Mathematics

 RESEARCH PROJECTS 
 ARTICLES, BOOKS AND REPORTS 
 ABSTRACTS, PRESENTATIONS AND PROCEEDINGS 
 MISCELLANEOUS 
 MASTER'S THESES 
 
 

RESEARCH PROJECTS
 

Zero-free regions for entire functions and sections of their power series

F. Abi-Khuzam.
 
 

A note on rings with constraints involving potent and noncentral elements

We give the structure of weakly periodic rings such that the potent elements form an ideal. Indeed, we prove that a weakly periodic ring with commuting nilpotents, such that the set of potent elements form an ideal of R, is a subdirect sum of finite fields and nil commutative rings. This result generalizes a well known result of Jacobson. We also prove that a ring having at most finitely many noncentral elements is either commutative or finite. This project is completed and accepted for publication. H. Abu-Khuzam.
 
 

A commutativity theorem for rings with conditions involving nilpotents and the Jacobson Radical

Let n be a fixed positive integer, and let R be n(n-1) -torsion-free ring with identity 1. Let N be the set of nilpotents, and J the Jacobson Radical of R. It is proved that if (xy)ⁿ = xⁿyⁿ for all x, y in R\(N È J), then R is commutative. This generalizes an earlier result of the author, and an earlier result of Bell and Yaqub. This project is completed and accepted for publication. H. Abu-Khuzam.
 
 

On the closed form of the covariance matrix and its inverse of a stationary ARMA process

Submitted to the Journal of Time Series Analysis, (1998). J.N. Haddad.
 
 

On robust estimation in the first order autoregressive processes

Submitted to Communications in Statistics (1999). J.N. Haddad.
 
 

Modeling chemical contents of the Litani river

The water quality of the Litani river has been investigated for the past three consecutive years. Samples were collected from six major sites situated along the initial stretch of the river. The sampling sites (stations) were selected mainly due to their easy access and secondly due to their proximity of major urban centers. The study has been focusing on a number of chemical contaminants such as nitrate, pH level, and salinity. It is evident from data that nitrate is present in all sites and follow a slow downward trend from site to site. On the other hand, the pH and salinity levels do not seem to be problem at any site. A technical report is in preparation to the Lebanese National Research Council for Scientific Research. J.N. Haddad and M.N. Nimah. (Supported by LNCSR.)
 
 

Order selection in moving average processes using the coefficient of determination

Submitted to the Journal of Royal Statistical Society (1998). J. N. Haddad, S.B. Provost*, and Y.H. Cheong*.
 
 

On an existence theory for the double series inverse

The variational double series solution to sampled Fredholm integral equations of the first kind, advanced previously by this author, belongs fundamentally in a certain nonlinear weighted Sobolev space. We report in this work on a measure-theoretic proof of the existence of this solution. Research completed and a paper will appear during 1999 in Applied Mathematics and Computation.N. Haidar.
 
 

Time-localized harmonic analysis of almost periodic processes

Variations in harmonic amplitudes of one-dimensional signals, which are locally almost periodic, are analyzed over finite length records by single Fourier series. Two algorithms, one differentiation-invoking and the other differentiation-free, are advanced for the design of a universal selective time-localized harmonic analyzer for such real signals. Research completed and a paper has been conditionally accepted for publication in the Japan Journal of Industrial and Applied Mathematics. N. Haidar.
 
 

Mathematics of temporally optimized neutron radiotherapy

Composite-surface finite integral transforms are applied to formulate the optimal ballistic property for a temporally tuned multi-beam neutron cancer 3D therapy as a single-valued dynamical system. By invoking Pontryagin’s maximum principle, with the operation functions of the beams constituting the control vector, it is proved that for every operation function of any one of the neutron beams, there exists a local optimal control vector satisfying an a priori system of homogeneous Volterra integral equations of the first kind. The possibility of designing a globally quazi-optimal tuned multi-beam 3D irradiation therapy is shown, in an inverse problem formulation, to result from a solution to a related semi-homogeneous dual system of integral equations of the first kind. A criterion for the controllability of this optimization problem has also been established. Research completed and was invited for publication in one applied mathematics journal. A paper has however been submitted to the Journal of Mathematical Biology. N. Haidar.
 
 

A rational fraction triplic form for the exponential

We derive a rational fraction triplic form for the exponential function on the real line. This form turns out to be a simpler equivalent to the vertical sequence of the related Pade’ table and with comparable summation features. Research completed and a paper has been submitted for publication. N. Haidar.
 
 

The global diffusion of the internet project

The objective of this project is to reach detailed framework of analysis describing the diffusion and absorption of the Internet in a country, to explain how the current capacity came into being, and how it is evolving giving particular attention to business and public policy issues with regard to the spread and control of the Internet in any country. Work in Progress. GITAG lead by S. Goodman*, G. Jalloul.
 
 

Concurrent object oriented systems: modeling and implementation

Coupling concurrency and object-orientation in a single framework faces many challenging issues. A key challenge is how to carry through the key advantage of reuse to concurrent environments. We have addressed this challenge by developing Communicating Sequential Systems (CSS) which is a model for concurrent object oriented programming and a corresponding implementation framework. Also a pattern language for designing concurrent object oriented programming languages was reached. G. Jalloul. (Supported by URB.)
 
 

A Meta language for patterns

The importance of patterns and pattern languages in promoting reuse and reducing development time and cost has been recognized and addressed in various domains by the object oriented community. However there is a gap between the manner in which patterns are described and between UML, which is by and large becoming the standard for expressing object oriented designs. This project aims at bridging the gap between patterns and UML. G. Jalloul. (Supported by URB.)
 
 

Training in object oriented software engineering

Training professionals as well as students in concepts of object orientation especially when combined with software engineering suffers of a gap between theory and practice, lack of well documented case studies as well approaches for an effective paradigm shift oriented. This project opts to find resolutions for these problems. Project has been on going for more than three years. G. Jalloul.
 
 

Object oriented software engineering methodologies for small projects

The intensive iterative nature of object oriented methodologies makes these methodologies inappropriate for developing small projects. This project opts to reach a methodology for small projects. A methodology has been reached and a paper is being drafted. Partially Completed. G. Jalloul.
 
 

Nonconforming finite element methods and the evolution Navier-Stokes equations

Nonconforming finite element methods have been proposed and analyzed for the stationary Navier-Stokes equations. These methods are based on a non-standard variational formulation of the equations, in which continuity conditions on the trial functions, which are required by standard formulations, are relaxed. Thus the velocity vector field is approximated using piecewise solenoidal functions that are totally discontinuous across inter-element boundaries, but which are point-wise divergence free on each element. These methods will be extended to the time-dependent Navier-Stokes equations. In addition, the computational aspects of the resulting numerical schemes are studied. In particular, it is investigated whether these numerical schemes can take advantage of any parallel features, due to the de-coupling of the elements across inter-element boundaries. W. Jureidini.
 
 

Frameworks abstracted from software applications

The study of frameworks explores the reuse of design in computer science. A framework is an abstract application described by a set of abstract classes and/or design patterns that constitute a well-defined set of services and provide a sound way in which instances of these classes cooperate. Design patterns are recurring design structures that can be used over and over again in describing software designs. Frameworks have gained their importance because they provide significant saving in the time and effort required for application development. In this project, frameworks for a variety of software application areas are investigated. The application areas considered are those in which the researchers have had practical experience, especially with design matters. The practicality of using some CASE tools in this respect is also addressed. W. Jureidini, G. Jalloul
 
 

Univalent harmonic mappings of ring domains onto puntured domains

The project studies the geometry and analysis of harmonic mappings of ring domains onto puntured domains. A. Lyzzaik.
 
 

Univalent harmonic mappiand a conjecture of J.C.C. Nitsche

The object of the work is to find a lower bound for the module of the image annulus of a univalent harmonic mapping of an annulus. The result obtained is the first substantial solution for a 39- year open problem that was originally raised in l962 by J.C.C. Nitsche. The problem originated from the study of minimal surfaces. Whether this bound is best possible is sharp remains an open problem. A. Lyzzaik.
 
 

On Vandermonde matrices

An alternate proof for the inverse of the Vandermonde matrix has been obtained, using the result from complex analysis that the sum of the residues in the extended complex plane of any rational function is zero. An explicit expression for the determinant of generalized Vandermonde matrices was obtained by considering the generalized Vandermonde matrix as a partition of a Vandermonde matrix. This has lead to an explicit expression for the inverse of generalized Vandermonde matrices. Manuscript is being prepared for submission to the American Mathematical Monthly. M.J. Monsour.
 
 

Representation of autoregressive processes

A general solution of the difference equation for an autoregressive process of any order was obtained. This has led to an explicit expression of the covariance matrix for the finite sample process, an expression for the inverse of the covariance matrix and also a representation of any p^th order process in terms of p artificial first order process. These results are currently being prepared for submission to a journal. M.J. Monsour.
 
 

Solid modeling by recursive subdivision surface

The incorporation of free-form surfaces in solid modeling systems is well established. Recursive subdivision provides attractive techniques in this domain. A solid can be defined by a configuration of vertices, edges and facets, where the facets need not be planar. A surface that bounds a solid can be generated from a configuration as the limit of a refinement procedure. The originality of the methods lies in the ability to define surfaces over arbitrary topology. In this project we consider the problem of curve interpolation by the limit surface. We developed methods for generating free-form curves from strip complexes and applied this theory to curve interpolation. The system is running on a Silicon Graphics machine. A. Nasri. (Supported by URB.)
 
 

Piecewise circular spline by recursive subdivision

We present a subdivision algorithm for generating a piecewise circular spline curve from a given control polygon. The algorithm can be easily adopted in a recursive subdivision surface scheme to generate surfaces with circular boundaries and parameterized by arc length, a property not yet available. As an application, a modified version of Doo-Sabin subdivision algorithm is outlined making it possible to blend a subdivision surface with other surfaces having circular boundaries such as cylinders. A paper was submitted for publication. A. Nasri, K. V. Overveld*, B. Wyvill.* (Partially supported by URB.)
 
 

Computing higher moments of solids enclosed by recursive subdivision surfaces for physics-based animation

Moments, such as volume, center of mass and inertia tensor, of a solid enclosed by a recursive subdivision surface can be approximated based on the closed-form representation of regular parts of the subdivision surface and a tight estimate of the local convex hull near extraordinary points. We describe algorithms for computing such moments for physics-based animation. A. Nasri. J. Peters*. (Supported by URB.)
 
 

Taxonomy of interpolation in recursive subdivision surfaces

The work describes and classifies the various situations, which any complete study of interpolation constraints for a recursive subdivision surface needs to consider. It does so in the form of a systematic taxonomy of situations, first of curve interpolation conditions, and then of surfaces. Known results are classified and open questions identified. The work is completed and a paper was submitted for publication. A. Nasri. M. Sabin*. (Supported by LNCSR and Royal Society, UK, and URB.)
 
 

Recursive subdivision in computer-Aided geometric design

The theory of recursive division has two roots: the divide and conquer algorithms, and the definition of surfaces over meshes of arbitrary topology. This book builds on both aspects, describing (with analysis) first the ways in which the recursive subdivision approach can be applied efficiently and robustly to many interrogations of curves and surfaces, not just intersections, and then the many different refinement constructions, which give limit curves and surfaces. A. Nasri, M. Sabin*. (Partially supported by LNCSR and Numerical Geometry Ltd., Cambridge UK.)
 
 

Nonisotropic strongly singular integral operators

We extend the results concerning the boundedness of strongly singular integral operators to the nonisotropic setting. We also describe a geometric property of the underlying space, which helps us show that our results are sharp. Research completed and paper submitted for publication. B. Shayya.
 
 

Nonhomogeneous strongly singular convolution operators

We study the boundedness properties of convolution operators when the singularity of the kernel and the cancellation it possesses have different types of homogeneity. In progress. B. Shayya.
 
 
 
 

ARTICLES, BOOKS AND REPORTS

 

Abi-Khuzam, F., The distribution and multiplicity of values of entire functions of small growth. Complex Variables, 1999.

Al-Jadir, L. and Leonard, M., Evolution de schema de bases de donnees a l’aide d’un systeme homogene. Technique et Science Informatiques, 18 (3), 275-296, 1999.

Haddad, J.N., A simple method for computing the covariance matrix and its inverse of a stationary auto-regressive process. Communications in Statistics: Simulation and Computation, 27, 617-623, (1998).

Haidar, N., The collocational double series inverse in quasi-linear regularizer form. Journal of Inverse and Ill-Posed Problems, 7(2), 127-144, 1999.

Haidar, N., A Green’s function approach to invertibility. Mathematical Japonica, 50(1), 10-18, 1999.

Haidar, N., A nonlinear method for a severely ill-posed generalized moment problem. Journal of Inverse and Ill-Posed Problems, 5 (4), 323-335, 1997.

Jalloul, G., Computing in Lebanon. Communications of the ACM, 42 (22), February 1999.

Jureidini, W.N. and Karakashian, O.A.*, A nonconforming finite element method for the stationary Navier-Stokes equations. SIAM Journal on Numerical Analysis, 35(1), 93-120, 1998.

Lyzzaik, A., Covering theorems for open continuous mappings having two valences between orientable surfaces. Pacific Journal of Mathematics, 178(20), 307-329, 1997.

———, Univalence criteria for harmonic mappings in multiply-connected domains. Journal of the London Mathematical Society, 58 (2), 163-171, 1998.

———, A note on the valence of harmonic maps. Journal of Mathematical Analysis an Applications, 218, 611-620,1998.

———, An existence theorem for open continuous functions having two valences in the unit disc. Complex Variables, Theory and Applications, 1-18, 1998.

———, Univalent harmonic mappings. Annales Uniwersytet Mariae Curie-Sklodowska, 3, June 1999.

Monsour, M.J. and Mikulski, P.W.*, On limiting distributions in explosive autoregressive processes. Statistics and Probability Letters, 37, 141-147, 1998.

Nasri, A., A 4-sided approach to curve interpolation by recursive subdivision surfaces. The Visual Computer, Springer Verlag, 14(7), 343-353, 1998.

Nasri, A. and Peters, J.*, Computing the volume of solids enclosed by subdivision surfaces. Computer Graphics Forum, 16(3), C89-C94, 1997.

Nikiel, J., Purisch, S.* and Treybig, L.B.*, Separable, zero-dimensional spaces which are continuous images of ordered compacta. Houston Journal of Mathematics, 24, 45-56, 1998.

Nikiel, J., Treybig, L.B.* and Tuncali, H.M.*, Local connectivity and maps onto non-metrizable arcs. International Journal of Mathematics and Mathematical Sciences, 20, 681-688, 1997.

Shamsuddin, A., Rings with Krull dimension one. Communications in Algebra, 26(7), 2147-2158, 1998.
 
 
 
 

Haidar, N., Temporally optimized neutron therapy: an inverse problem formulation (abstract). 6th SIAM Conference on Optimization, jointly with the 1999 SIAM Annual Meeting, Atlanta, Georgia, USA, 1999.

Haddad, J.N., Estimation of the first-order autoregressive process parameters by the median substitute method. Proceedings of the American Statistical Association, Business and Economic Statistics Section, 93-97, 1997.

Jalloul, G., Teaching concepts effectively: a practice oriented approach. Accepted to appear in the Proceedings of Educators’ Symposium, OOPSLA’ 99.

Lyzzaik, A., Recent results for open continuous mappings having two valences between orientable surfaces. Computational Methods and Function Theory 1997, Nicosia, Cyprus, 1997.

———, The module of the image annuli under univalent harmonic mappings and a conjecture of J.C.C. Nitsche. XII Conference on Analytic Functions, Lublin, Poland, 1998.

Nasri, A., Interpolation of open curves by recursive subdivision surfaces, 173-188, 1997. The Mathematics of Surfaces VII, Dundee, U.K., 1997.

———, Free-form curve generation by recursive subdivision of polygonal complexes. The 5th SIAM Conference on Geometric Design, Nashville, USA, 1997.

———, Computing moments of solids enclosed by subdivision surfaces. Symposium on Computational Science, American University of Beirut, Center for Advanced Mathematical Sciences, Beirut, Lebanon, 1998.

———, An algorithm for interpolating intersecting curves by recursive subdivision surfaces. Shape Modeling International’99, University of Aizu-Wakamatsu, Japan, 1999.

———, An algorithm for interpolating intersecting curves by recursive subdivision surfaces, 130-137, 1999. Proceedings of Shape Modeling International ’99, IEEE, University of Aizu, Japan.

Nasri, A. and Peters, J.*, Computing the volume of solids enclosed by subdivision surfaces. 16, Eurographics, 97, Budapest, Hungary, 1997.

Nikiel, J., Tuncali*, H.M. and Tymchatyn*, E.D., Spaces of inverse limits. 12th Summer Conference on General Topology and its Applications, North Bay, Ontario, Canada, 1997.
 
 
 
 

 
 
 
 

Dankar, F., Harmonic mappings in multiply connected domains (1997). A. Lyzzaik.