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

RESEARCH PROJECTS

Value distribution for functions of small growth

For an entire function f with non-negative Maclaurin coefficients we investigate the relationship between the growth of Hayman’s function b(r) and the location of the real values of f. The following best possible result is obtained: If limsup b(r) ? 0.5 then, for each real w, all but a finite number of the roots of the equation f(z) = w are simple, real and negative. F. Abi-Khuzam. 

Note on the width-conjecture

The width-conjecture is a statement about the location of zeros of sections of power series having positive order l . In this work we show that l ought to be replaced by another well-defined index of growth in order to obtain a statement that is sharp and covers all cases including the case l =0. F. Abi-Khuzam. 

Compiling devices

We presented a framework elsewhere for characterizing and computing minimal diagnoses given a structured system description. Moreover, we presented an algorithm for computing minimal diagnoses when the system structure is a tree. In this paper, we pursue this structure-based framework in three important directions. First, we present an algorithm and pseudocode for computing minimal diagnoses when the system structure is a directed acyclic graph. Second, we analyze the computational complexity of the given algorithm, showing the effect of system structuand system observation on this complexity. Third, we provide a very simple extension to our algorithm which allows one to compile a given device into a parameterized propositional sentence. This compiled sentence can then be processed using a simple procedure to generate the minimal diagnoses of any system observation. A. Darwiche.

The effect of observations on the complexity of model-based diagnosis

This paper shows how to efficiently diagnose systems by making use of observations. In particular, we present two theorems concerning the effect of observations on the complexity of Model-Based Diagnosis. The first theorem shows how the presence of certain observations allows us to decompose a diagnostic reasoning task into independent reasoning tasks on subsystems. The second theorem shows how the absence of certain observations allows us to ignore parts of a system during diagnostic reasoning. Another main contribution of this paper is an application of these theorems to diagnosing discrete-event systems. In particular, we identify observability and modularity characteristics of discrete-event systems that make them amenable to the presented theorems and, hence, to any diagnostic approach that employs these theorems effectively. A. Darwiche and G. Provan*. 

Standard approach for optimizing belief network inference using Query DAGs

This paper proposes a novel, algorithm-independent approach to optimizing belief network inference. Rather than designing optimizations on an algorithm by algorithm basis, we argue that one should use an unoptimized algorithm to generate a Q-DAG, a compiled graphical representation of the belief network, and then optimize the Q-DAG and its evaluator instead. We present a set of Q-DAG optimizations that supplant optimizations designed for traditional inference algorithms, including zero compression, network pruning and caching. We show that our Q-DAG optimizations require time linear in the Q-DAG size, and significantly simplify the process of designing algorithms for optimizing belief network inference. A. Darwiche and G. Provan*. (Supported by URB.) 

G1 surface interpolation over irregular network

The geometric continuity between two adjacent triangular Bezier patches can be controlled by manipulating their control points. In this project, we study the use of these patches in generating G1 surfaces through a predefined irregular mesh of data points. The algorithm is under implementation. G. Farin*, A. Nasri and K. Tae-wan*. 

Modeling chemical contents of the Litani River

The quality of the Litani River is to be investigated over equally spaced intervals in time and space (distance). Its physical, chemical, and biological composition will be considered. In particular, PH level, salinity level, turbidity average time, and nitrogen concentration level will be measured and monitored over a period of one year as an initial phase of the study. Samples will be collected at the rate of twice per month from different equidistant locations along the river. The data produced will be analyzed in a time series farlion. The target of this research project is to set up a statistical model for the observed data for future prediction and contamination control. J. Haddad. (Supported by LNCSR.) 

Three papers

Estimation of the first order autoregressive process parameters by the Median substitute method; submitted to the Australian Journal of Statistics. A simple method for computing the covariance matrix and its inverse of a stationary autoregressive process; submitted to Communications in Statistics. On the derivation of the theoretical autocovariance function of the autoregressive moving average process; submitted to Statistics and Probability Letters. J. Haddad.

Severely ill-posed moment problems

We propose an inverse problem formulation for the generalized moment problem

with a “one-dimensional” kernel K and a parametrized solution

By assuming that there exists a linearly independent sampled solution , and utilizing the collocational double series inverse in a multiplexed form, we derive for it a regularizing system of nonlinear integrodifferential equations. The solution of this second-order differential system turns out to be a direct deconvolution of the considered severely ill-posed moment problem. Research completed and a paper will appear during summer 1997 in the Journal of Inverse and Ill-Posed Problems. N. Haidar.

Mathematics of cancer therapy with neutrons

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 unique 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 to result from a solution to a related semi-homogeneous dual system of integral equations of the first kind. The controllability of this optimization problem has also been established. Research completed and a paper has been submitted for publication. N. Haidar.

Object-oriented concurrent programming

The work focused on integrating concurrency into object-oriented programming. It resulted in reaching and realizing an abstract model for concurrent object-oriented programming that supports novel resolutions for conflicts arising as a result of coupling the two paradigms. The work described has been completed and is being currently drafted as a book that will act as a reference for people working in this area. This direction has been taken as a result of unsolicited strong recommendations given by referees of the work. G. Jalloul.

A framework for concurrent object-oriented programming

This project focuses on developing a framework that can be reused to implement concurrent language extensions for object-oriented programming languages. A framework is an abstract application which can be reused to reach a variety of applications that belong to the domain of the framework by effecting the abstractions composing the framework. The advantage of reaching the framework under investigation is easing the task of the design and implementation of concurrent object-oriented extensions. G. Jalloul.

An object-oriented framework for generating GUI interfaces

This work focuses on developing a framework in Eiffel that can be reused to build GUI interfaces for software applications. The resulting framework should allow users to generate OO GUI interfaces for their applications and to link the resulting interface with commands in the application. The advantage of this framework is easing the development task so that developers can focus on aspects specific to the application under development. G. Jalloul.

Reusable control mechanisms for concurrent object-oriented programming

Investigatation of control mechanisms in the area of concurrent programming. It opts to reach mechanisms that can be reused compatibly with inheritance. Two mechanisms have been identified. The applicability of these mechanisms for Eiffel, C++ and Java is currently being investigated. G. Jalloul.

Three papers

Univalence criteria for harmonic mappings in multiply connected domains; accepted for publication in Journal of the London Mathematical Society. An existence theorem for open continuous mappings having two valences in the unit disc; submitted to Complex Variables, Theory and Applications. A note on the valency of harmonic mapping; submitted to Journal of Mathematical Analysis and Applications. A. Lyzzaik. 

Computer visualization techniques in atom orbitals education

Atomic orbitals require a lot of imagination in conceiving the spatial arrangement and orientation of electron density regions. This source of difficulty can be dissolved using computer visualization techniques where students can sethe unseen. We have described several techniques to visualize such an abstract concept, offering a range of choices for educators to implement a visualization system suitable to their taste or needs and using the available hardware and software. A. El-Maiss, A. Nasri and R. Sultan. 

Basic groups of Lie algebras and Hopf algebras

Let H(L) be the Hopf algebra of a Lie algebra L and let B(L) be the Hochschild basic group of L.

We extend a result of M. Moskowitz which determines the Lie algebras whose (all) associated analytic groups have faithful representations. So we find a similar criterion for a single analytic group to have a faithful representation. This work will appear in the Proceedings of the American Mathematical Society. N. Nahlus. 

Solid modeling by recursive subdivision

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.) 

Interpolation conditions on recursive subdivision surfaces — a taxonomy

We describe and classify 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. Although the context of 3D Euclidean geometry is used, the mapping to projective geometry modeled by homogenous coordinates is straightforward A. Nasri and M. Sabin*. (Supported in part by the Royal Society; LNCSR.)

Computing volumes and higher order moments of solids enclosed by recursive subdivision surfaces

One of the noted drawbacks of subdivision surfaces for quantitative applications is their lack of a finite closed-form representation. This makes it difficult to generate objects that hold a given volume and do not topple over because the center of mass is outside the support. We have developed an algorithm for measuring volumes of solids enclosed by subdivision surfaces. This is to be extended to higher order moments such as center of mass and the inertia tensor. A. Nasri and J. Peter*. (Supported by URB; National Science Foundation.) 

A rational subdivision algorithm for curve generation

Chaikin developed an algorithm for generating curves by cutting off the corners of an initial control polygon. One drawback of this algorithm is that it does not generate circles, neither its rational form. In this project, we developed a rational modified version of Chaikin's algorithm that generates piecewise ellipses. When the legs of the control polygons are equals, the conics are circles and are parameterized by arc length. A. Nasri and G. Farin*. 

Local connectivity and maps onto non-metrizable arcs

A paper has been accepted in the International Journal of Mathematics and Mathematical Sciences. J. Nikiel, L.B. Treybig* and H.M. Tuncali*. 

Separable zero-dimensional spaces which are continuous images of ordered compacta

A paper has been submitted. J. Nikiel, S. Purich* and L.B. Treybig*. 

Spaces of inverse limit spaces

A paper is being written. J. Nikiel, H.M. Tuncali* and E.D. Tymchatyn*. 

Convex metrics on locally connected spaces

Research in progress. J. Nikiel, H.M. Tuncali* and E.D. Tymchatyn*. 

Nonisotropic strongly singular integral operators

We consider a class of convolution operators whose kernels are strongly singular at the origin, and extend the results concerning the L^p boundedness of these operators to the nonisotropic setting. We also describe a geometric property of the underlying space which helps us show that our results are sharp. B. Shayya.
 
 

ARTICLES, BOOKS AND REPORTS
 

Abu-Khuzam, H., Hasanali, M. and Yaqub, A., Weakly periodic rings with conditions on commutators. Acta Mathematica Hungarica, 71, 145-153, 1996. 

Darwiche, A. and Pearl, J.*, On the logic of iterated belief revision. Artificial Intelligence, 89 (1-2), 1-29, 1997. 

Darwiche, A. and Provan*, G., Query-DAGs: A practical paradigm for implementing belief network inference. Journal of Artificial Intelligence Research, 6, 147-176, 1997. 

Haddad, J., The recursive property of the inverse of the covariance matrix of a moving-average process of general order. Journal of Time Series Analysis, 16 (6), 551-554, 1995. 

Haidar, N., On sampled fourier sums. Digital Signal Processing, 6 (3), 179-184, 1996. 

———, Recursive pseudo-inversion of the Laplace transform on the real line. Applied Mathematics and Computation, 84 (2-3), 213-220, 1997. 

———, On a boundary value problem posed by cancer therapy with neutron beams. Journal of the Australian Mathematical Society, Ser. B, 39 (1), 47-61, 1997. 

Herbera, D. and Shamsuddin, A., Modules with semi-local endomorphism ring. Proceedings of the American Mathematical Society, 123 (12), 3593-3600, 1995. 

———, On self injective perfect rings. Canadian Mathematical Bulletin, 39 (1), 55-58, 1996. 

Lyzzaik, A., Covering properties of open continuous mappings having two valences between Riemann surfaces, Mathematical Proceedings of the Cambridge Philosophical Society, 115, 321-340, 1995. 

———, The geometry of open continuous mappings having two valences between Riemann surfaces. Mathematical Proceedings of the Cambridge Philosophical Society, 120, 309-329, 1996. 

———, Covering theorems for open continuous mappings having two valences between orientable surfaces. Pacific Journal of Math, 3 (1), 113-135, 1997. 

Nahlus, N., Homomorphisms of Lie algebras of algebraic groups and analytic groups. Canadian Mathematical Bulletin, 38 (3), 352-359, 1995. 

Nasri, A., Developing a hypermedia system for computer systems education. Computer Science Education, 6 (1), 33-47, 1995. 

———, Curve interpolation in recursively generated B-spline surfaces over arbitrary topology. Computer Aided Geometric Design, 14 (1), 13-30, 1997. 

Nikiel, J., Dendrons, dendritic spaces and uniquely arcwise connected spaces. Proceedings of the New York Academy of Sciences, 788, 170-174, 1996. 

Nikiel, J. and Treybig*, L.B., Null-families of subsets of monotonically normal compacta. Colloquium Mathematicum, 70, 87-92, 1996. 

Shamsuddin, A., Homological properties of SF rings. Bulletin of the Australian Mathematical Society, 55, 327-333, 1997. 

Umphrey*, G.J. and Haddad, J., Maximum likelihood estimation of Allele frequencies in the Eusocial hymenoptera using offspring genotypes. Biometrics, 52, 1026-1032, 1996.
 
 
 

ABSTRACTS, PRESENTATIONS AND PROCEEDINGS
 

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

Nasri, A., A recursive subdivision surface through an irregular mesh of curves. The Fourth SIAM Conference on Geometric Design, Nashville, USA, 1995. 

———, Recursive subdivision in simulation and scientific visualization. First LAAS International Conference on Computers and Simulation, 337-334, Beirut, Lebanon, 1995. 

———, Interpolation of Open B-spline curves by recursive subdivision surfaces. The Seventh Conference of the Institute of Math and its Applications on the Mathematics of Surfaces, Dundee, Scotland, UK, 1996.
 
 
 

MASTER’S THESES
 

El-Dana, L., Normal spaces (1995). Advisor: J. Nikiel. 

Eido, F., Different approaches to Noether-Lasker decomposition theorem (1995). Advisor: A. Shamsuddin. 

Duhaini, H., On first kind Fredholm integral equations (1996). Advisor: N. Haidar. 

Hunayni, L., An operational study of monomials over [0,1] (1995). Advisor: N. Haidar. 

Mahfoud, Z., Analytic coverings and some major applications in complex function theory (1996). Advisor: A. Lyzzaik. 

Mouzeihem, Z., On self-injective rings (1996). Advisor: A. Shamsuddin.
 
 
 
 
 

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