Computer Algebra in Scientific Computing 

Abstract: The vast area of Scientific Computing, which is concerned with the computer aided simulation of various processes in engineering, natural, economical, or social sciences, now enjoys rapid progress owing to the development of new efficient symbolic, numeric, and symbolic/numeric algorithms. There has already been for a long time a worldwide recognition of the fact that the mathematical term algorithm takes its origin from the Latin word algo ritmi, which is in turn a Latin transliteration of the Arab name "AI Khoresmi" of the Khoresmian mathematician Moukhammad Khoresmi, who lived in the Khoresm khanate during the years 780 - 850. The Khoresm khanate took sig nificant parts of the territories of present-day TUrkmenistan and Uzbekistan. Such towns of the Khoresm khanate as Bukhara and Marakanda (the present day Samarkand) were the centers of mathematical science and astronomy. The great Khoresmian mathematician M. Khoresmi introduced the Indian decimal positional system into everyday's life; this system is based on using the famil iar digits 1,2,3,4,5,6,7,8,9,0. M. Khoresmi had presented the arithmetic in the decimal positional calculus (prior to him, the Indian positional system was the subject only for jokes and witty disputes). Khoresmi's Book of Addition and Subtraction by Indian Method (Arithmetic) differs little from present-day arith metic. This book was translated into Latin in 1150; the last reprint was produced in Rome in 1957.

Abstract: We present an algorithm for solving polynomial equations, which uses generalized eigenvalues and eigenvectors of resultant matrices. We give special attention to the case of two bivariate polynomials and the Sylvester or Bezout resultant constructions. We propose a new method to treat multiple roots, detail its numerical aspects and describe experiments on tangential problems, which show the efficiency of the approach. An industrial application of the method is presented at the end of the paper. It consists in recovering cylinders from a large cloud of points and requires intensive resolution of polynomial equations.

Abstract: Dynamical systems arising from chemical reaction networks with mass action kinetics are the subject of chemical reaction network theory (CRNT). In particular, this theory provides statements about uniqueness, existence, and stability of positive steady states for all rate constants and initial conditions. In terms of the corresponding polynomial equations, the results guarantee uniqueness and existence of positive solutions for all positive parameters.

Abstract: In the present paper we consider the Navier---Stokes equations for the two-dimensional viscous incompressible fluid flows and apply to these equations our earlier designed general algorithmic approach to generation of finite-difference schemes In doing so, we complete first the Navier---Stokes equations to involution by computing their Janet basis and discretize this basis by its conversion into the integral conservation law form Then we again complete the obtained difference system to involution with eliminating the partial derivatives and extracting the minimal Grobner basis from the Janet basis The elements in the obtained difference Grobner basis that do not contain partial derivatives of the dependent variables compose a conservative difference scheme By exploiting arbitrariness in the numerical integration approximation we derive two finite-difference schemes that are similar to the classical scheme by Harlow and Welch Each of the two schemes is characterized by a 5×5 stencil on an orthogonal and uniform grid We also demonstrate how an inconsistent difference scheme with a 3×3 stencil is generated by an inappropriate numerical approximation of the underlying integrals