Mathematics Commons^™

Do The Coefficients Of A Modular Form Really "Encode Arithmetic Data"?, Ken Mcmurdy, Hari Ravindran

Mathematical Sciences Technical Reports (MSTR)

Language and terminology are so critical to the understanding of modern math- ematics that it is often difficult for even very good mathematicians from different fields to discuss their work in any detail. As a result, common phrases often evolve within each discipline which attempt to capture the avor of some impor- tant idea while avoiding technicality and jargon. For example, when algebraic number theorists are asked why they are so interested in modular forms, it has become common to say with enthusiasm that the coefficients of a modular form "encode arithmetic data". If pressed further, one might go on …

Octic 2-Adic Fields, John W. Jones, David P. Roberts

Mathematics Publications

We compute all octic extensions of Q₂ and find that there are 1823 of them up to isomorphism. We compute the associated Galois group of each field, slopes measuring wild ramification, and other quantities. We present summarizing tables here with complete information available at our online database of local fields.

Rethinking Pythagorean Triples, William J. Spezeski

Applications and Applied Mathematics: An International Journal (AAM)

It has been known for some 2000 years how to generate Pythagorean Triples. While the classical formulas generate all of the primitive triples, they do not generate all of the triples. For example, the triple (9, 12, 15) can’t be generated from the formulas, but it can be produced by introducing a multiplier to the primitive triple (3, 4, 5). And while the classical formulas produce the triple (3, 4, 5), they don’t produce the triple (4, 3, 5); a transposition is needed. This paper explores a new set of formulas that, in fact, do produce all of the triples …

Signed Decomposition Of Fully Fuzzy Linear Systems, Tofigh Allahviranloo, Nasser Mikaeilvand, Narsis A. Kiani, Rasol M. Shabestari

Applications and Applied Mathematics: An International Journal (AAM)

System of linear equations is applied for solving many problems in various areas of applied sciences. Fuzzy methods constitute an important mathematical and computational tool for modeling real-world systems with uncertainties of parameters. In this paper, we discuss about fully fuzzy linear systems in the form AX = b (FFLS). A novel method for finding the non-zero fuzzy solutions of these systems is proposed. We suppose that all elements of coefficient matrix A are positive and we employ parametric form linear system. Finally, Numerical examples are presented to illustrate this approach and its results are compared with other methods.

A Statistical Look At Maps Of The Discrete Logarithm, Nathan Lindle

Mathematical Sciences Technical Reports (MSTR)

Cryptography is being used today more than it ever has in the past. Millions of transactions are being conducted every hour using encrypted channels, most of which use the Internet as their medium. It is taken for granted by the average user that these transaction are secure, but mathematicians and computer scientists alike are constantly testing the algorithms being used. Several of these cryptosystems use the transformation

g^x = y (mod n)

The appeal of this transformation is that it is quite simple to calculate gx mod n; exponentiation by squaring is fairly simple and quick even using …

An Improved Multi-Set Algorithm For The Dense Subset Sum Problem, Andrew Shallue

Scholarship

No abstract provided.

An Improved Multi-Set Algorithm For The Dense Subset Sum Problem, Andrew Shallue

Andrew Shallue

No abstract provided.

Number Fields Ramified At One Prime, John W. Jones, David P. Roberts

Mathematics Publications

For G a finite group and p a prime, a G-p field is a Galois number field K with Gal(K/Q)≅G and disc(K)=±p^a for some a. We study the existence of G-p fields for fixed G and varying p.

Multiplicative Properties Of Integral Binary Quadratic Forms, A. G. Earnest, Robert W. Fitzgerald

Articles and Preprints

In this paper, the integral binary quadratic forms for which the set of represented values is closed under k-fold products, for even positive integers k, will be characterized. This property will be seen to distinguish the elements of odd order in the form class group of a fixed discriminant. Further, it will be shown that this closure under k-fold products can always be expressed by a k-linear mapping from (Z²)^k to Z². In the case k = 2, this resolves a conjecture of Aicardi and Timorin.

Some Identities Between Basic Hypergeometric Series Deriving From A New Bailey-Type Transformation, James Mclaughlin, Peter Zimmer

Mathematics Faculty Publications

We prove a new Bailey-type transformation relating WPBailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.

Some New Families Of Tasoevian- And Hurwitzian Continued Fractions, James Mclaughlin

Mathematics Faculty Publications

We derive closed-form expressions for several new classes of Hurwitzian- and Tasoevian continued fractions, including [0; p − 1, 1, u(a + 2nb) − 1, p − 1, 1, v(a + (2n + 1)b) − 1 ]∞ n=0, [0; c + dmn] ∞n=1 and [0; eun, fvn] ∞n=1. One of the constructions used to produce some of these continued fractions can be iterated to produce both Hurwitzian- and Tasoevian continued fractions of arbitrary long quasi-period, with arbitrarily many free parameters and whose limits can be determined as ratios of certain infinite series. We also derive expressions for arbitrarily long finite …

N- Linear Algebra Of Type I And Its Applications, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

With the advent of computers one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure namely n-linear algebras of type I are introduced in this book and its applications to n-Markov chains and n-Leontief models are given. These structures can be thought of as the generalization of bilinear algebras and bivector spaces. Several interesting n-linear algebra properties are proved. This book has four chapters. The first chapter just introduces n-group which is essential for the definition of nvector spaces and n-linear algebras of type I. Chapter two gives the notion of n-vector spaces and several …

Special Classes Of Set Codes And Their Applications, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce the notion of set codes, set bicodes and set n-codes. These are the most generalized notions of semigroup n-codes and group n-codes. Several types of set ncodes are defined. Several examples are given to enable the reader to understand the concept. These new classes of codes will find applications in cryptography, computer networking (where fragmenting of codes is to be carried out) and data storage (where confidentiality is to be maintained). We also describe the error detection and error correction of these codes. The authors feel that these codes would be appropriate to the …

Chinese Neutrosophy And Taoist Natural Philosophy, Florentin Smarandache, Jiang Zhengjie

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.