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Factorization Patterns in F_q [x]

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Advances in Pure Mathematics 2022年第2期12卷 70-79页

作者： thomas beatty Nicole Legge Department of Mathematics Florida Gulf Coast University Fort Myers USA

The finite field Fq has q elements, where q = pk for prime p and k∈N. Then Fq[x] is a unique factorization domain and its polynomials can be bijectively associated with their unique (up to order) factorizations into irreducibles. Such a factorization for a polynomial of degree n can be viewed as conforming to a specific template if we agree that factors with higher degree will be written before those with lower degree, and factors of equal degree can be written in any order. For example, a polynomial f(x) of degree n may factor into irreducibles and be written as (a)(b)(c), where deg a ≥ deg b ≥deg c. Clearly, the various partitions of n correspond to the templates available for these canonical factorizations and we identify the templates with the possible partitions. So if f(x) is itself irreducible over Fq, it would belong to the template [n], and if f(x) split over Fq, it would belong to the template [n] Our goal is to calculate the cardinalities of the sets of polynomials corresponding to available templates for general q and n. With this information, we characterize the associated probabilities that a randomly selected member of Fq[x] belongs to a given template. Software to facilitate the investigation of various cases is available upon request from the authors.

关键词： Finite Field Factorization Polynomial Degree Irreducible Splitting Partition Probability

On Irrational Points in the Plane

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Advances in Pure Mathematics 2017年第4期7卷 299-305页

作者： thomas beatty Timothy Jones Florida Gulf Coast University Fort Myers Florida USA

This paper summarizes research intended to develop a pedagogically friendly argument that establishes the fact that (x,ex ) is never a rational point in the plane. A point (x, y)∈R2 is rational if both x and y are rational. Applying a method based on Hurwitz polynomials, the research establishes simple irrationality proofs for nonzero rational powers of e and logarithms of positive rationals (excluding one).

关键词： Rational Point Irrationality Sum of Derivatives Prime Divisibility Hurwitz Polynomial

Pringsheim Convergence and the Dirichlet Function

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Advances in Pure Mathematics 2016年第6期6卷 441-445页

作者： thomas beatty Bradley Hansen Department of Mathematics Florida Gulf Coast University Ft. Myers FL USA

Double sequences have some unexpected properties which derive from the possibility of commuting limit operations. For example, may be defined so that the iterated limits and exist and are equal for all x, and yet the Pringsheim limit does not exist. The sequence is a classic example used to show that the iterated limit of a double sequence of continuous functions may exist, but result in an everywhere discontinuous limit. We explore whether the limit of this sequence in the Pringsheim sense equals the iterated result and derive an interesting property of cosines as a byproduct.

关键词： Convergence Pointwise Limit Double Sequence Pringsheim Dirichlet Function Baire Category Theorem Cosine

Structured Shuffles and the Josephus Problem

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Open Journal of Discrete Mathematics 2012年第4期2卷 138-141页

作者： Shaun Sullivan thomas beatty Florida Gulf Coast University Fort Myers USA

The Australian Shuffle consists of placing a deck of cards onto a table according to this rule: put the top card on the table, the next card on the bottom of the deck, and repeat until all the cards have been placed on the table. A natural question is “Where was the very last card placed located in the original deck?” Card trick magicians have known empirically for years that the fortieth card from the top of a standard fifty-two card deck is the final card placed by this shuffle. The moniker “Australian” comes from putting every other card “Down Under”. We develop a formula for the general case of N cards, and then extend that generalization further to cases involving the discard of k cards before or after putting one on the bottom of the deck. Finally, we discuss the connection of the Australian Shuffle and its generalizations to the famous Josephus problem.

关键词： Josephus Shuffling

On Conditional Probabilities of Factoring Quadratics

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Advances in Pure Mathematics 2020年第3期10卷 114-124页

作者： thomas beatty Gabriela von Linden Department of Mathematics Florida Gulf Coast University Fort Myers FL USA

Factoring quadratics over Z is a staple of introductory algebra and textbooks tend to create the impression that doable factorizations are fairly common. To the contrary, if coefficients of a general quadratic are selected randomly without restriction, the probability that a factorization exists is zero. We achieve a specific quantification of the probability of factoring quadratics by taking a new approach that considers the absolute size of coefficients to be a parameter n. This restriction allows us to make relative likelihood estimates based on finite sample spaces. Our probability estimates are then conditioned on the size parameter n and the behavior of the conditional estimates may be studied as the parameter is varied. Specifically, we enumerate how many formal factored expressions could possibly correspond to a quadratic for a given size parameter. The conditional probability of factorization as a function of n is just the ratio of this enumeration to the total number of possible quadratics consistent with n. This approach is patterned after the well-known case where factorizations are carried out over a finite field. We review the finite field method as background for our method of dealing with Z [x]. The monic case is developed independently of the general case because it is simpler and the resulting probability estimating formula is more accurate. We conclude with a comparison of our theoretical probability estimates with exact data generated by a computer search for factorable quadratics corresponding to various parameter values.

关键词： Factorization Polynomial Quadratic Integers Rational Numbers Monic Modular Arithmetic Conditional Probability

Generalized Series of Bernoulli Type

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Advances in Pure Mathematics 2023年第9期13卷 537-542页

作者： thomas beatty Nicholas Bianco Nicole Legge Department of Mathematics Florida Gulf Coast University Fort Myers FL USA

The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.

关键词： Bernoulli Series Convergence Sum Recursion Formula Zeta-Function Sine Maclaurin Series Infinite Product

A Geometric Proof of Fermat’s Little Theorem

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Advances in Pure Mathematics 2018年第1期8卷 41-44页

作者： thomas beatty Marc Barry Andrew Orsini Florida Gulf Coast University Fort Myers USA

We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in using the idea of colorings applied to regular polygons to establish a number-theoretic result. A lemma traditionally, if ambiguously, attributed to Burnside provides a critical enumeration step.

关键词： Fermat Carmichael Number Group Permutation Burnside’s Lemma Action Invariant Set Orbit Stabilizer Coloring Pattern Prime Regular Polygon Cyclic Group

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e^(p/q)是无理数的一个简单证明

数学译林 2022年第2期41卷 188-189页

作者： thomas beatty Timothy W.Jones 陆柱家(译) 陆昱(校) 不详

利用中值定理的一个简单应用,我们证明了e的有理数幂是无理数.

关键词：中值定理无理数有理数简单应用

Tuning beneficial calcineurin phosphatase activation to counterα-synuclein toxicity in a yeast model of Parkinson’s disease

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Neural Regeneration Research 2025年第1期20卷 199-200页

作者： Srishti Chawla Mikael Molin thomas Nystrom Institute for Biomedicine Sahlgrenska AcademyCentre for Ageing and Health-AgeCapUniversity of GothenburgGothenburgSweden Department of Life Sciences Chalmers University of TechnologyGothenburgSweden

Calcineurin(CN)is a calcium-and calmodulindependent serine/threonine that has been studied in many model organisms including yeast,filamentous fungi,plants,and *** biological functions range from ion homeostasis and virulence in lower eukaryotes to T-cell activation in humans by human nuclear factors of activated *** is a heterodimeric protein consisting of a catalytic subunit,calcineurin A(Cna1p),which contains an active site with a dinuclear metal center,and a regulatory Ca^(2+) binding subunit called calcineurin B(Cnb1p)required to activate *** calcineurin B subunit has been highly conserved through evolution:For example,the mammalian calcineurin B shows 54%identity with calcineurin B from Saccharomyces cerevisiae.

关键词： activation dinuclear conserved