Catálogo de publicaciones - libros

Dynamics of a cosymmetric system of nonlinear parabolic equations is studied to model of population kinetics. Computer algebra system Maple is applied to perform some stages of analytical investigation and develop a finite-difference scheme which respects the cosymmetry property. We present different scenarios of evolution for coexisted nonstationary regimes and families of equilibria branched off of the state of rest.

We consider the integers using the language of ordered rings extended by ternary symbols for congruence and incongruence. On the logical side we extend first-order logic by bounded quantifiers. Within this framework we describe a weak quantifier elimination procedure for univariately nonlinear formulas. Weak quantifier elimination means that the results possibly contain bounded quantifiers. For fixed choices of parameters these bounded quantifiers can be expanded into finite disjunctions or conjunctions. In univariately nonlinear formulas all congruences and incongruences are linear and their modulus must not contain any quantified variable. All other atomic formulas are linear or contain only one quantified variable, which then may occur there with an arbitrary degree. Our methods are efficiently implemented and publicly available within the computer logic system redlog , which is part of reduce . Various application examples demonstrate the applicability of our new method and its implementation.

A common way of implementing multivariate polynomial multiplication and division is to represent polynomials as linked lists of terms sorted in a term ordering and to use repeated merging. This results in poor performance on large sparse polynomials. In this paper we use an auxiliary heap of pointers to reduce the number of monomial comparisons in the worst case while keeping the overall storage linear. We give two variations. In the first, the size of the heap is bounded by the number of terms in the quotient(s). In the second, which is new, the size is bounded by the number of terms in the divisor(s). We use dynamic arrays of terms rather than linked lists to reduce storage allocations and indirect memory references. We pack monomials in the array to reduce storage and to speed up monomial comparisons. We give a new packing for the graded reverse lexicographical ordering. We have implemented the heap algorithms in C with an interface to Maple. For comparison we have also implemented Yan’s “geobuckets” data structure. Our timings demonstrate that heaps of pointers are comparable in speed with geobuckets but use significantly less storage.

Ruppert and Sylvester matrices are very common for computing irreducible factors of bivariate polynomials and computing polynomial greatest common divisors, respectively. Since Ruppert matrix comes from Ruppert criterion for bivariate polynomial irreducibility testing and Sylvester matrix comes from the usual subresultant mapping, they are used for different purposes and their relations have not been focused yet. In this paper, we show some relations between Ruppert and Sylvester matrices as the usual subresultant mapping for computing (exact/approximate) polynomial GCDs, using Ruppert matrices.

We propose a new and efficient approach for solving the tasks of the microobjects recognition based on using the neural network (NN) and work out a computer system of image visualization, recognition, and classification of the microobjects on the samples of the pollen grains. The technology is developed for a preliminary processing of images of the microobjects on the basis of the “Snake” model. The principle of teaching of formal neuron and mathematical model of teaching multilayer perceptron for recognition of the microobjects is proposed. An algorithm is developed for teaching the NN of the returning distribution, subject domain, and methods of classes of computer system.

A rheological linear model for a certain generalized Bingham fluid with rheological memory, which flows in a movable tube is proposed and analytically solved. The model is a system of two linear and coupled partial differential equations with integral memory. We apply the Laplace transform method making the inverse transform by means of the Bromwich integral and the theorem of residues and the analytical solution are obtained using computer algebra. We deduce the explicit forms of the velocity and stress profiles for the generalized Bingham fluid in terms of Bessel and Struve functions. Various limit cases are obtained and the standard Hagen-Poiseuille and Buckingham-Reiner equations are recovered from more general equations. This works shows the powerful of Maple to solve complex rheological problems in an analytical form as it is presented here by the first time.

New elimination methods are applied to compute polynomial relations for the coefficients of the characteristic polynomial of certain families of matrices such as tensor squares.

We find a full system of invariants with respect to gauge transformations L → g ^− 1 L g for third-order hyperbolic linear partial differential operators on the plane. The operators are considered in a normalized form, in which they have the symbol Sym_ L  = ( pX  +  qY ) XY for some non-zero bivariate functions p and q . For this normalized form, explicit formulae are given. The paper generalizes a previous result for the special, but important, case p  =  q  = 1.

An efficient CAS helps the user to develop different Symbolic calculus problems, a clear example of this aid consist in the solution of the diffusion equation with and without memories, and its stability analysis working with Maple software package; the software gives the symbolic solution to this problem, but to do it, some basic definitions had to be implemented in the software, the stability analysis was not made automatically by the software, and when the problem was solved the necessity of an automatic solver were found.

We obtain new inequalities on the real roots of a univariate polynomial with real coefficients. Then we derive estimates for the largest positive root, which is a key step for real root isolation. We discuss the case of classic orthogonal polynomials. We also compute upper bounds for the roots of orthogonal polynomials using new inequalities derived from the differential equations satisfied by these polynomials. Our results are compared with those obtained by other methods.