Catálogo de publicaciones - libros

E.A.Grebenikov and A.N.Prokopenya proved that rhombus-like central configuration in Newton 5-body problem is unstable. In this article, the problem of existence and stability of the rhombus-like central configurations in Newton 9-body problem, which consists of two homothetic rhombuses, is studied. It is proved that these central configurations are unstable. All computations are executed by means of computer algebra system Mathematica.

Let be a group given by generators and relations. It is possible to compute a presentation matrix of a module over a ring through Fox’s differential calculus. We show how to use Gröbner bases as an algorithmic tool to compare the chains of elementary ideals defined by the matrix. We apply this technique to classical examples of groups and to compute the elementary ideals of Alexander matrix of knots up to 11 crossings with the same Alexander polynomial.

Sudoku is a logic-based placement puzzle. We recall how to translate this puzzle into a 9-colouring problem which is equivalent to a (big) algebraic system of polynomial equations. We study how far Gröbner bases techniques can be used to treat these systems produced by Sudokus. This general purpose tool can not be considered as a good solver, but we show that it can be useful to provide information on systems that are —in spite of their origin— hard to solve.

In this paper, we deal with the problem of the computation of the homology of a finite simplicial complex after an “elementary simplicial perturbation” process such as the inclusion or elimination of a maximal simplex or an edge contraction. To this aim we compute an algebraic topological model that is a special chain homotopy equivalence connecting the simplicial complex with its homology (working with a field as the ground ring).

The computer algebra methods are effective means for the search of approximate and exact solutions of differential equations of theoretical physics, celestial mechanics, astrodynamics, and other natural sciences. Before appearance of Programming Systems such as etc., we knew for classical Newtonian three-body problem only Euler exact collinear and Lagrange triangular solutions, for many-body problem – the rotating regular tetragon solution found by A. Dziobek [1] and the general homographic solution theory developed by A. Winter [2] in the 30es of the 20th century. An amount of similar research [3,4,5,6,7,8,9,10] has grown recently due to the fact that the existence of central configurations of the many-body problem is eventually reduced to the solution of the systems of nonlinear algebraic-irrational equations, which can be solved only by the computer algebra methods, thanks to exceptional properties of them.

The planar central configurations in the newtonian problem of four bodies are studied with the computer algebra system . We have shown that in the case of two equal masses there can exist central configurations in the form of isosceles triangle with three bodies being in its vertices and the fourth body being situated in the axis of symmetry inside or outside the triangle. The number of possible configurations in such cases depends on the masses of the bodies and may be equal to ten, six or two. We have provided evidence numerically that there exist one-parametric family of central configurations in the form of antiparallelogram. We have shown also that central configuration may be deformed continuously by means of changing masses of the bodies and found two-parametric family of central configurations in the neighborhood of the square.

The boundary-value problem in spherical coordinates for the Shrödinger equation describing a hydrogen-like atom in a strong magnetic field is reduced to the problem for a set of radial equations in the framework of the Kantorovich method. The effective potentials of these equations are given by integrals over the angular variable between the oblate angular spheroidal functions depending on the radial variable as a parameter and their derivatives with respect to the parameter. A symbolic-numerical algorithm for evaluating the oblate spheroidal functions and corresponding eigenvalues which depend on the parameter, their derivatives with respect to the parameter and matrix elements is presented. The efficiency and accuracy of the algorithm and of the numerical scheme derived are confirmed by computations of eigenenergies and eigenfunctions for the low-excited states of a hydrogen atom in the uniform magnetic field.

In this paper we present algorithmic considerations and theoretical results about the relation between the orders of certain groups associated to the components of a polynomial and the order of the group that corresponds to the polynomial, proving it for arbitrary tame polynomials, and considering the case of rational functions.

In this paper we obtain an analog of Schultz decomposition for arbitrary convex smooth functions. We prove existence of adapted coordinate systems for analytic convex functions. We show that the oscillation index of oscillatory integrals with analytic phases is defined by the distance between Newton polyhedron constructed in adapted coordinate systems and the origin.

The cellular automata with local permutation invariance are considered. We show that in the two-state case the set of such automata coincides with the generalized Game of Life family. We count the number of equivalence classes of the rules under consideration with respect to permutations of states. This reduced number of rules can be efficiently generated in many practical cases by our C program. Since a cellular automaton is a combination of a local rule and a lattice, we consider also maximally symmetric two-dimensional lattices. In addition, we present the results of compatibility analysis of several rules from the Life family.