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My interests are graph theory and its applications. One major direction deals with coverings and decompositions. In particular, decompositions of graphs into paths, cycles and trees, covering weighted graphs with cycles, 1-factorizations and general isomorphic factorizations are being studied. Such factorizations can be and have been utilized in solving scheduling problems. Another major direction deals with the existence of Hamilton cycles and paths in vertex-trasitive graphs. Such problems are of interest in interconnection networks and parallel architectures. Another area of interest is the interaction of graph theory and design theory. Particular instances of this are the study of orthogonal factorizations of graphs and the study of matching designs. Cayley graphs is another major topic in which I am interested. They have become an extremely active area of research because of their interest in network probelms. Circulant graphs and digraphs is a particular family of Cayley graphs in which I am interested. Circulant graphs have interesting interactions with group theory because of isomorphism questions. Circulant graphs also appear in many graph-theoretic settings.
Most of my work centers on the study of the mathematical sciences in ancient Greece and medieval Islam, and I am presently publishing, with James Evans at the University of Puget Sound, a translation of a Greek treatise titled Introduction to the Phenomena. The 'phenomena' refers to things seen in the heavens concerning the Sun, Moon, stars and planets, and the author of the book,Geminos, intended it as an introduction to astronomy as it was known in the first century B.C., written for the educated Greek reader. I am also engaged in the translation and study of the geometrical works of the 10th century Persian scientist, Abu Sahl al-Kuhi, whose works continue the Greek geometric heritage of rigorous solution of difficult problems of the sort found in Archimedes and Apollonius. Among his major works are studies of centres of gravity, the geometry of astrolabes, and the construction of a special kind of geometrical compass for drawing conic sections.
My research interest is focused on control strategies applied to population dynamics and robot manipulators. It is based on ordinary differential equations; phase-space study, Liapunov stability, bifurcation, control, and numerical simulation. In population dymamics ecological and bioeconomical systems are subjected to changes due to various causes which leads to undesirable large fluctuation of the size of the populations (consumers, resources). This may trigger in general two types of responses: 1) the populations may change their behavior abruptly in order to damper the fluctuations; 2) human influence from outside the system may be imposed to restrict the fluctuations of the populations. The control objective is finding behavioral policies that result in dampening large fluctuations of populations in bioeconomical systems. Robot manipulators can be modeled as kinetic chains of connected material links. The manipulator has to reach and capture targets in specified work space coordinating its links and avoiding collisions with stationary or moving objects. This requires the establishment of coordination strategies based on adaptive control laws.
I work in a subfield of combinatorics called "Ramsey Theory." I am mostly interested in questions related to "van der Waerden's theorem on arithmetic progressions," which states that "for each k, there exists a (smallest) n = n(k) such that whenever [1, 2, ... , n] is partitioned into two parts, in any way whatsoever, then at least one of these parts must contain a k-term arithmetic progression." (It is known that n(3)=9, n(4)=35, n(5)=178.) For example, one can look for upper and lower bounds for the function n(k) and related functions (allowing partitions into more than two parts, for example). Or, one can replace the set of all k-term arithmetic progressions by a larger or smaller or different set. Or, one can restrict the class of allowable partitions.
Our main research interest is in the mathematical theory of general relativity. We are currently investigating the spherically symmetric collapse of an anisotropic fluid (one possible model of a star) into a black hole. In the solution, we would like to satisfy some of the energy conditions (weak, dominant, or strong). We would like to obtain an analytic equation for the collapsing boundary, and satisfy one of three jump conditions (Synge, Synge-O'Brien, or Darmoir) across the surface of discontinuity. We wish to match the interior solution to an exterior Schwarzschild (or Kruskal) or to a radiating Vaidya metric. Our goal is to obtain a reasonable, mathematically rigorous model for which the collapse of a spherically symmetric star into the final singularity can be pursued analytically in a doubly-null coordinate chart.
We have another research interest in the mathematical foundation of quantum theory of interacting fields. The present formulation is plagued by severe divergence difficulties. In 1960, we eliminated divergence problems by using partial difference equations (which presuppose a fundamental length). Unfortunately, the theory was not Lorentz-covariant. Recently, we have been formulating relativistic lattice wave field equations (in terms of partial difference equations) in order to obtain divergence-free S-matrix expansions.
Malgorzata Dubiel is dedicated to teaching mathematics at all levels, to all ages, in all kinds of settings, from classroom to market place. An educational leader and a 3M National Teaching Fellow, Malgorzata Dubiel’s outreach work has transformed the teaching of mathematics and math readiness throughout the entire province.
A variety of algebras is a class of algebras closed under products, homomorphic images, and subalgebras. Equivalently, a variety is a class of algebras defined by some family of identities or equations (for example the variety of abelian groups is the class of all groups satisfying the familiar commutative identity xy = yx). For any given type of algebra (group, semigroup, etc.) the varieties of algebras of that type constitute a lattice with respect to the usual order relation, where one says that U ≤ V if and only if every algebra in U also belongs to V. One of the main thrusts of my research is to study this lattice for various classes of algebras, such as lattice ordered groups, semigroups, regular semigroups, inverse semigroups, etc. One question that arises in relation to varieties is whether or not two expressions or words will always have the same values in a particular variety; for example, in the variety of commutative groups, the words xyx andxxy always have the same value. This problem is referred to as the word problem for the variety and, in general, is quite difficult. In order to study this problem, it is often useful to know something about the structure of the free object in the variety. The free object in a variety over a set X is an algebra in the variety generated by X and such that every mapping of X into any other algebra in the variety can be extended to a homomorphism of the free object into that algebra. Free objects have many interesting properties. A considerable part of my research programme is devoted to the study of word problems and the structure of free objects.
My main focus is on the numerical solution of differential equations. A major focus has been the investigation of adaptive (moving mesh) methods for solving partial differential equations (PDEs). These methods are essential to numerically solve most complex physical problems, where both high speed parallel processors and sophisticated computer graphics play key roles as well. Recent emphasis has been on the study of adaptive methods for PDEs in higher spatial dimensions and the development of corresponding mathematical software. Another interest is in dynamical systems (where qualitative studies of solutions to differential equations is the main concern). This area of research, applicable many problems in science and engineers, relates to the study of highly nonlinear physical processes and necessitates numerical investigations.
My research interest lies in the development of efficient algorithms to perform large-scale computations in electromagnetic wave propagation, with emphasis on applications to scattering, radar imaging, and microwave heating in radar absorbing materials or biological tissues. The issue of numerical convergence and error estimation is also investigated. This numerical work is often complemented by the necessary theoretical analysis on electromagnetic wave phenomena, which typically involves potential theoretical techniques, integral equations, partial differential equations, and functional analysis.
My most recent research involves the development of efficient algorithms to extract high resolution radar images from segmented and incomplete data. Computer codes based on these algorithms have been tested on supercomputers to yield good results. These techniques, although developed primarily for radar imaging, can also be adapted to use in medical imaging.
My general interest is in the area of classical real analysis. The best introduction to this field and to its many current practitioners can be obtained by perusing the Real Analysis Exchange. For the last dozen or so years this journal has played the role that the Fundamenta Mathematica did in the period between the wars when real analysis was much in vogue.
More specifically and narrowly my current research program is the study of the properties of real functions in terms of symmetry properties. This leads to a variety of problems in generalized derivatives (symmetric derivatives, approximate symmetric derivatives, higher order Riemann derivatives) and various notions of continuity. A host of symmetric integrals arise in this way too. These ideas have intimate connections with many problems in trigonometric series.