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Euler's Infinite Product Formula Of Sine, Sarah Joy Kinnison Aug 2023

Euler's Infinite Product Formula Of Sine, Sarah Joy Kinnison

Student Theses and Dissertations

Euler’s infinite product formula of sine is a classical math problem that was discovered by Euler in 1748 and is still generating interest and being researched by mathematicians today. The goal of this research is to analyze current published proofs for the formula and find those proofs that are most accessible to upper level undergraduate math students. Three different proofs are found to be suitable for our purpose, and two these proofs are modified so that they are easier to understand. Some well-known applications of the Euler’s formula are the Basel problem and Bernoulli numbers.

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Differential Calculus: From Practice To Theory, Eugene Boman, Robert Rogers Aug 2023

Differential Calculus: From Practice To Theory, Eugene Boman, Robert Rogers

Milne Open Textbooks

Differential Calculus: From Practice to Theory covers all of the topics in a typical first course in differential calculus. Initially it focuses on using calculus as a problem solving tool (in conjunction with analytic geometry and trigonometry) by exploiting an informal understanding of differentials (infinitesimals). As much as possible large, interesting, and important historical problems (the motion of falling bodies and trajectories, the shape of hanging chains, the Witch of Agnesi) are used to develop key ideas. Only after skill with the computational tools of calculus has been developed is the question of rigor seriously broached. At that point, the …

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Double Barrier Backward Doubly Stochastic Differential Equations, Tadashi Hayashi Aug 2023

Double Barrier Backward Doubly Stochastic Differential Equations, Tadashi Hayashi

Journal of Stochastic Analysis

No abstract provided.

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Euler Archive Spotlight: Translations Of Euler's Works To Languages Other Than English, Michael P. Saclolo Aug 2023

Euler Archive Spotlight: Translations Of Euler's Works To Languages Other Than English, Michael P. Saclolo

Euleriana

The Euler Archive is home to almost all of Euler’s original publications and roughly a quarter of them have accompanying translations to English. A small number of Euler’s works also have translations to a handful of other languages, namely, Dutch, French, German, Italian, Portuguese, and Turkish that are available in the archive. We put a spotlight on these non-English translations in this issue.

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Perfect Numbers, Uwe Hassler Aug 2023

Perfect Numbers, Uwe Hassler

Euleriana

We provide a selected review of results known or conjectured before Euler entered and changed the quest for perfect numbers. Then we discuss his contributions to the determination of Mersenne primes that fueled research on primality tests.

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Euler’S Variational Approach To The Elastica, Sylvio R. Bistafa Aug 2023

Euler’S Variational Approach To The Elastica, Sylvio R. Bistafa

Euleriana

The history of the elastica is examined through the works of various contributors, including those of Jacob and Daniel Bernoulli, since its first appearance in a 1690 contest on finding the profile of a hanging flexible cord. Emphasis will be given to Leonhard Euler’s variational approach to the elastica, laid out in his landmark 1744 book on variational techniques. Euler’s variational approach based on the concept of differential value is highlighted, including the derivation of the general equation for the elastica from the differential value of the first kind, from which nine shapes adopted by a flexed lamina under different …

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Euler’S First Proof Of Stirling’S Formula, Alexander Aycock Aug 2023

Euler’S First Proof Of Stirling’S Formula, Alexander Aycock

Euleriana

We present a proof given by Euler in his paper “De serierum determinatione
seu nova methodus inveniendi terminos generales serierum” [4] (E189:“On
the determination of series or a new method of finding the general terms
of series”) for Stirling’s formula. Euler’s proof uses his theory of difference
equations with constant coefficients. This theory outgrew from his ear-
lier considerations on inhomogeneous differential equations with constant
coefficients of finite order that he tried to extend to the case of infinite
order.

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On Euler's Solution Of The Simple Difference Equation, Alexander Aycock Aug 2023

On Euler's Solution Of The Simple Difference Equation, Alexander Aycock

Euleriana

In this note we will discuss Euler's solution of the simple difference equation that he gave in his paper{\it ``De serierum determinatione seu nova methodus inveniendi terminos generales serierum"} \cite{E189} (E189:``On the determination of series or a new method of finding the general terms of series") and also present a derivation for the values of the Riemann $\zeta$-function at positive integer numbers based on Euler's ideas.

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Euler And The Legendre Polynomials, Alexander Aycock Aug 2023

Euler And The Legendre Polynomials, Alexander Aycock

Euleriana

In this note we will present how Euler's investigations on various different subjects lead to certain properties of the Legendre polynomials. More precisely, we will show that the generating function and the difference equation for the Legendre polynomials was already written down by Euler in at least two different papers. Furthermore, we will demonstrate that some familiar expressions for the Legendre polynomials are corollaries of the before-mentioned works. Finally, we will show that Euler's ideas on continued fractions lead to an integral representation for the Legendre polynomials that seems to be less generally known.

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Solution Of The Diophantine Equation (Maa+Nbb)=Cd(Mcc+Ndd) Using Rational Numbers, Georg Ehlers Aug 2023

Solution Of The Diophantine Equation (Maa+Nbb)=Cd(Mcc+Ndd) Using Rational Numbers, Georg Ehlers

Euleriana

This paper (E716) was published in Nova acta Academiae scientiarum imperialis petropolitanae, Volume 13 (1795/96), pp. 45-63. It was also included in Commentationes Arithmeticae, Volume II, as Number LXVIII, pp. 281-293 (E791). Euler starts with Fermat's Last Theorem and mentions the proofs for the cases n=3 and n=4 which he had completed himself earlier. He then moves on to make the sum of powers conjecture, which was later disproved in the second half of the 20th century. In this context he discusses his discovery of 134^4+133^4=158^4+59^4, which he calls unexpected. Euler derives the title equation from A^4+B^4=C^4+D^4, generalizing it to …

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On The Motion Of The Nodes Of The Moon And The Variation Of Its Inclination To The Ecliptic (An English Translation Of De Motu Nodorum Lunae Eiusque Inclinationis Ad Eclipticam Variatione), Patrick T. Headley Aug 2023

On The Motion Of The Nodes Of The Moon And The Variation Of Its Inclination To The Ecliptic (An English Translation Of De Motu Nodorum Lunae Eiusque Inclinationis Ad Eclipticam Variatione), Patrick T. Headley

Euleriana

In this paper Euler attempts to explain some features of the motion of the Moon using Newton’s inverse-square law of gravity. He describes the evidence in favor of Newton’s theory but also the lack of progress in the study of lunar motion due to the difficulty of the three-body problem, arising here since both the Sun and the Earth have large effects on the Moon. He proceeds to investigate the line of intersection between the planes of the Earth's orbit and the Moon's orbit, as well as the angle between the two planes.

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The Wide Scope Of Euler’S Work, Christopher Goff, Erik R. Tou Aug 2023

The Wide Scope Of Euler’S Work, Christopher Goff, Erik R. Tou

Euleriana

An introduction to the contents in Issue 2, Volume 3 of Euleriana.

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The Gamma-Signless Laplacian Adjacency Matrix Of Mixed Graphs, Omar Alomari, Mohammad Abudayah, Manal Ghanem Aug 2023

The Gamma-Signless Laplacian Adjacency Matrix Of Mixed Graphs, Omar Alomari, Mohammad Abudayah, Manal Ghanem

Theory & Applications of Graphs

The α-Hermitian adjacency matrix Hα of a mixed graph X has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number α. This enables us to define an incidence matrix of mixed graphs. Consequently, we define a generalization of line graphs as well as a generalization of the signless Laplacian adjacency matrix of graphs. We then study the spectral properties of the gamma-signless Laplacian adjacency matrix of a mixed graph. Lastly, we characterize when the signless Laplacian adjacency matrix of …

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Undetermined Coefficients With Hyperbolic Sines And Cosines, Laurie A. Florio, George L. Fischer Aug 2023

Undetermined Coefficients With Hyperbolic Sines And Cosines, Laurie A. Florio, George L. Fischer

CODEE Journal

The method of undetermined coefficients is commonly applied to solve linear, constant coefficient, non-homogeneous ordinary differential equations when the forcing function is from a selected class of functions. Often the hyperbolic sine and cosine functions are not explicitly included in this list of functions. Through a set of guided examples, this work argues that the hyperbolic sine and cosine ought to be included in the select class of functions. Careful explanation is provided for the necessary treatment of the cases where the argument of the hyperbolic sine and/or cosine functions matches one or both of the roots of the characteristic …

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Stability Analyses On The Effect Of Vaccination And Contact Tracing In Monkeypox Virus Transmission, Solomon Eshun, Richmond Essieku, James Ladzekpo Aug 2023

Stability Analyses On The Effect Of Vaccination And Contact Tracing In Monkeypox Virus Transmission, Solomon Eshun, Richmond Essieku, James Ladzekpo

School of Mathematical & Statistical Sciences Faculty Publications

Monkeypox is a significant health concern due to its potential for morbidity and occasional mortality. Vaccination and effective contact tracing play pivotal roles in controlling infectious diseases, including monkeypox. This study aims to contribute to our understanding of monkeypox dynamics by developing a comprehensive mathematical model that incorporates key factors such as vaccination, quarantining, and contact tracing. Through rigorous sensitivity analysis, we explore the impact of varying vaccination coverage and contact tracing on the disease’s dynamics. In particular, we investigate the dynamics of the disease in relation to variable vaccination coverage and contact tracing. Our findings highlight the critical role …

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A Review Of Recent Gene Expression-Based And Dna Methylation-Based Mathematical Cell Type Deconvolution Methods, Chenxiao Tian Aug 2023

A Review Of Recent Gene Expression-Based And Dna Methylation-Based Mathematical Cell Type Deconvolution Methods, Chenxiao Tian

Arts & Sciences Graduate Student Theses and Dissertations

In recent years, many cell type deconvolution methods based on DNA methylation data and gene expression data have been developed. Both of these two methods have its special advantages and disadvantages, e.g., DNA methylation-based methods’ data source is usually more stable than gene expression and DNA methylation is easier to measure in FFPE tissues or formalin-fixed paraffin-embedded, while some gene-expression data like scRNA-seq data usually has high cost and complexity. On the other hand, gene expression-based deconvolution methods currently have many more available methods than DNA methylation-based deconvolution methods, which leads to DNA methylation-based methods in many cases can learn …

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A Variational Theory For Integral Functionals Involving Finite-Horizon Fractional Gradients, Javier Cueto, Carolin Carolin, Hidde Schönberger Aug 2023

A Variational Theory For Integral Functionals Involving Finite-Horizon Fractional Gradients, Javier Cueto, Carolin Carolin, Hidde Schönberger

Department of Mathematics: Faculty Publications

The center of interest in this work are variational problems with integral functionals depending on nonlocal gradients with finite horizon that correspond to truncated versions of the Riesz fractional gradient. We contribute several new aspects to both the existence theory of these problems and the study of their asymptotic behavior. Our overall proof strategy builds on finding suitable translation operators that allow to switch between the three types of gradients: classical, fractional, and nonlocal. These provide useful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, which is the natural …

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Symmetric Functions Algebras (Sfa) Iii: Stochastic And Constant Row Sum Matrices, Philip Feinsilver Aug 2023

Symmetric Functions Algebras (Sfa) Iii: Stochastic And Constant Row Sum Matrices, Philip Feinsilver

Journal of Stochastic Analysis

No abstract provided.

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Topological Data Analysis Using The Mapper Algorithm, Jessica Girard Aug 2023

Topological Data Analysis Using The Mapper Algorithm, Jessica Girard

Electronic Theses and Dissertations, 2020-2023

Topological data analysis is an expanding field that attempts to obtain qualitative information from a data set using topological ideas. There are two common methods of topological data analysis: persistent homology and the Mapper algorithm; the focus of this thesis is on the latter. In this thesis, we will be discussing the key ideas behind the Mapper algorithm, following the flow from Morse Theory to Reeb graphs to the topological version of the algorithm and finally to the statistical version. Lastly, we will present an application of Mapper to the USAIR97 data set using the RTDAmapper package.

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Efficient And Secure Digital Signature Algorithm (Dsa), Nissa Mehibel, M'Hamed Hamadouche Aug 2023

Efficient And Secure Digital Signature Algorithm (Dsa), Nissa Mehibel, M'Hamed Hamadouche

Emirates Journal for Engineering Research

The digital signature is used to ensure the integrity of messages as well as the authentication and non-repudiation of users. Today it has a very important role in information security. Digital signature is used in various fields such as e-commerce and e-voting, health, internet of things (IOT). Many digital signature schemes have been proposed, depending on the computational cost and security level. In this paper, we analyzed a recently proposed digital signature scheme based on the discrete logarithm problem (DLP). Our analysis shows that the scheme is not secure against the repeated random number attack to determine the secret keys …

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Approaches To The Erdős–Straus Conjecture, Ivan V. Morozov Aug 2023

Approaches To The Erdős–Straus Conjecture, Ivan V. Morozov

Publications and Research

The Erdős–Straus conjecture, initially proposed in 1948 by Paul Erdős and Ernst G. Straus, asks whether the equation 4/n = 1/x + 1/y + 1/z is solvable for all n ∈ N and some x, y, z ∈ N. This problem touches on properties of Egyptian fractions, which had been used in ancient Egyptian mathematics. There exist many partial solutions, mainly in the form of arithmetic progressions and therefore residue classes. In this work we explore partial solutions and aim to expand them.

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Using Geographic Information To Explore Player-Specific Movement And Its Effects On Play Success In The Nfl, Hayley Horn, Eric Laigaie, Alexander Lopez, Shravan Reddy Aug 2023

Using Geographic Information To Explore Player-Specific Movement And Its Effects On Play Success In The Nfl, Hayley Horn, Eric Laigaie, Alexander Lopez, Shravan Reddy

SMU Data Science Review

American Football is a billion-dollar industry in the United States. The analytical aspect of the sport is an ever-growing domain, with open-source competitions like the NFL Big Data Bowl accelerating this growth. With the amount of player movement during each play, tracking data can prove valuable in many areas of football analytics. While concussion detection, catch recognition, and completion percentage prediction are all existing use cases for this data, player-specific movement attributes, such as speed and agility, may be helpful in predicting play success. This research calculates player-specific speed and agility attributes from tracking data and supplements them with descriptive …

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Secondary Features Of Importance For A Url Ranking, Atajan Abdyyev Aug 2023

Secondary Features Of Importance For A Url Ranking, Atajan Abdyyev

Dissertations and Theses

This paper investigates the impact of secondary ranking factors on webpage relevance and rankings in the context of Search Engine Optimization (SEO), focusing on the jewelry domain within the United States e-commerce market. By generating a keyword list related to jewelry and retrieving top URLs from Google's search results, the study employs machine learning models including XGBoost, CatBoost, and Linear Regression to identify key features influencing webpage relevance and rankings.The findings highlight specific optimal ranges for features like Outlinks, Unique Inlinks, Flesch Reading Ease Score, and others, indicating their significant impact on better rankings. Notably, Random Forest model performed best …

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Signings Of Graphs And Sign-Symmetric Signed Graphs, Ahmad Asiri Aug 2023

Signings Of Graphs And Sign-Symmetric Signed Graphs, Ahmad Asiri

Theses and Dissertations

In this dissertation, we investigate various aspects of signed graphs, with a particular focus on signings and sign-symmetric signed graphs. We begin by examining the complete graph on six vertices with one edge deleted ($K_6$\textbackslash e) and explore the different ways of signing this graph up to switching isomorphism. We determine the frustration index (number) of these signings and investigate the existence of sign-symmetric signed graphs. We then extend our study to the $K_6$\textbackslash 2e graph and the McGee graph with exactly two negative edges. We investigate the distinct ways of signing these graphs up to switching isomorphism and demonstrate …

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Anomaly Detection In The Molecular Structure Of Gallium Arsenide Using Convolutional Neural Networks, Timothy Roche, Aihua W. Wood, Philip Cho, Chancellor Johnstone Aug 2023

Anomaly Detection In The Molecular Structure Of Gallium Arsenide Using Convolutional Neural Networks, Timothy Roche, Aihua W. Wood, Philip Cho, Chancellor Johnstone

Faculty Publications

This paper concerns the development of a machine learning tool to detect anomalies in the molecular structure of Gallium Arsenide. We employ a combination of a CNN and a PCA reconstruction to create the model, using real images taken with an electron microscope in training and testing. The methodology developed allows for the creation of a defect detection model, without any labeled images of defects being required for training. The model performed well on all tests under the established assumptions, allowing for reliable anomaly detection. To the best of our knowledge, such methods are not currently available in the open …

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Examining The Effectiveness Of Using Point-Of-View Video Modeling On Mathematics Improvement In Students With Learning Disabilities In Saudi Arabia, Tirad Alsaluli Aug 2023

Examining The Effectiveness Of Using Point-Of-View Video Modeling On Mathematics Improvement In Students With Learning Disabilities In Saudi Arabia, Tirad Alsaluli

Electronic Theses and Dissertations

Video Modeling (VM) is one of the most widely used approaches by researchers to improve many skills, such as academic skills in students with Learning Disabilities (LD; Boon et al., 2020). As the incidence rate of individuals with LD in Saudi Arabia increase (Almedlij & Rubinstein-Ávila, 2018), the need for evidence-based math interventions focused on the math development of individuals with LD also increases. Although VM is recognized as an Evidence-based Practice (EBPs), a limited number of studies have implemented VM as an intervention to improve mathematic skills. Implementing VM as a math intervention strategy would explore its effects on …

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Cookie(X) = 1/2, Lawrence M. Lesser Aug 2023

Cookie(X) = 1/2, Lawrence M. Lesser

Journal of Humanistic Mathematics

This poem applies the concept of expected value, denoted E(X), to the context of any limited resources two parties desire. Usually, "you divide, I choose" keeps pieces equal enough to preempt charges of unfairness. But if one piece is much larger, many distrust the unbiased (in expected value) process of a coin flip giving each person the same chance at the bigger piece and the same cookie amount on average: E(X) = (1/2)p + (1/2)(1-p) = 1/2

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Ahab's Mercy, David Sheskin Aug 2023

Ahab's Mercy, David Sheskin

Journal of Humanistic Mathematics

A tale in which Captain Ahab and his chief mate Starbuck confront a classic problem in probability theory: the Monty Hall Problem.

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Zeno Of Elea: A Dichotomy, Joseph Chaney Aug 2023

Zeno Of Elea: A Dichotomy, Joseph Chaney

Journal of Humanistic Mathematics

This two-part poem interprets Zeno's paradoxes as dimensions of a paradoxical view of reality.

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I Am A Math Professor, Cacey L. Wells Aug 2023

I Am A Math Professor, Cacey L. Wells

Journal of Humanistic Mathematics

Original poem depicting the life of a math professor.

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