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A Tale Of Two Workshops: Two Workshops, Three Papers, New Ideas, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Arithmetic Jet Spaces And Modular Forms, Arnab Saha

Mathematics & Statistics ETDs

In this thesis, we would look into the theory of arithmetic jet spaces and its application in modular forms. The arithmetic jet spaces can be thought of as an analogue of jet spaces in differential algebra. In the case of arithmetic jet spaces, a derivation is replaced by $p$-derivation $\\d$. This theory was initiated by A. Buium in \\cite{char}. The results in the first chapter are concerning the connection between arithmetic jet spaces and Witt vectors. Let $R = \\widehat{\\Z_p^{ur}}$ be the $p$-adic completion of the maximal unramified extension of $\\Z_p$. If $A$ is an $R$-algebra and we denote $J^nA$ …

Unveiling Of Geometric Generation Of Composite Numbers Exactly And Completely, Stein E. Johansen

Applied Mathematics & Information Sciences

We present a certain geometrical interpretation of the natural numbers, where these numbers appear as joint products of 5- and 3-multiples located at specified positions in a revolving chamber. Numbers without factors 2, 3 or 5 appear at one of eight such positions, after a specified amount of rotations of the chamber. Our approach determines the sets of rotations constituting primes at the respective eight positions, as the complements of the sets of rotations constituting composite numbers at the respective eight positions. These sets of rotations constituting composite numbers are exhibited in strict rotation regularities from a basic 8£8-matrix of …

An Exponential Open Hashing Function Based On Dynamical Systems Theory, Chaouki T. Abdallah, Bradley J. Smith, Gregory L. Heileman

Electrical & Computer Engineering Faculty Publications

In this paper an efficient open addressing hash function called exponential hashing is developed using concepts from dynamical systems theory and number theory. A comparison of exponential hashing versus a widely used double hash function is performed using an analysis based on Lyapunov exponents and entropy. Proofs of optimal table parameter choices are provided for a number of hash functions. We also demonstrate experimentally that exponential hashing nearly matches the performance of an optimal double hash function for uniform data distributions and performs significantly better for nonuniform data distributions. We show that exponential hashing exhibits a higher integer Lyapunov exponent …

Prove It!, Kenny W. Moran

Journal of Humanistic Mathematics

A dialogue between a mathematics professor, Frank, and his daughter, Sarah, a mathematical savant with a powerful mathematical intuition. Sarah's intuition allows her to stumble into some famous theorems from number theory, but her lack of academic mathematical background makes it difficult for her to understand Frank's insistence on the value of proof and formality.

Orthogonal Polynomials, George Gevork Antashyan

Theses Digitization Project

This thesis will show work on Orthogonal Polynomials. In mathematics, the type of polynomials that are orthogonal to each other under inner product are called orthogonal polynomials. Jacobi polynomials, Laguerre polynomials, and Hermite polynomials are examples of classical orthogonal polynomials that was invented in the nineteenth century. The theory of rational approximations is one of the most important applications of orthogonal polynomials.

Leonhard Euler's Contribution To Infinite Polynomials, Jack Dean Meekins

Theses Digitization Project

This thesis will focus on Euler's famous method for solving the infinite polynomial. It will show how he manipulated the sine function to find all possible points along the sine function such that the sine A would equal to y; these would be roots of the polynomial. It also shows how Euler set the infinite polynomial equal to the infinite product allowing him to determine which coefficients were equal to which reciprocals of the roots, roots squared, roots cubed, etc.

An Introduction To Modern Cryptology Within An Algebraic Framework, John Szwast

EWU Masters Thesis Collection

No abstract provided.

Special Quasi Dual Numbers And Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce a new notion called special quasi dual number, x = a + bg.

Among the reals – 1 behaves in this way, for (– 1)2 = 1 = – (– 1). Likewise –I behaves in such a way (– I)2 = – (– I). These special quasi dual numbers can be generated from matrices with entries from 1 or I using only the natural product ×n. Another rich source of these special quasi dual numbers or quasi special dual numbers is Zn, n a composite number. For instance 8 in Z12 is such that …

Solutions To A Generalized Pell Equation, Kyle Christopher Castro

Theses Digitization Project

This study aims to extend the notion of continued fractions to a new field Q (x)*, in order to find solutions to generalized Pell's Equations in Q [x] . The investigation of these new solutions to Pell's Equation will begin with the necessary extensions of theorems as they apply to polynomials with rational coefficients and fractions of such polynomials in order to describe each "family" of solutions.

Prouhet-Tarry-Escott Problem, Juan Manuel Gutierrez

Theses Digitization Project

The purpose of this research paper is to gain a deeper understanding of a famous unsolved mathematical problem known as the Prouhet-Terry-Escott Problem. The Prouhet-Terry-Escott Problem is a complex problem that still has much to be discovered. This fascinating problem shows up in many areas of mathematics such as the study of polynomials, graph theory, and the theory of integral quadratic forms.