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2022

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Articles 181 - 210 of 1375

Full-Text Articles in Mathematics

Mathematical Musings On The External Anatomy Of The Novel Coronavirus. Part 3: Spherical Triangles, Jyotirmoy Sarkar, Mamunur Rashid Sep 2022

Mathematical Musings On The External Anatomy Of The Novel Coronavirus. Part 3: Spherical Triangles, Jyotirmoy Sarkar, Mamunur Rashid

Mathematics Faculty Publications

What is the shape of the novel coronavirus (n-CoV) which has turned our world upside down? Even though under a microscope, it looks dull, unattractive, and even disgusting, creative artists have attributed to it bright colors, made it look pretty, and depicted it as a thing of beauty. What can a mathematician contribute to this effort? We take a purist’s point of view by imposing on it a quasi-symmetry and then deriving some consequences. In an idealistic world, far removed from reality but still constrained by the rules of mathematics, anyone can enjoy this ethereal beauty of the mind’s creation, …

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Classification Of Pixel Tracks To Improve Track Reconstruction From Proton-Proton Collisions, Kebur Fantahun, Jobin Joseph, Halle Purdom, Nibhrat Lohia Sep 2022

Classification Of Pixel Tracks To Improve Track Reconstruction From Proton-Proton Collisions, Kebur Fantahun, Jobin Joseph, Halle Purdom, Nibhrat Lohia

SMU Data Science Review

In this paper, machine learning techniques are used to reconstruct particle collision pathways. CERN (Conseil européen pour la recherche nucléaire) uses a massive underground particle collider, called the Large Hadron Collider or LHC, to produce particle collisions at extremely high speeds. There are several layers of detectors in the collider that track the pathways of particles as they collide. The data produced from collisions contains an extraneous amount of background noise, i.e., decays from known particle collisions produce fake signal. Particularly, in the first layer of the detector, the pixel tracker, there is an overwhelming amount of background noise that …

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A First-Passage Problem For Exponential Integrated Diffusion Processes, Mario Lefebvre Sep 2022

A First-Passage Problem For Exponential Integrated Diffusion Processes, Mario Lefebvre

Journal of Stochastic Analysis

No abstract provided.

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“Math Talks Are Like An Alarm Clock Waking You Up”: Language’S Crucial Role In Mathematics, Gabriella M. Wasser Sep 2022

“Math Talks Are Like An Alarm Clock Waking You Up”: Language’S Crucial Role In Mathematics, Gabriella M. Wasser

Journal of Practitioner Research

Whole group math talks, or number talks, are a common practice to get students talking about their own understanding of mathematical concepts. The purpose of this study was to implement math talks in small group settings to see what would happen, specifically to students’ conceptual understanding as well their general perceptions of math talks. This study took place in a fourth-grade math classroom, and math talks were implemented with the whole class for a week and then moved to small groups for the remaining three weeks of the study. During the study, a pre-and post-assessment was given, field notes were …

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Euler's Navigation Variational Problem, Sylvio R. Bistafa Sep 2022

Euler's Navigation Variational Problem, Sylvio R. Bistafa

Euleriana

In a 1747 publication, De motu cymbarum remis propulsarum in fluviis (“On the motion of boats propelled by oars in rivers”), Leonhard Euler (1707-1783) works out various instances of a boat moving at constant speed across a stream flowing in straight streamlines at assigned speeds, in which one of these gives rise to a variational problem consisting of finding the quickest crossing path between two points on opposite side of the river banks, which is generally known as the navigation variational problem. This problem together with the well-known catenary and brachistochrone problems, are considered classical examples in the calculus …

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Basel Problem: Historical Perspective And Further Proofs From Stochastic Processes, Uwe Hassler, Mehdi Hosseinkouchack Sep 2022

Basel Problem: Historical Perspective And Further Proofs From Stochastic Processes, Uwe Hassler, Mehdi Hosseinkouchack

Euleriana

In this note, we offer a historical perspective on solutions of the Basel problem. In particular, we have a closer look at some of the less famous results by Euler E41 and provide a review of a selection of the assemblage of earlier proofs. Moreover, we show how to generate further proofs using Karhunen-Lo\`{e}ve expansions of stochastic processes.

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Answer To A Question Concerning Euler's Paper "Variae Considerationes Circa Series Hypergeometricas", Alexander Aycock Sep 2022

Answer To A Question Concerning Euler's Paper "Variae Considerationes Circa Series Hypergeometricas", Alexander Aycock

Euleriana

We solve a problem concerning Euler's paper "Variae considerationes circa series hypergeometricas" (\cite{E661}), as suggested by G. Faber in the preface to Volume 16,2 of the first series of Euler's Opera Omnia. Our solution employs methods introduced by Euler at other places.

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Euler Archive Spotlight, Cynthia Huffman Ph.D. Sep 2022

Euler Archive Spotlight, Cynthia Huffman Ph.D.

Euleriana

This issue we spotlight the translations of Jordan Bell, one of the most prolific translators for the Euler Archive.

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Learning Mathematics From The Master: A Collection Of Euler-Based Primary Source Projects For Today’S Students, Part Ii, Janet H. Barnett, Dominic Klyve, Dave Ruch Sep 2022

Learning Mathematics From The Master: A Collection Of Euler-Based Primary Source Projects For Today’S Students, Part Ii, Janet H. Barnett, Dominic Klyve, Dave Ruch

Euleriana

This article and its prequel together highlight a set of nine classroom-ready projects that draw on the remarkable writing of Leonhard Euler (1707--1783) as a means to help students develop an understanding of standard topics from today’s undergraduate mathematics curriculum. Part of a larger collection of primary source projects intended for use in a wide range of undergraduate mathematics courses, these projects are freely available to students and their instructors. We provide a general description of the pedagogical design underlying these projects, more detailed descriptions of the individual projects themselves, and instructions for obtaining downloadable copies for classroom use.

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On The Motion Of Boats Propelled By Oars In Rivers, Sylvio R. Bistafa Sep 2022

On The Motion Of Boats Propelled By Oars In Rivers, Sylvio R. Bistafa

Euleriana

Euler considers the following problem: A boat with a perfect rudder moves at constant speed across a stream flowing in straight streamlines at assigned speeds. Assuming that the downstream velocity of the boat equals that of the river, how should the rudder be set so that the boat traverses a given path? He works out various instances, one of which gives rise to a variational problem, in detail. (From Clifford Truesdell's An idiot's fugitive essays on science: methods, criticisms, training, circumstances.)

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A Method For Calculating The Equation Of Noon (An English Translation Of Methodus Computandi Aequationem Meridiei), Patrick T. Headley Sep 2022

A Method For Calculating The Equation Of Noon (An English Translation Of Methodus Computandi Aequationem Meridiei), Patrick T. Headley

Euleriana

In this paper Euler presents a method for determining solar noon, the time at which the Sun crosses the meridian. The method requires the times of two observations of the Sun, one in the morning and one in the afternoon, at equal altitudes above the horizon. Solar noon is approximately the midpoint between two such observations, but, since the declination of the Sun will have changed during the day, a correction term, called the equation of noon, is required. Euler explains that this term is too large to ignore and discusses the table of values constructed by de la Hire; …

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Review Of The History Of Mathematics: A Source-Based Approach (Vol. 2), Part I, Erik R. Tou Sep 2022

Review Of The History Of Mathematics: A Source-Based Approach (Vol. 2), Part I, Erik R. Tou

Euleriana

Review of The History of Mathematics: A Source-Based Approach (Vol. 2), Part I, by June Barrow-Green, Jeremy Gray, and Robin Wilson. MAA Press, 2022, 330 + xiv pages.

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Variety And Variation, Christopher Goff, Erik R. Tou Sep 2022

Variety And Variation, Christopher Goff, Erik R. Tou

Euleriana

Welcome to Volume 2 of Issue 2 of Euleriana. While we occasionally discover themes that emerge from the articles and translations presented in a given issue, the wide range of Euler’s work more often results in a variety of topics for each issue. This is no less true for Issue 2.

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Generalizations Of Commutativity In Dihedral Groups, Noah A. Heckenlively Sep 2022

Generalizations Of Commutativity In Dihedral Groups, Noah A. Heckenlively

Rose-Hulman Undergraduate Mathematics Journal

The probability that two elements commute in a non-Abelian finite group is at most 5 8 . We prove several generalizations of this result for dihedral groups. In particular, we give specific values for the probability that a product of an arbitrary number of dihedral group elements is equal to its reverse, and also for the probability that a product of three elements is equal to a permutation of itself or to a cyclic permutation of itself. We also show that for any r and n, there exists a dihedral group such that the probability that a product of n …

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On The Linear Independence Of Finite Gabor And Wavelet Systems, Abdelkrim Bourouihiya Sep 2022

On The Linear Independence Of Finite Gabor And Wavelet Systems, Abdelkrim Bourouihiya

Mathematics Colloquium Series

Gabor and Wavelet Systems are some of the most important families of integrable functions with great potential in applications. Those applications include numerical analysis, signal processing (sound, images), and many other areas of physics and engineering. In this talk, we will present some partial results on a conjecture that states each finite Gabor system is linearly independent. We will also present cases of linearly independent and cases of linearly dependent finite wavelet systems.

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Applications Of The Harary-Sachs Theorem For Hypergraphs, Gregory J. Clark, Joshua N. Cooper Sep 2022

Applications Of The Harary-Sachs Theorem For Hypergraphs, Gregory J. Clark, Joshua N. Cooper

Faculty Publications

The Harary-Sachs theorem for k-uniform hypergraphs equates the codegree-d coefficient of the adjacency characteristic polynomial of a uniform hypergraph with a weighted sum of subgraph counts over certain multi-hypergraphs with d edges. We begin by showing that the classical Harary-Sachs theorem for graphs is indeed a special case of this general theorem. To this end we apply the generalized Harary-Sachs theorem to the leading coefficients of the characteristic polynomial of various hypergraphs. In particular, we provide explicit and asymptotic formulas for the contribution of the k-uniform simplex to the codegree-d coefficient. Moreover, we provide an explicit formula for the leading …

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Eigen’S Paradox And The Quasispecies Model In A Non-Archimedean Framework, Wilson A. Zuniga-Galindo Sep 2022

Eigen’S Paradox And The Quasispecies Model In A Non-Archimedean Framework, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

In this article we present a new p -adic generalization of the Eigen–Schuster model where the genomes (sequences) are represented by words written in the alphabet { 0 , 1 , … , p − 1 } , where p is a prime number, with a time variable length. The time evolution of the concentration of a given sequence is controlled by a p -adic evolution equation. The long term behavior of the concentration of a sequence depends on a fitness function f , a mutation measure Q , and an initial concentration distribution. The new model provides essentially two …

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Norm And Numerical Radius Of Single Operators Through Tools And Techniques From Multivariable Operator Theory, Raul E. Curto, Sang Hoon Lee, Jasang Yoon Sep 2022

Norm And Numerical Radius Of Single Operators Through Tools And Techniques From Multivariable Operator Theory, Raul E. Curto, Sang Hoon Lee, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

We employ tools and techniques from multivariable operator theory to obtain new proofs and extensions of well known inequalities regarding the norm and the numerical radius of elementary operators defined on the C⁎" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; position: relative;">C⁎C⁎–algebra of all bounded operators on Hilbert space, or on the ⁎–ideal of Hilbert-Schmidt operators. In the process, we provide new insights on the study of Heinz-type inequalities related to the arithmetic-geometric mean inequality, and generalize results of several authors, including R. Bhatia, G. Corach, C. Davis, F. Kittaneh, and M.S. Moslehian. To estimate the norm, our approach …

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Leveraging The "Large" In Large Lecture Statistics Classes, Kady Schneiter, Kimberleigh Felix Hadfield, Jenny Lee Clements Sep 2022

Leveraging The "Large" In Large Lecture Statistics Classes, Kady Schneiter, Kimberleigh Felix Hadfield, Jenny Lee Clements

Mathematics and Statistics Faculty Publications

Being a teacher or a student in a class with a large enrollment can be intimidating. Often, teachers view comforts that are common to small classes as unattainable in a larger class, including knowing students’ names, using active learning, employing group work, and creating group discussion. Students in large classes may find that the class size leads to isolation. At Utah State University, we offer introductory statistics classes for various audiences using a large lecture format. The authors have collectively led these large lectures dozens of times and found that, despite its shortcomings, the large lecture format can be an …

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On Cantor Sets Defined By Generalized Continued Fractions, Danielle Hedvig, Masha Gorodetski Sep 2022

On Cantor Sets Defined By Generalized Continued Fractions, Danielle Hedvig, Masha Gorodetski

Rose-Hulman Undergraduate Mathematics Journal

We study a special class of generalized continuous fractions, both in real and complex settings, and show that in many cases, the set of numbers that can be represented by a continued fraction for that class form a Cantor set. Specifically, we study generalized continued fractions with a fixed absolute value and a variable coefficient sign. We ask the same question in the complex setting, allowing the coefficient's argument to be a multiple of \pi/2. The numerical experiments we conducted showed that in these settings the set of numbers formed by such continued fractions is a Cantor set for large …

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The New Era In American Mathematics, 1920-1950 (Book Review), Calvin Jongsma Sep 2022

The New Era In American Mathematics, 1920-1950 (Book Review), Calvin Jongsma

Faculty Work Comprehensive List

Reviewed Title: The New Era in American Mathematics, 1920-1950 by Karen Hunger Parshall. Princeton University Press, 2022. 640 pages. ISBN: 9780691197555.

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Sd Prism: Visualizing The Standard Deviation As The Size Of A Prism Using R/Rstudio, Hieu Nguyen, Tom Nguyen, Mamunur Rashid, Jyotirmoy Sarkar Sep 2022

Sd Prism: Visualizing The Standard Deviation As The Size Of A Prism Using R/Rstudio, Hieu Nguyen, Tom Nguyen, Mamunur Rashid, Jyotirmoy Sarkar

Student Research

SD Prism is a graphical R package used for visualizing the standard deviation of a data set. Given a raw data set, the standard deviation (SD) is defined as the square-root of twice the mean square of all pairwise half deviations between any two sample observations. This interpretation leads to a geometric visualization of the sample SD, and a more elementary explanation as to why the denominator in the sample variance is one less than the sample size. In this article we will explain step by step how to understand it mathematically and how the package implements the methodology to …

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Numerical Analysis Of A Second Order Ensemble Method For Evolutionary Magnetohydrodynamics Equations At Small Magnetic Reynolds Number, John Carter, Nan Jiang Sep 2022

Numerical Analysis Of A Second Order Ensemble Method For Evolutionary Magnetohydrodynamics Equations At Small Magnetic Reynolds Number, John Carter, Nan Jiang

Mathematics and Statistics Faculty Research & Creative Works

We study a second order ensemble method for fast computation of an ensemble of magnetohydrodynamics flows at small magnetic Reynolds number. Computing an ensemble of flow equations with different input parameters is a common procedure for uncertainty quantification in many engineering applications, for which the computational cost can be prohibitively expensive for nonlinear complex systems. We propose an ensemble algorithm that requires only solving one linear system with multiple right-hands instead of solving multiple different linear systems, which significantly reduces the computational cost and simulation time. Comprehensive stability and error analyses are presented proving conditional stability and second order in …

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Why Seneca Effect?, Sean R. Aguilar, Vladik Kreinovich Sep 2022

Why Seneca Effect?, Sean R. Aguilar, Vladik Kreinovich

Departmental Technical Reports (CS)

Already ancients noticed that decrease is usually faster than growth -- whether we talk about companies or empires. A modern researcher Ugo Bardi confirmed that this phenomenon is still valid today. He called it Seneca effect, after the ancient philosopher Seneca -- one of those who observed this phenomenon. In this paper, we provide a natural explanation for the Seneca effect.

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Why Smaller-Size Objects Affect The Flow Much More Than Larger Ones: A Geometric Explanation With Applications Ranging From Volcanoes And Tornadoes To Blood, Fish, And Building Preservation, Laxman Bokati, Vladik Kreinovich Sep 2022

Why Smaller-Size Objects Affect The Flow Much More Than Larger Ones: A Geometric Explanation With Applications Ranging From Volcanoes And Tornadoes To Blood, Fish, And Building Preservation, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, the larger the object, the larger should be its effect on the surroundings -- in particular, the larger should be its effect on the surrounding flow. However, in many practical situations, we observe the opposite effect: smaller-size particles affect the flow much more than larger-size particles. This seemingly counterintuitive phenomena has been observed in many situations: lava flow in the volcanoes, air circulation in tornadoes, blood flow in a body, the effect of fish on water circulation in the ocean, and the effect of added particles on seeping water that damages historic buildings. In this paper, we …

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What Is The Most Adequate Fuzzy Methodology?, Noah Velasco, Olga Kosheleva, Vladik Kreinovich Sep 2022

What Is The Most Adequate Fuzzy Methodology?, Noah Velasco, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In practice, there is often a need to describe the relation y = f(x) between two quantities in algorithmic form: e.g., we want to describe the control value y corresponding to the given input x, or we want to predict the future value y based on the current value x. In many such cases, we have expert knowledge about the desired dependence, but experts can only describe their knowledge by using imprecise ("fuzzy") words from a natural language. Methodologies for transforming such knowledge into an algorithm y = f(x) are known as fuzzy methodologies. There exist several fuzzy methodologies, a …

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A General Commonsense Explanation Of Several Medical Results, Olga Kosheleva, Vladik Kreinovich Sep 2022

A General Commonsense Explanation Of Several Medical Results, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that many recent experimental medical results about the effect of different factors on our health can be explained by common sense ideas.

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On Some Mixing Properties Of Copula-Based Markov Chains, Martial Longla, Hamadou Mous-Abou, Isidore Seraphin Ngongo Sep 2022

On Some Mixing Properties Of Copula-Based Markov Chains, Martial Longla, Hamadou Mous-Abou, Isidore Seraphin Ngongo

Faculty and Student Publications

This paper brings some insights of ψ′-mixing, ψ∗-mixing and ψ-mixing for copula-based Markov chains and the perturbations of their copulas. We provide new tools to check Markov chains for ψ-mixing or ψ′-mixing. We show that perturbations of ψ′-mixing copula-based Markov chains are ψ′-mixing while perturbations of ψ-mixing Markov chains are not necessarily ψ-mixing Markov chains, even when the perturbed copula generates ψ-mixing. The Farlie–Gumbel–Morgenstern, gaussian and Ali-Mikhail-Haq copula families are considered among other examples. A statistical study is provided to emphasize the impact of perturbations on copula-based Markov chains in a simulation study. Moreover, we provide a correction to a …

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Why Best-Worst Method Works Well, Sean Aguilar, Vladik Kreinovich Sep 2022

Why Best-Worst Method Works Well, Sean Aguilar, Vladik Kreinovich

Departmental Technical Reports (CS)

In many cases, experts are much more accurate when they estimate the ratio of two quantities than when they estimate the actual values. For example, if it difficult to accurately estimate the height of a person on a photo, but if we have two people standing side by side, we can easily estimate to what extent one of them is taller than the other one. To get accurate estimates, it is therefore desirable to use such ratio estimates. Empirical analysis shows that to obtain the most accurate results, we need to compare all the objects with either the "best" object …

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How Hot Is Too Hot, Sofia Holguin, Vladik Kreinovich Sep 2022

How Hot Is Too Hot, Sofia Holguin, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent study has shown that the temperature threshold -- after which even young healthy individuals start feeling the effect of heat on their productivity -- is 30.5 ± 1 C. In this paper, we use decision theory ideas to provide a theoretical explanation for this empirical finding.

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