopher Caruvana, Marion Cohen, Lawrence M. Lesser, Dan May, Vanessa Sun, and Michele Willman, ordered alphabetically by poet last name.

Mathematical Rigor From Within, Lowell Abrams

Journal of Humanistic Mathematics

There is a certain feel that is unique to the rarefied context of rigorous mathematics. These poems constitute an exploration of my experience of mathematical rigor when I am in the midst of exercising my skills as a research mathematician.

Examining The Modeling Framework Of Crime Hotspot Models In Predictive Policing, Heidi Goodson, Alanna Hoyer-Leitzel

Journal of Humanistic Mathematics

Predictive policing has its roots in crime hotspot modeling. In this paper we give an example of what goes into mathematical crime hotspot modeling and show that the modeling assumptions perpetuate systemic racism in policing. The goal of this paper is to raise objections to this field of research, not on its mathematical merit, but on the scope of the problem formation. We do not make any suggestions about how to incorporate race into predictive policing models in order to make them less biased.

Ubiratan D’Ambrosio: Celebrating His Life And Legacy, Milton Rosa, Daniel Clark Orey

Journal of Humanistic Mathematics

On May 12^th, Brazil and the world said goodbye to one of its biggest names in mathematics education, Ubiratan D’Ambrosio (December 8, 1932 – May 12, 2021). D’Ambrosio is recognized both throughout Brazil and internationally for having been the creator of ethnomathematics, a concept that acknowledges, values, and respects different mathematical ideas, procedures, and practices developed by the members of distinct cultural groups. D’Ambrosio analyzed the history of explanations of life and of natural evolution in different cultures. In the last years, his motivation had been the pursuit of peace, in all its four dimensions: individual, social, environmental, …

The Genius Box, James Propp

Journal of Humanistic Mathematics

Who gets called a genius, and why? What effect does having a special category of people called "geniuses" have on an intellectual community and on individuals within it? Drawing on my own experience, and reflecting on writings by Moon Duchin, Elizabeth Gilbert, and others, I try to find some answers.

The Mystery Of The Dancing Men, Manmohan Kaur

Journal of Humanistic Mathematics

In this paper I describe an activity based on a 1903 Sherlock Holmes murder mystery, in which a substitution cipher is used to encrypt secret messages. The story provides a fun and interesting way to talk about frequency analysis, and can be used as a segue into mathematical constructs such as modular arithmetic and computation. The activity is accessible to ages twelve and above, and has been successfully used in mathematics outreach and popularization efforts as well as in general education and mathematics courses.

The Little Prince -- The Lost Chapter, Igor Podlubny

Journal of Humanistic Mathematics

A "lost chapter" from Antoine de Saint-Exupéry's Le Petite Prince about the Little Prince visiting a mathematician, written in French in the style of the original work, is presented along with several translations.

Felix Hausdorff’S Poem “Den Ungeflügelten”, Benjamin Elkins

Journal of Humanistic Mathematics

In 1900, Felix Hausdorff published Ekstasen (Ecstasy) under the pseudonym Paul Mongré. The book is comprised of 157 poems (70 sonnets, 32 rondels, 25 "mixed poems", where Hausdorff effortlessly combines different types of poetic styles, and 30 more poems). Den Ungeflügelten (To The Wingless Ones) is the first poem in this book and provides an interesting self-portrait of Hausdorff as he embraces his muse with confidence. Here I present an English translation of this poem without (much) commentary.

Teaching Mathematics: Heuristics Can And Ought To Lead The Way, Marshall Gordon

Journal of Humanistic Mathematics

In contrast to problem-solving procedures that are the “bricks and mortar” of demonstrations in mathematics textbooks, heuristics, defined by Polya as “the study of means and methods of problem solving”, are those mental actions that enable the practitioner to make progress when it is not clear how to solve problems directly. Yet, as essential as heuristic tools are, they tend not to be included in presentations in mathematics textbooks. The overarching problem can be understood in terms of students’ not developing productive means for engaging problems. A few mathematics problems are included to argue for the validity, if not the …

Archimedes Of Syracuse And Sir Isaac Newton: On The Quadrature Of A Parabola, Wyatte C. Hooper

Journal of Humanistic Mathematics

Good mathematics stands the test of time. As culture changes, we often ask different questions, bringing new perspectives, but modern mathematics stands on ancient discoveries. Isaac Newton’s discovery of calculus (along with Leibniz) may seem old but is predated by Archimedes’ findings. Current mathematics students should be familiar with parabolas and simple curves; in our introductory calculus courses, we teach them to compute the areas under such curves. Our modern approach derives its roots from Newton’s work; however, we have filled in many of the gaps in the pursuit of mathematical rigor. What many students may not know is that …

Fantasy At The Service Of Mathematics, Clara Ziskin, Esther Williams, Alla Shmukler

Journal of Humanistic Mathematics

This article aims to introduce the reader to a book published in 2016 under the title “Amazing Tales from the Magic Wood and Famous Problems of Mathematics” by Elli Shor and Clara Ziskin. The book offers an original method of presenting mathematical facts and history through a fantasy narrative. The book’s two authors, Clara Ziskin and Alla Shmukler (Elli Shor), together with consultant psychologist Esther Williams, share here several excerpts taken from the first part of the book as well as related illustrations and mathematical riddles, so that the reader can form an informed impression of the book, its structure, …

The List: Proverbs For Calculus, Bruce H. Pourciau

Journal of Humanistic Mathematics

Topics chosen from first-year calculus illustrate a number of “sayings” or “proverbs,” the first three, for example, being: be awed, like a child; meaning before truth; and act with intention. Many are proverbs for life as well as mathematics.