Preface

At present, the international computer algebra community is using several dozens of both general-purpose and problem oriented computer algebra systems (CASs). But there was the time, in the late fifties and early sixties of the 20th century, when the concept of a computer algebra system was still vague. It has been the Member of the USSR Academy of Sciences and Professor Victor M. Glushkov, director of the Institute of Cybernetics of the Academy of Sciences of Ukraine, Kyiv, who was one of the first mathematicians to recognize the practical feasibility of computer algebra systems. Furthermore, in the early sixties of the twentieth century, i.e., long before the appearance of the first personal computer, he already developed the new principles for the design of structures of small desktop computers for engineering computations. One of the general ideas of Prof. V.M. Glushkov was to couple the electronic design and architecture of computers with the algebraic tasks to be solved. For this purpose, Prof. V.M. Glushkov developed the concept of structural programming which combines the technological devices oriented towards the development of large algorithms and computer codes and the so-called algorithmic algebras (the term proposed by V.M. Glushkov) as the theoretical underpinning of the structured algorithms and programs.

The interactive software system OPTIMA developed during the sixties, for both analytic computations and a multi-stage verification of newly developed large computer programs, was one of the outcomes of these investigations and developments. Thus, one of the first computer algebra systems in the world has been developed in the Ukraine. In recognition of this fact, one of the International Symposia on Symbolic and Algebraic Computation (ISSAC'93) was held in Kyiv, the capital of Ukraine.

Therefore, our wish to organize CASC'2002, the fifth conference devoted to theoretical and applied aspects of using CASs in Scientific Computing, in Yalta, Ukraine, is by no means incidental. The four earlier conferences in this sequence, CASC'98, CASC'99, CASC'2000, and CASC'2001, took place in St. Petersburg, Russia, in Munich, Germany, in Samarkand, Uzbekistan, and in Konstanz, Germany, and they all proved to be very successful.

We have to thank the program committee, listed overleaf, for a tremendous job in soliciting and providing reviews for the submitted papers. There were more than three reviews per submission on average. The result of this job is reflected in the present volume, which contains accordingly revised versions of the accepted papers. The collection of papers included in the proceedings covers various topics of computer algebra methods, algorithms and software applied to scientific computing.

In particular, several papers deal with the methods for the solution of ordinary differential equations (ODEs) such as the methods using a piecewise linear vector field, algorithms for computing the Liouvillian solution of linear ODEs, approximate solution of linear ODEs with polynomial coefficients, the use of the Newton polyhedra for investigation of complex bifurcations of periodic solutions in certain systems of ODEs, or the asymptotic solution of higher-order boundary value problems.

Also, quite a number of papers is devoted to quantifier elimination.

A traditional theme of the CASC conferences, namely Gröbner bases, is covered by contributions on the construction of differential Gröbner bases, or on non-commutative Gröbner bases in Poincaré-Birkhoff-Witt extensions. A novel theme in this area, which was not considered in the earlier CASC conferences, is the parallelization of the computation of involutive Gröbner bases.

A traditional theme of the CASC conferences, namely Gröbner bases, is covered by contributions on the construction of differential Gröbner bases, or on non-commutative Gröbner bases in Poincaré-Birkhoff-Witt extensions. A novel theme in this area, which was not considered in the earlier CASC conferences, is the parallelization of the computation of involutive Gröbner bases.

The computation of cohomology is discussed in two papers. They describe, in particular, incremental algorithms and algorithms for the computation of the cohomology of Lie algebras of Hamiltonian vector fields.

Several papers deal with sparse matrix calculations, differential ideals, moduli spaces of low dimension, and Puiseux expansions by Hensel's lemma.

As in the foregoing CASC conference, an enhanced emphasis has been put on engineering applications of computer algebra. In particular, applied problems are considered, like the motion of an artificial satellite with a gravitation stabilizer, algorithms for non-linear signal processing, problems in atomic and laser physics, the motion of a particle with variable mass in the field of a massive geoid, a new strategy for the incremental solution of inviscid fluid dynamics problems, tools for the structured representation of physical objects with the aid of the Mathematica CAS, and so on.

The invited lecture by A. Weber shows in detail how the questions of stability of polynomial vector fields can be reduced to quantifier elimination problems on real closed fields. For the common case of equilibrium points with nonzero Jacobian determinant, it is shown that there is a computationally well suited description that can serve as an infrastructure for more efficient methods.

The CASC'2002 workshop is supported financially by a generous grant from the Deutsche Forschungsgemeinschaft (DFG). We are grateful to W. Meixner and M. Mnuk for their technical help in the preparation of the camera ready manuscript for this volume, and in organizing the meeting.

Our particular thanks are due to the members of the local organizing committee at the Institute for Mathematics of the National Academy of Sciences of Ukraine, who have ably handled local arrangements in this particularly pleasant location in the southern part of the Crimea peninsula, Ukraine, about 15 km to the south of Yalta (Simeiz).