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Professeur Badih GHUSAYNI |
Name : Dr. Badih
Ghusayni
Email : bgou@ul.edu.lb, badih@future-in-tech.net
Faculty of Science -1, Department of Mathematics, Lebanese University
Managing Editor, International Journal of mathematics and computer science
http://ijmcs.future-in-tech.net/index.htm
Research Interests and Specialties
- Complex and Harmonic Analysis :
Entire functions and Fourier Transforms, Representation of Entire
Functions by Series and Integrals.
- Approximation Theory : Approximation by a Nonfundamental Sequence of translates
- Analytic Number Theory : Tauberian Theorems, Distribution of Primes, Twin
Primes, Perfect Numbers, The Zeta Function at Odd Arguments, Factorization
and Primality.
- Computerized Instruction : Maple
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Publications |
Books
·
Number Theory from an analytic point of view
ISBN 9953-0-0282-7 : Badih Ghusayni
Paperback 198 pages
Contents
1.
Overview of Complex Numbers and Functions.
2.
Hadamard Factorization Theorem and Entire
Functions of Order One and Infinite Type.
3.
The Goldbach and Twin Prime
Conjectures.
4.
Zeta of 3.
5.
Maple Explorations.
6.
Function Characterizations.
7.
Exploring New Identities with Maple as a Tool.
8.
Mersenne Primes, Perfect Numbers, and
Friendly Numbers.
9.
The Prime Number Theorem from an Analytic Point of View.
10. Cryptography.
This book has emerged from the author's interest in Number theory which began
in 1980 when the author wrote his masters thesis on Tauberian
Theorems and the Prime Number Theorem. This interest turned out to be an
increasing function of time.
Some results were discovered by using the Computer Algebra System Maple and
then proved mathematically thus providing new venues of mathematical research.
To each chapter, I have supplied exercises which range from simple to unsolved
(needless to say, I would of course let the reader know which problem remains
unsolved but hopefully, by doing so, the reader's interest in trying to solve
it won't diminish).
This is among the reasons why the author thinks that this book is targeted
towards amateurs and professionals alike. At the end of some chapters, we shed
some light on lives of relevant mathematicians which the author feels attracts
the interest of readers and may put things in perspective.
Each chapter has its own references.
·
Théorie de Nombre
d'un Point de Vue Analytique
ISBN 9953-0-0283-5 Badih Ghusayni
Table des matières
1.
Vue d'ensemble
des Nombres et des Fonctions
Complexes .
2.
Théorème de Factorization du Hadamard et
les Fonctions Entières d'Ordre Un et Type Infini.
3.
Conjectures de Goldbach et nombres premiers Jumeaux.
4.
Zeta de 3.
5.
Exploration du Maple,
6.
Caractérisations des Fonctions,
7.
Exploration de Identités Nouvelles avec Maple comme un Outil,
8.
Mersenne Premiers, Nombres
Parfaits, et Nombres Amicaux.
9.
Le Théorème de Nombre Premier d'un Point de Vue Analytique
10. Cryptographie.
Ce livre est paru
motivé par l'intérêt que son auteur a manifesté pour
la théorie des nombres. L'auteur a découvert ce domaine
des mathématiques pendant la préparation
de son mémoire de master sur
les Théorèmes Tauberiens et
le Théorème des Nombres
Premiers. Cet intérêt s'est avéré
une fonction croissante du temps.
Quelques résultats on tété découverten utilisant le système informatique d'algèbre de Maple puis prouvés mathématiquement. Ce procédé, l'utilisation
des logiciels mathématiques
pour deviner des comportements
des nombres, s'inscrit parmi les nouveles techniques de
la recherche mathématique.
Pour chaque chapitre, j'ai fourni des exercices de difficulté variée, qui s'étendent de simple à non résolu (bien
sur, les questions ouvertes
sont signalées et les lecteurs intéressés sont invités a y réfléchir avec bon espoir de les résoudre). C'est pourquoi, l'auteur pense que ce livre peut intéresser
les amateurs tout comme les professionnels.
A la fin de quelques chapitres,
nous avons donné un aperçu
rapide sur la vie de mathématiciens célèbres directement concernés par ce domaine de mathématique.
Ceci est
dans le but de divertir et d'informer le lecteur sur le déroulement historique des découvertes dans l'espoir de mettre chaque résultat
dans sa perspective historique.
Author of
"Online Course in Complex Analysis" as part of Avicenna Virtual
Campus, supported by UNESCO
Papers
·
"Maple explorations, perfect numbers and Mersenne
primes", The International Journal of Mathematics Education in Science and
Technology Vol. 36,
No. 6, 2005, 643-654.
·
"A Collection of Number and Function
Characterizations", WSEAS Transactions on Mathematics, Vol
4, Issue 1, January 2005, 12-17.
·
"Exploring new identities with Maple as a tool",
WSEAS Transactions on Information Science and Applications, Vol . 1, Issue 5, November
2004, 1151-1157.
·
"Characterizations of Arithmetical Progression Series
with some Counterexamples on Interpolation",
·
"Euler-type Formula using Maple", Palma Research
Journal, Vol. 7, 2001, 175-180.
·
"Perfect Numbers and some of their properties,
Proceedings of the International Conference on Scientific Computations held at
the
Abstract.
Perfect numbers have fascinated people for a very long time and continue to do
so. In this paper we look at some of their interesting properties and mention
some questions that still await answers. A good venue, nowadays, is numerical
computation.
·
"Some Representations of zeta of 3", Vol. 10,
Missouri Journal of Mathematical Sciences, (1998), 169-175.
Abstract.
We find a simple representation of zeta of 3 in terms of a single integral. We
also obtain a series representation for zeta of 3.
·
"On Approximation by a nonfundamental
sequence of translates" , Vol. 199, Journal of Mathematical Analysis and
Applications), (1996), 469-477.
Abstract.
If a function and its transform satisfy some growth conditions and if a
sequence of distinct real numbers satisfies a certain separation condition, we
represent those functions which are in the closure of the linear span of a nonfundamental sequence of translates. A result about the
degree of approximation is also proved.
·
"Products and sums with applications", Vol. 9,
Missouri Journal of Mathematical Sciences, (1997), 90-94.
Abstract.
The twin prime conjecture states that the number of twin primes is infinite.
Many attempts to prove or disprove the conjecture have failed. The objective of
this note is to tie the twin prime conjecture to complex variable theory and
prove some results that make it possible to consider the conjecture from a
complex variable point of view rather than from a purely number theoretic one.
·
"Entire functions of order one and infinite
type", Vol. 10, Missouri Journal of Mathematical Sciences, (1998), 20-27.
Abstract.
In this paper we first prove an auxiliary result that an entire function of
order one and infinite type must have infinitely many zeros. We then give an
explicit canonical representation for those functions. We apply the
representation to prove a result and its converse about entire functions of
order one and infinite type. Next, we mention a few interesting examples of entire
functions of order one and infinite type. Finally, we formulate and disprove a
conjecture which serves as an analogue to Paley-Wiener
theorem for entire functions of order one and infinite type.
·
"Integral Representation of 2-pi periodic and
trigonometrically convex functions , Vol. 14, Complex Analysis, (1990),
129-138.
Abstract.
The integral representation given in Levin's book "Distribution of Zeros
of Entire Functions" of 2-pi periodic and r-trigonometrically convex
functions which are indicators of holomorphic
functions of non-zero order r is incorrect. Counterexamples are given here as
well as a corrected version of the representation.
Selected Presentations
·
Integral representation of 2-pi periodic and
trigonometrically convex functions, South-eastern conference, Clemson, 1985.
·
Entire functions and Fourier transforms, AMS-MAA Annual
meeting,
·
The Order of an entire Function and the twin prime
conjecture conjecture, International Conference on
Analytic Number Theory,
·
The Order of a function and the twin prime conjecture,
AMS-MAA Annual meeting,
·
Entire functions of order one and infinite type, AMS-MAA
Annual meeting,
