Mathematics Commons^™

Incomplete? Or Indefinite? Intuitionism On Gödel’S First Incompleteness Theorem, Quinn Crawford

The Yale Undergraduate Research Journal

This paper analyzes two natural-looking arguments that seek to leverage Gödel’s first incompleteness theorem for and against intuitionism, concluding in both cases that the argument is unsound because it equivocates on the meaning of “proof,” which differs between formalism and intuitionism. I argue that this difference explains why “proof” has definite extension for the formalist but not for the intuitionist. Sections 1-3 summarize various philosophies of mathematics and Gödel’s result. Section 4 argues that, because the Gödel sentence of a formal system is provable to the intuitionist, they are neither aided nor attacked by Gödel’s first incompleteness theorem. Section 5 …

Bernstein-Sato Polynomials In Commutative Algebra, Josep Àlvarez Montaner, Jack Jeffries, Luis Núñez-Betancourt

Department of Mathematics: Faculty Publications

This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra.

Euler's De Serie Lambertina, Translated From Latin To English With Supplementary Notes, Sam Gallagher

Euleriana

Originally published in 1779, Euler's De Serie Lambertina provides one of the early examples of the Lambert W function, a special function used in the solution to certain transcendental equations. Following the work of Johann Heinrich Lambert in 1759, who discussed a series solution to the general polynomial in series, and then particularly the solution of the general trinomial, Euler describes a symmetric form of the trinomial and its series solution. Euler investigates the series' special cases and general properties, and its use in solving certain transcendental equations. He provides several proofs of the validity of the series expansion to …

Translation Of Euler's Paper E421, Alexander Aycock

Euleriana

This is paper is the result of Euler’s findings on the Eulerian integral of second kind, i.e. the Γ-function: It summarises results and formulas on and properties of the integral in the title that Euler had obtained up to this point in his career and offers more elegant proofs of those before-mentioned results, formulas and properties. The results include a derivation of the integral in the title from an algebraic integral, the reflection formula for the Γ-function and finally a formula equivalent to the Gaußian multiplication formula for the Γ-function, expressed by Euler using mere integrals of algebraic functions.

Euler And The Multiplication Formula For The Gamma Function, Alexander Aycock

Euleriana

We show that an apparently overlooked result of Leonhard Euler (1707-1783) from [E421] is essentially equivalent to the general multiplication for- mula for the Γ-function that was proven by Carl Friedrich Gauss (1777-1855) in [Ga28].

On The Rectilinear Motion Of Three Bodies Mutually Attracting Each Other, Sylvio R. Bistafa

Euleriana

This is an annotated translation from Latin of E327 -- De motu rectilineo trium corporum se mutuo attrahentium (“On the rectilinear motion of three bodies mutually attracting each other”). In this publication, Euler considers three bodies lying on a straight line, which are attracted to each other by central forces inversely proportional to the square of their separation distance (inverse-square law). Here Euler finds that the parameter that controls the relative distances among the bodies is given by a quintic function.

Euler's Miracle, William Dunham

Euleriana

This article features some genuine Eulerian magic. In 1748, Leonhard Euler considered a modification of the harmonic series in which negative signs were attached to various terms by a rule that was far from self-evident. With his accustomed flair, he determined its sum, and the result was utterly improbable. There are a few occasions in mathematics when the term “breathtaking” is not too strong. This is one of them.

Ed Sandifer: An Eulerian Marathoner, Robert E. Bradley

Euleriana

Ed Sandifer was the founding secretary of the Euler Society. He published a remarkable quantity of Euler scholarship at the time of Euler’s Tercentenary in 2007.

Spotlight On The Euler Archive, Cynthia J. Huffman Ph.D.

Euleriana

A spotlight on the Euler Archive with a special emphasis on contributions by Dr. C. Edward Sandifer.

Review: A History Of Mathematics In The United States And Canada (Vol. 1), By David Zitarelli, Lawrence D'Antonio

Euleriana

This is a review of the 2019 text by David ZItarelli, A History of Mathematics in the United States and Canada. Volume 1: 1492 - 1900

Sharing Contributions To Euler Scholarship, Erik R. Tou, Christopher Goff

Euleriana

A summary of this issue's contents.

Hessian Formulas And Estimates For Parabolic Schrödinger Operators, Xue-Mei Li

Journal of Stochastic Analysis

No abstract provided.

Remembering Kunita-San, Ken-Iti Sato

Journal of Stochastic Analysis

No abstract provided.

On \Delta^(K)-Colouring Of Powers Of Paths And Cycles, Merlin Thomas Ellumkalayil Ms, Sudev Naduvath

Theory & Applications of Graphs

In an improper vertex colouring of a graph, adjacent vertices are permitted to receive same colours. An edge of an improperly coloured graph is said to be a bad edge if its end vertices have the same colour. A near-proper colouring of a graph is a colouring which minimises the number of bad edges by restricting the number of colour classes that can have adjacency among their own elements. The δ (k) - colouring is a near-proper colouring of G consisting of k given colours, where 1 ≤ k ≤ χ(G) – 1, which minimises the number of bad …

On The Application Of Principal Component Analysis To Classification Problems, Jianwei Zheng, Cyril Rakovski

Mathematics, Physics, and Computer Science Faculty Articles and Research

Principal Component Analysis (PCA) is a commonly used technique that uses the correlation structure of the original variables to reduce the dimensionality of the data. This reduction is achieved by considering only the first few principal components for a subsequent analysis. The usual inclusion criterion is defined by the proportion of the total variance of the principal components exceeding a predetermined threshold. We show that in certain classification problems, even extremely high inclusion threshold can negatively impact the classification accuracy. The omission of small variance principal components can severely diminish the performance of the models. We noticed this phenomenon in …

An Adaptive Cryptosystem On A Finite Field, Awnon Bhowmik, Unnikrishnan Menon

Publications and Research

Owing to mathematical theory and computational power evolution, modern cryptosystems demand ingenious trapdoor functions as their foundation to extend the gap between an enthusiastic interceptor and sensitive information. This paper introduces an adaptive block encryption scheme. This system is based on product, exponent, and modulo operation on a finite field. At the heart of this algorithm lies an innovative and robust trapdoor function that operates in the Galois Field and is responsible for the superior speed and security offered by it. Prime number theorem plays a fundamental role in this system, to keep unwelcome adversaries at bay. This is a …

Chemistry Card Games - Games For Learning And Enjoying Chemistry, Michelle M. Clark

Sabbatical Projects

Many chemistry students struggle due to a lack of basic math skills. My goal over sabbatical is to develop active learning modules that improve math literacy. I will be collaborating with Dr. Joan Cotter, a renowned math curriculum author whose comprehensive math program utilizes an innovative approach that could benefit chemistry students who have poor math literacy.

Globally Conservative Solutions For The Modified Camassa–Holm (Moch) Equation, Zhaonan Luo, Zhijun Qiao, Zhaoyang Yin

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we present the globally conservative solutions to the Cauchy problem for the modified Camassa–Holm (MOCH) equation. First, we transform the equation into an equivalent semi-linear system under new variables. Second, according to the standard ordinary differential equation theory with the aid of the conservation law, we give the global solutions of the semi-linear system. Finally, returning to the original variables, we obtain the globally conservative solutions to the MOCH equation.

Community Detection In Complex Networks, Zhenqi Lu

McKelvey School of Engineering Graduate Student Theses & Dissertations

Network science plays a central role in understanding and modeling complex systems in many disciplines, including physics, sociology, biology, computer science, economics, politics, and neuroscience. By studying networks, we can gain a deep understanding of the behavior of the systems they represent. Many networks exhibit community structure, i.e., they have clusters of nodes that are locally densely interconnected. These communities manifest the hierarchical organization of the objects in systems, and detecting communities greatly facilitates the study of the organization and structure of complex systems.

Most existing community-detection methods consider low-order connection patterns, at the level of individual links. But high-order …

Smooth Ica Under Time Pattern Assumptions, Jiayi Fu

Arts & Sciences Graduate Student Theses and Dissertations

Independent component analysis (ICA) is wildly used in differently areas. As traditional ICA models make no assumptions on time pattern, they do not take time domain information into consideration. In this thesis, we introduced new assumptions that allow local dependence over time, and we built smooth ICA models to utilize the smoothness information for sources signals. Based on the local dependence assumptions, constrained optimization problems with smoothing penalty were discussed. Then we introduced smooth ICA estimators and estimating equations. Under local dependence assumptions, we gave proofs about the consistency and asymptotic normality of these estimators. We derived the Newton iterative …