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Let Λ be a commutative ring, an augmented differential graded algebra over Λ (briefly, DGA-algebra) and be a relatively free resolution of Λ over . The standard bar resolution of Λ over , denoted by (), provides an example of a resolution of this kind. The comparison theorem gives inductive formulae : ()→ and : →() termed comparison maps. In case that =1 and is connected, we show that is endowed a -tensor product structure. In case that is in addition commutative then (,) is shown to be a commutative DGA-algebra with the product =*(⊗) (* is the shuffle product in ()). Furthermore, and are algebra maps. We give an example in order to illustrate the main results of this paper.

A general scheme of a symbolic-numeric approach for solving the eigenvalue problem for the one-dimensional Shrödinger equation is presented. The corresponding algorithm of the developed program EWA using a conventional pseudocode is described too. With the help of this program the energy spectra and the wave functions for some Schrödinger operators such as quartic, sextic, octic anharmonic oscillators including the quartic oscillator with double well are calculated.

Homological Perturbation Theory [11,13] is a well-known general method for computing homology, but its main algorithm, the , presents, in general, high computational costs. In this paper, we propose a general strategy in order to reduce the complexity in some important formulas (those following a specific pattern) obtained by this algorithm. Then, we show two examples of application of this methodology.

Relation algebra is well suited for dealing with many problems on ordered sets. Introducing lattices via order relations, this suggests to apply it and tools for its mechanization for lattice-theoretical problems, too. We combine relation algebra and the specific purpose Computer Algebra system to solve some algorithmic problems.

In Gröbner bases computation, as in other algorithms in commutative algebra, a general open question is how to guide the calculations coping with numerical coefficients and/or not exact input data. It often happens that, due to error accumulation and/or insufficient working precision, the obtained result is not one expects from a theoretical derivation. The resulting basis may have more or less polynomials, a different number of solution, roots with different multiplicity, another Hilbert function, and so on. Augmenting precision we may overcome algorithmic errors, but one does not know in advance how much this precision should be, and a trial–and–error approach is often the only way to follow. Coping with initial errors is an even more difficult task. In this experimental work we propose the combined use of syzygies and interval arithmetic to decide what to do at each critical point of the algorithm.

13P10, 65H10, 90C31.

The algorithm is realized (with the help of computer algebra methods) for construction of numeric-analytical quasi-periodic solutions of precise(!) equations of restricted planar circular three-body problem (Sun–Jupiter-small planet) for an arbitrary sufficiently wide variety of initial data. This algorithm and corresponding exe-code allows us to obtain solutions in automatic mode (certainly, approximate but satisfying the motion equations with user-specified high precision) represented by twofold Fourier polynomials. Besides, the development of so-called perturbation function is not required (essential fact). These solutions are valid in principle for infinite time interval unlike known classical solutions of such problem. Such solutions are obtained for the first time.

This paper presents a framework and prototype implementation for preprocessing quantified input formulas that are intended as input for quantifier elimination algorithms. The framework loosely follows the AI search paradigm — exploring the space of formulas derived from the input by applying various rewriting operators in search of a problem formulation that will be good input for the intended Q.E. program. The only operator provided by the prototype implementation presented here is substitution for variables constrained by equations in which they appear linearly, supported by factorization and a simple check for non-vanishing of denominators in substitutions. Yet we present examples of quantified formulas which can be reduced by our preprocessing method to problems solvable by current quantifier elimination packages, whereas the original formulas had been inaccessible to those.

The kinetic integral equation for a quasi-one-dimensional organic crystal is solved exactly, and the expression for the electrical conductivity is presented as a transport integral. The latter has two singularities depending on crystal parameters. The possibilities to obtain analytic expressions in some particular cases and the general numerical calculations with pronounced singularities are analysed.

Automatic generation of smooth, non-overlapping meshes on arbitrary regions is the well-known problem. Considered as optimization task the problem may be reduced to finding a minimizer of the weighted combination of so-called length, area, and orthogonality functionals. Unfortunately, it has been shown that on the one hand, certain weights of the individual functionals do not admit the unique optimizer on certain geometric domains. On the other hand, some combinations of these functionals lead to the lack of ellipticity of corresponding Euler-Lagrange equations, and finding the optimal grid becomes computationally too expensive for practical applications. Choosing the right functional for the particular geometric domain of interest may improve the grid generation very much, but choosing the functional parameters is usually done in the trial and error way and depends very much on the geometric domain. This makes the automatic and robust grid generation impossible. Thus, in the present paper we consider the way to compute certain approximations of minimizer of grid functionals independently of the particular domain. Namely, we are looking for the approximation of the minimizer of the individual grid functionals in the local sense. This means the functional has to be satisfied on the possible largest parts of the domain. In particular, we shall show that the so called method of envelopes, otherwise called the method of rolling circle, that has been proposed in our previous paper, guarantees the optimality with respect to the area and orthogonality functionals in this local sense. In the global sense, the grids computed with the aid of envelopes, can be considered as approximations of the optimal solution. We will give the comparison of the method of envelopes with well established Winslow generator by presenting computational results on selected domains with different mesh size.

Many computer algebra systems lack modern user-friendly software environment. Poorly designed interface depreciates rich mathematical ideas implemented in calculation engine. It obstructs extensive usage of such systems because of requiring special knowledge and skills, e.g., in programming, to use them. Another problem of computer algebra systems is multitude of data formats and the implied difficulty in simultaneous usage of different systems. We discuss basics of and requirements to interfaces for computer algebra systems and techniques of their implementation. Modern software engineering approaches permit to provide a toolkit for semi-automated development of software environments for computer algebra systems.