Catálogo de publicaciones - libros

We produce a parallel algorithm realizing the Laplace transform method for symbolic solution of differential equations. In this paper we consider systems of ordinary linear differential equations with constant coefficients, nonzero initial conditions, and the right-hand sides reduced to the sums of exponents with the polynomial coefficients.

This work considers the basic issues of the theory of involutive divisions, namely, the property of constructivity which assures the existence of minimal involutive basis. The work deals with class of ≻ -divisions which possess many good properties of Janet division and can be considered as its analogs for orderings different from the lexicographic one. Various criteria of constructivity and non-constructivity are given in the paper for these divisions in terms of admissible monomial orderings ≻ . It is proven that Janet division has the advantage in the minimal involutive basis size of the class of ≻ -divisions for which ≻ ≻ ... ≻ holds. Also examples of new involutive divisions which can be better than Janet division in minimal involutive basis size for some ideals are given.

In this note we present a symbolic-numeric method to the problem of tube modeling in CAD systems. Our approach is based on the Kirchhoff kinetic analogy which allows us to find analytic solutions to the static Kirchhoff equations for rods under given boundary conditions.

In this paper we propose two methods for the computation of upper bounds of the real roots of univariate polynomials with real coefficients. Our results apply to polynomials having at least one negative coefficient. The upper bounds of the real roots are expressed as functions of the first positive coefficients and of the two largest absolute values of the negative ones.

We address various aspects of our computer algebra-based computer logic system . There are numerous examples in the literature for successful applications of to practical problems. This includes work by the group around the developers as well as by many others. is, however, not at all restricted to the real numbers but comprises a variety of other domains. We particularly point at the immense potential of quantifier elimination techniques for the integers. We also address another new domain, which is queues over arbitrary basic domains. Both have most promising applications in practical computer science, viz. automatic loop parallelization and software security.

We wish to work with polynomials where the exponents are not known in advance, such as – 1. There are various operations we will want to be able to do, such as squaring the value to get  − 2+1, or differentiating it to get 2. Expressions of this sort arise frequently in practice, for example in the analysis of algorithms, and it is very difficult to work with them effectively in current computer algebra systems.

We consider the case where multivariate polynomials can have exponents which are themselves integer-valued multivariate polynomials, and we present algorithms to compute their GCD and factorization. The algorithms fall into two families: algebraic extension methods and interpolation methods. The first family of algorithms uses the algebraic independence of , etc, to solve related problems with more indeterminates. Some subtlety is needed to avoid problems with fixed divisors of the exponent polynomials. The second family of algorithms uses evaluation and interpolation of the exponent polynomials. While these methods can run into unlucky evaluation points, in many cases they can be more appealing. Additionally, we also treat the case of symbolic exponents on rational coefficients (e.g. ) and show how to avoid integer factorization.

The current primality test in use for Mersenne primes continues to be the Lucas-Lehmer test, invented by Lucas in 1876 and proved by Lehmer in 1935. In this paper, a practical approach to an elliptic curve test of Gross for Mersenne primes, is discussed and analyzed. The most important advantage of the test is that, unlike the Lucas-Lehmer test which requires arithmetic operations and bit operations in order to determine whether or not =2–1 is prime, it only needs arithmetic operations and bit operations, with ≪. Hence it is more efficient than the Lucas-Lehmer test, but is still as simple, elegant and practical.