Mathematics Commons^™

Exploring Topological Phonons In Different Length Scales: Microtubules And Acoustic Metamaterials, Ssu-Ying Chen

Dissertations

The topological concepts of electronic states have been extended to phononic systems, leading to the prediction of topological phonons in a variety of materials. These phonons play a crucial role in determining material properties such as thermal conductivity, thermoelectricity, superconductivity, and specific heat. The objective of this dissertation is to investigate the role of topological phonons at different length scales.

Firstly, the acoustic resonator properties of tubulin proteins, which form microtubules, will be explored The microtubule has been proposed as an analog of a topological phononic insulator due to its unique properties. One key characteristic of topological materials is the …

Geometry Through Architectural Design, Maureen T. Carroll, Elyn Rykken

LASER Journal

In her 1912 geometry book, Mabel Sykes surveys complex and beautiful architectural designs from around the world to inspire exercises on geometric proof, construction and computation. In over 1800 exercises, Sykes analyzes geometric patterns from ornamental and structural features found in tile mosaics, parquet floors, Gothic windows, trusses and arches. As Sykes' writes, ``Geometry gives, as no other subject can give, an appreciation of form as it exists in the material world" . We have chosen four examples to illustrate how her appealing designs and the accompanying exercises of this hidden gem can be incorporated into any geometry course.

Topics In Absolute And Relative Treatment Effects In Medical Data Models And Applications With A Continuous Response, Sangit Regmi

Student Theses and Dissertations

Medical data models aim to evaluate the effectiveness of different treatments in a patient population, and to determine the relative and absolute treatment effects. The aim of this thesis is to evaluate the use of the traditional model when relative treatment effects exists and further develops a new model to account for both the absolute and relative treatment effects for a continuous response. The new model can be seen as a generalization of the traditional model by incorporating the relative treatment effect. It is shown in the simulation studies and real data set that the new model is able to …

An Analysis Of The Sequence Xn+2 = I M Xn+1 + Xn, David Duncan, Prashant Sansgiry, Ogul Arslan, Jensen Meade

Journal of the South Carolina Academy of Science

We analyze the sequence Xn+2 = imXn+1 + Xn, with X1 = X2 = 1 + i, where i is the imaginary number and m is a real number. Plotting the sequence in the complex plane for different values of m, we see interesting figures from the conic sections. For values of m in the interval (−2, 2) we show that the figures generated are ellipses. We also provide analysis which prove that for certain values of m, the sequence generated is periodic with even period.

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.

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 …

Double Barrier Backward Doubly Stochastic Differential Equations, Tadashi Hayashi

Journal of Stochastic Analysis

No abstract provided.

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.

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.

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 …

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.

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.

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.

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 …

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.

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.

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

Theory & Applications of Graphs

The α-Hermitian adjacency matrix H_α 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 …

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 …

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 …

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 …