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Full-Text Articles in Number Theory
Mapping The Discrete Logarithm, Daniel R. Cloutier
Mapping The Discrete Logarithm, Daniel R. Cloutier
Mathematical Sciences Technical Reports (MSTR)
The discrete logarithm is a problem that surfaces frequently in the field of cryptog- raphy as a result of using the transformation ga mod n. This paper focuses on a prime modulus, p, for which it is shown that the basic structure of the functional graph is largely dependent on an interaction between g and p-1. In fact, there are precisely as many different functional graph structures as there are divisors of p-1. This paper extracts two of these structures, permutations and binary functional graphs. Estimates exist for the shape of a random permutation, but …
Some Effective Diophantine Results Over Q-Bar, Lenny Fukshansky
Some Effective Diophantine Results Over Q-Bar, Lenny Fukshansky
CMC Faculty Publications and Research
In his 1999 paper D. W. Masser talks about effective search bounds for polynomial equations over integers and rationals. This discussion can also be extended over number fields. Unfortunately, as illustrated by Matiasevich's negative answer to Hilbert's 10-th problem, search bounds in general probably do not exist. Some special cases are understood, but in general very little is known. I will talk about effective search bounds for solutions of polynomial equations over Q-bar with some additional arithmetic conditions. This discussion also naturaly ties into the realm of "absolute" diophantine results, like Siegel's lemma of Roy and Thunder. I will try …
A Convergence Theorem For Continued Fractions Of The Form K_{N=1}^{\Infty}A_{N}/1, James Mclaughlin, Nancy Wyshinski
A Convergence Theorem For Continued Fractions Of The Form K_{N=1}^{\Infty}A_{N}/1, James Mclaughlin, Nancy Wyshinski
Mathematics Faculty Publications
In this paper we present a convergence theorem for continued fractions of the form K∞n=1an/1. By deriving conditions on the an which ensure that the odd and even parts of K∞n=1an/1 converge, these same conditions also ensure that they converge to the same limit. Examples will be given.
Ramanujan And The Regular Continued Fraction Expansion Of Real Numbers, James Mclaughlin, Nancy Wyshinski
Ramanujan And The Regular Continued Fraction Expansion Of Real Numbers, James Mclaughlin, Nancy Wyshinski
Mathematics Faculty Publications
In some recent papers, the authors considered regular continued fractions of the form [a0; a, · · · , a | {z } m , a2 , · · · , a2 | {z } m , a3 , · · · , a3 | {z } m , · · · ], where a0 ≥ 0, a ≥ 2 and m ≥ 1 are integers. The limits of such continued fractions, for general a and in the cases m = 1 and m = 2, were given as ratios of certain infinite series. However, these formulae can be derived …
Sums Of Gauss Sums And Weights Of Irreducible Codes, Robert W. Fitzgerald, Joseph L. Yucas
Sums Of Gauss Sums And Weights Of Irreducible Codes, Robert W. Fitzgerald, Joseph L. Yucas
Articles and Preprints
We develop a matrix approach to compute a certain sum of Gauss sums which arises in the study of weights of irreducible codes. A lower bound on the minimum weight of certain irreducible codes is given.
Factors Of Dickson Polynomials Over Finite Fields, Robert W. Fitzgerald, Joseph L. Yucas
Factors Of Dickson Polynomials Over Finite Fields, Robert W. Fitzgerald, Joseph L. Yucas
Articles and Preprints
We give new descriptions of the factors of Dickson polynomials over finite fields.
Highly Degenerate Quadratic Forms Over Finite Fields Of Characteristic 2, Robert W. Fitzgerald
Highly Degenerate Quadratic Forms Over Finite Fields Of Characteristic 2, Robert W. Fitzgerald
Articles and Preprints
Let K/F be an extension of finite fields of characteristic two. We consider quadratic forms written as the trace of xR(x), where R(x) is a linearized polynomial. We show all quadratic forms can be so written, in an essentially unique way. We classify those R, with coefficients 0 or 1, where the form has a codimension 2 radical. This is applied to maximal Artin-Schreier curves and factorizations of linearized polynomials.
Real Numbers With Polynomial Continued Fraction Expansions, James Mclaughlin, Nancy Wyshinski
Real Numbers With Polynomial Continued Fraction Expansions, James Mclaughlin, Nancy Wyshinski
Mathematics Faculty Publications
In this paper we show how to apply various techniques and theorems (including Pincherle’s theorem, an extension of Euler’s formula equating infinite series and continued fractions, an extension of the corresponding transformation that equates infinite products and continued fractions, extensions and contractions of continued fractions and the Bauer-Muir transformation) to derive infinite families of in-equivalent polynomial continued fractions in which each continued fraction has the same limit. This allows us, for example, to construct infinite families of polynomial continued fractions for famous constants like π and e, ζ(k) (for each positive integer k ≥ 2), various special functions evaluated at …
Powers Of A Matrix And Combinatorial Identities, James Mclaughlin, B. Sury
Powers Of A Matrix And Combinatorial Identities, James Mclaughlin, B. Sury
Mathematics Faculty Publications
In this article we obtain a general polynomial identity in k variables, where k ≥ 2 is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a k × k matrix. Finally, we use these results to derive various combinatorial identities.
Fuzzy And Neutrosophic Analysis Of Women With Hiv/Aids, Florentin Smarandache, W.B. Vasantha Kandasamy
Fuzzy And Neutrosophic Analysis Of Women With Hiv/Aids, Florentin Smarandache, W.B. Vasantha Kandasamy
Branch Mathematics and Statistics Faculty and Staff Publications
Fuzzy theory is one of the best tools to analyze data, when the data under study is an unsupervised one, involving uncertainty coupled with imprecision. However, fuzzy theory cannot cater to analyzing the data involved with indeterminacy. The only tool that can involve itself with indeterminacy is the neutrosophic model. Neutrosophic models are used in the analysis of the socio-economic problems of HIV/AIDS infected women patients living in rural Tamil Nadu. Most of these women are uneducated and live in utter poverty. Till they became seriously ill they worked as daily wagers. When these women got admitted in the hospital …