Physical Sciences and Mathematics Commons^™

Euler Archive Spotlight: Multiple Search Options, Christopher Goff

Euleriana

The Euler Archive houses PDF versions of almost all of Euler's original publications. While most visitors search the archive via a work's Eneström number, the Archive can be searched via source publication name, date written, or decade of publication. The Archive also provides context for Euler's publications through short pieces of historical information.

Euler And A Proof Of The Functional Equation For The Riemann Zeta-Function He Could Have Given, Alexander Aycock

Euleriana

We explain how Euler could have proved a functional equation, which is equivalent to the one for the Riemann zeta-function, that he conjectured in his paper {\it ``Remarques sur un beau rapport entre les series des puissances tant directes que reciproques"} \cite{E352} (E352: ``Remarks on the beautiful relation between the series of the direct and reciprocal powers").

Euler And The Gaussian Summation Formula For The Hypergeometric Series, Alexander Aycock

Euleriana

We show that in his paper {\it ``Plenior expositio serierum illarum memorabilium, quae ex unciis potestatum binomii formantur"} \cite{E663} (E663: ``A more thorough exposition of those memorable series that are formed from the binomial coefficients") Euler could have found the Gaussian summation formula for the hypergeometric series from his own formulas in that same paper, if he actually set the task for himself.

Euler And Homogeneous Difference Equations With Linear Coefficients, Alexander Aycock

Euleriana

We present a method outlined by Euler in his paper{\it ``De fractionibus continuis observationes"} \cite{E123} (E123: ``Observations on continued fractions") that can be used to solve homogeneous difference equations with linear coefficients. We will illustrate his ideas by applying it to two familiar examples and explain how it can be understood from a more modern point of view.

On The Cases In Which The Formula X^4+Kxxyy+Y^4 Can Be Reduced To A Square, Georg Ehlers

Euleriana

Euler’s key idea for equating the Quartic in the title to a square is to set k=P+surd(Q). From this he derives P=f·x^2 and Q=4f·y^2+4 and solves the Pell equation for y. He then discusses various extensions to rational numbers that leave k an integer. Euler provides incomplete tables for integers k with |k|square.

Research On Arithmetic, Erik R. Tou

Euleriana

In this English translation, some of Joseph-Louis Lagrange's early number theory is presented. Here, he laid out a theory of binary quadratic forms with special attention to the representation problem: determining those integers which may be represented by a given form, and cataloguing the possible forms of their divisors.

Number Theory And More, Christopher Goff, Erik Tou

Euleriana

An introduction to the contents in Issue 1, Volume 4 of Euleriana.

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.

Production Functions Of Ncaa Men And Women Water Polo Matches, Joey Gullikson, Lewis R. Gale, John Mayberry, Lara Killick, John Kim

College of the Pacific Faculty Articles

Previous research has adapted the use of economic production functions to estimate the scoring production of teams in professional sports. Most of these studies have focused on professional male team sports, most notably, US baseball, basketball, and association football. This study adds to the literature by utilizing a new and distinctive data set of shooting statistics from 88 men’s and 38 women’s NCAA water polo contests to estimate production functions for United States’ collegiate water polo games and identify the most important variables for predicting margin of victory in such competitions. The results show that shots on goal, average shot …

Big Data Analysis And Programming For Engineers, Josh Steimel

Pacific Open Texts

This text serves to cover critical programming, data analysis, statistical analysis, and mathematical skills for engineers. In particular fundamental programming skills are demonstrated using Mathematica specifically the importing of data sets, loop structures, plotting and statistically analyzing data, image analysis, and machine learning. Critical engineering topics such as solid mechanics, vibrations, and engineering problems which require solving ODEs and PDEs are covered.

Shop Less, Shop Better: Ethical And Sustainability Concerns Throughout The Fast Fashion Supply Chain, Natali Ann Hoeg

Pacific Undergraduate Research and Creativity Conference (PURCC)

My capstone project is a series of short essays that describe how fast fashion is having negative impacts on local environments, communities, and supply chains. This research is a review of technical sources describing manufacturing of clothing and certification processes for alternative, sustainable textile production. Additionally, this review incorporates perspectives from journalists and documentarians, visiting and describing farms and factories that are involved with fast fashion production. This project was carried out over the course of a semester, as an independent study. The essays focus on the environmental and social impacts of fast fashion. Each of the six essays approaches …

Euler Archive Spotlight, Erik R. Tou

Euleriana

A survey of two translations posted to the Euler Archive in 2022.

Euler And Venus' Suspicious Moon, Michael P. Saclolo

Euleriana

This is a brief note on Leonhard Euler's published German translation from the French of two memoirs read by Armand Henri Baudouin de Guémadeuc to the Paris Academy of Sciences in 1761 and published the same year. The memoirs report on observations made of the planet Venus, performed in Limoges, France by Jacques Montaigne, where he claimed to have detected a moon orbiting the Morning and Evening Star.

Euler And The Duplication Formula For The Gamma-Function, Alexander Aycock

Euleriana

We show how the formulas in Euler’s paper "Variae considerationes circa series
hypergeometricas" [ 4] imply Legendre’s duplication formula for the Γ-function. This
paper can be seen as an Addendum to [2].

Euler Found The First Binary Digit Extraction Formula For Π In 1779, Nick Craig-Wood

Euleriana

In 1779 Euler discovered two formulas for π which can be used to calculate any binary digit of π without calculating the previous digits. Up until now it was believed that the first formula with the correct properties (known as a BBP-type formula) for this calculation was published by Bailey, Borwein and Plouffe in 1997.

Analytical Observations (Translation Of E326), Cynthia Huffman Ph.D.

Euleriana

Euler, in this publication with Eneström number E326, provides an induction fallacy which arises from analyzing a particular sequence. Euler wrote this work in 1763, one of only two papers he wrote on sequences and/or series in the 1760’s, out of a total of 79 papers on series during his career. His goal in E326 is to investigate the middle terms in the expansion of powers of quadratic trinomial expressions, beginning with the specific simple quadratic , before considering the general quadratic .

The induction fallacy shows up during the analysis of the simple case when Euler first finds an …

Euler's Anticipations, Christopher Goff, Erik Tou

Euleriana

Welcome to Volume 3 of Euleriana. This issue highlights occasions where Euler's work anticipated future results from other others, sometimes by decades or even centuries!

Making Sense Of Big (Kinematic) Data: A Comprehensive Analysis Of Movement Parameters In A Diverse Population, Naomi Wilma Nunis

University of the Pacific Theses and Dissertations

OBJECTIVE

The purpose of this study was to determine how kinematic, big data can be evaluated using computational, comprehensive analysis of movement parameters in a diverse population.

METHODS

Retrospective data was collected, cleaned, and reviewed for further analysis of biomechanical movement in an active population using 3D collinear resistance loads. The active sample of the population involved in the study ranged from age 7 to 82 years old and respectively identified as active in 13 different sports. Moreover, a series of exercises were conducted by each participant across multiple sessions. Exercises were measured and recorded based on 6 distinct biometric …

The Discovery And Characterization Of Novel Potent 5-Substituted 3, 3’, 4’, 7-Tetramethoxyflavonoid Dna Triplex Specific Binding Ligands, Vanessa Marie Rangel

University of the Pacific Theses and Dissertations

Chemotherapy works by killing fast dividing cells. Unfortunately, these drugs are not specific to cancer tissue and can damage normal cells. Chemotherapy is like taking poison and hoping it kills the cancer cells before it kills you. As an alternative, many researchers have investigated the use of antigene therapy to selectively target cancer causing genes to avoid off target effects. Although promising, the theory is limited by the stability of the triplex structure. Here, we report the discovery of potent triplex binding ligands derived from the natural product quercetin. Chemical derivatives of 5-substituted 3, 3’, 4’, 7-tetramethoxyquercetin derivatives were characterized …

Remote Ambient Aerosols Found In The Southern Great Plains And Eastern North Atlantic: A Comprehensive Analysis Of Optical Behavior And Aerosol-Cloud Interaction, Daniel J. Bonanno

University of the Pacific Theses and Dissertations

The data presented in this thesis showcases the investigation of various aerosol types and the role they play in the hydrological system and global climate models. Long-range transport aerosols, including atmospheric soot, mineral dust, and ammonium sulfate, in addition to near-site sources such as secondary organic aerosols (SOAs), sea spray, aliphatic hydrocarbons, and other organics, are prioritized in the analysis of two separate remote field studies. Each of these campaigns revealed the juxtaposition between dry summer months and wet winter months, and the connection between seasonal variability and sample collection. Two spectromicroscopic techniques were used to elucidate chemical composition, morphology …

Math 57: Applied Differential Equations I, John Mayberry

Pacific Open Texts

This book is designed for the fourth semester, “capstone” course in a calculus sequence with an emphasis on modeling with linear differential equations. Students will learn to translate verbal descriptions of physical problems into differential equation models, solve and visualize solutions to differential equations using MATLAB, calculate and investigate the behavior of analytic solutions to linear differential equations, discuss how solutions to differential equations depend on parameters, and interpret solutions to differential equations in the context of applications.

The Rock 2022, School Of Engineering And Computer Science

The Rock

No abstract provided.