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

 

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

·         Solved Problems in Analysis: A Companion to Math Majors
ISBN 978-9953-0-1306-0  Badih N. Ghusayni

Paperback 360 pages

Contents

  1. Number Systems.
  2. Numerical Sequences and Series.
  3. Limits and Continuity.
  4. Differentiation.
  5. Abstract Integration.
  6. Borel Measures.
  7. -Spaces.
  8. Hilbert Spaces.
  9. Banach Spaces.
  10. Complex Measures.
  11. Product Measures.
  12. Analytic Functions.
  13. Entire Functions of Exponential Type.
  14. Completeness of Sets of Complex Exponentials.


This book covers the wide area of Analysis and includes 525 problems and their complete solutions covering undergraduate Classical Analysis, Graduate Real and Complex Analysis, Functional Analysis and Nonharmonic Fourier Series. As a result this book is targeted for math majors at all levels.

 

·                    Classical Real Analysis
      ISBN 978-9953-0-1496-8

Paperback 127 pages

Contents:
1. The Real Number System
2. Numerical Sequences
3. Limits and Continuity
4. Differentiation
5. Usual Functions
6. Finite Expansions
7. Adjacent Sequences, Recursive Sequences, Limit Inferior and Limit Superior of a Sequence
8. Numerical Series
9. Series of Functions and the Special Case of a Power Series

This textbook covers Real Analysis from a classical point of view in the sense that we do not go over Cauchy's theory of analytic functions. This is not a Calculus textbook. The student is advised to take the material very seriously for not only it provides him/her with a solid background for other courses but also supplies
essential tools to increase his/her mathematical maturity and logic to even help in real-life situations like avoiding stereotyping which we'll give at the begining of the first chapter. Chapter 2 introduces convergence, an essential concept in Analysis with a wide spectrum, through the notion of the limit of a numerical sequence. Unlike Calculus which touches on the idea of "near" or "close to", Analysis (our state of affairs) validates this idea. Chapter 3 covers monotone functions, limit of a function, properties of limits of functions which will have direct impact on properties of continuous functions. We conclude this chapter with the important definition of a uniformly continuous function on an interval of (real) numbers and the Intermediate Value Theorem. In chapter 4 we take on the familiar and important concept of differentiation, discuss Mean Value Theorems, L'Hospital's Rule , and Taylor and Young formulas. Usual Functions are handled in chapter 5 and Finite Expansions in chapter 6. In chapter 7 we study Adjacent and Recursive Sequences, Cluster Points and Limit Inferior and Superior of a sequence. We then use our knowledge of Numerical Sequences to study Numerical Series in chapter 8. Finally, in chapter 9 we study Series of Functions in general but we concentrate on a special and important case called Power Series and use the concept of Uniform Convergence to derive results of utmost importance for applications. As a final note in this preface, the author would like to draw the attention of students to the fact that this book, like other mathematics books, contains "polished" proofs that will be best understood if the student makes the effort of reading "between the lines"; that is, to pause and ask oneself related questions in an effort to understand the mathematical argument. This may also require a blank piece of paper and a pen to jot a few lines.

 

 

Papers

 

·        Generalized Integration Formulas, Int. J.Math. Comput. Sci.,Vol. 5, No. 1, 2010, 7-14. The Value of the Zeta Function at an Odd Argument, Int. J. Math. Comput. Sci., Vol. 4, No. 1, 2009, 21-30

·        "Towards a Proof of the Twin Prime Conjecture", International Journal of Pure and Applied Mathematics, Vol. 47, No. 1, 2008, 31-40.

·         "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", Missouri J. Math. Sci. , Vol. 15, Issue 2, 2003, pp. 110-128.

·         "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 Lebanese American University, (1999), 117-126.
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, San Antonio, 1987.

·         The Order of an entire Function and the twin prime conjecture conjecture, International Conference on Analytic Number Theory, Allerton Park, University of Illinois, 1989.

·         The Order of a function and the twin prime conjecture, AMS-MAA Annual meeting, San Francisco, 1991.

·         Entire functions of order one and infinite type, AMS-MAA Annual meeting, Baltimore, 1992.

·         Towards a Proof of the Twin Prime Conjecture, AMS-MAA Annual meeting, San Diego, 2008.

·        The Value of The Zeta Function at an Odd Argument, AMS-MAA Annual meeting, San Francisco, 2010.