By definition, the coefficient sequence c = (c_n) of a d'Alembertian series — Taylor's or Laurent's — satisfies a linear recurrence equation with coefficients in C(n) and the corresponding recurrence operator can be factored into first order factors over C(n) (if this operator is of order 1, then the series is hypergeometric). Let L be a linear differential operator with polynomial coefficients. We prove that if the expansion of an analytic solution u(z) of the equation L(y) = 0 at an ordinary (i.e., non-singular) point z₀ of L is a d'Alembertian series, then the expansion of u(z) is of the same type at any ordinary point. All such solutions are of a simple form. However the situation can be different at singular points.

In this talk we shall discuss the relationship between the exponential parts of formal solutions of a differential system dY/dz = A(z)Y having a singularity of pole type at z = 0 and the singular parts of the eigenvalues of the matrix A(z) considered as Puiseux series in z. In particular, we will show that the exponential parts of a given differential system and the eigenvalues of its matrix do agree up to a certain order and that with suitable conditions on the matrix A(z) some formal invariants of the differential system dY/dz = A(z)Y can be computed from the characteristic polynomial of A(z).

A specialized computer algebra system GINV (abbreviation of Groebner INVolutive) will be presented. The system developed for investigating and solving systems of algebraic, differential and difference equations of polynomial type by their completion to involution. The central part of the system is implementation of authors involutive algorithms for construction of Janet, Janet-like and reduced Groebner bases for polynomial ideals and modules. It is planned in the nearest future to extend the built-in algorithms to linear differential and difference systems, and later on  to nonlinear differential systems. GINV consists of a C++ library which is organized as a module of Python and is distributed in accordance to regulations GPL2 v.1.2  quoted on the site http://invo.jinr.ru/ginv/ . The system GINV is used also as an external library for the Maple package Janet implementing the involutive algorithms for commutative algebra.

Special case A = B, M g x0 = 1, y0 = z0 = 0 of the equations of Euler--Poisson, describing movements of a heavy rigid body with the fixed point is considered. Near to one two-parametric family of their stationary points the normal forms are studied. On this family one-parametric families with the fixed resonances 1:2 and 1:3 are allocated. For them the structure of a normal form and the first integrals are investigated. There is a sequence of necessary conditions of existence of an additional formal first integral at various values of a parameter. Any of these conditions is a sufficient condition of absence of a formal integrability of the system and, thus, absence of a local, and consequently also a global integrability. By calculations of a normal form it is established, that the conditions necessary for existence of additional local first integral, are not satisfied in all cases, except for classical cases of a global integrability. Also stationary points near to which the system is locally integrable are found.

The methods of formal and algebraic representations in logic are considered on the basis of Planner language. In the formal logic approach some productive and re-writing rules are introduced. When applying to the predefined assertions, axioms, they allow sometimes to achieve a given goal. In the algebraic approach some Boolean expressions are evaluated. Among them the tautologies play important role. In this paper the Planner-B program, which joins these approaches, is proposed. Such a joint approach gives a tool for possible explanation of some logical paradoxes.

Symbolic-numerical algorithm for solving the two-dimensional boundary problem in a parametric basis is presented. The algorithm is implemented in Maple-Fortran and is used for the reduction of a singular two-dimensional boundary problem for the elliptic second-order partial differential equation to a regular boundary problem for a system of second-order ordinary differential equations using the Kantorovich method and to the corresponded algebraic problem using finite element method.

A C program for group analysis of discrete dynamical systems and lattice models in statistical mechanics is presented. The program constructs in particular phase portraits of discrete dynamical systems modulo groups of their symmetries and investigates structural properties of these phase portraits, searches dynamical systems possessing specific properties, e.g., reversibility, computes microcanonical partition functions and searches phase transitions in mesoscopic systems. Some computational results and observations are presented. We point out also general restrictions imposed by symmetries on behavior of deterministic discrete systems.

In this paper we consider improved versions of Buchberger algorithm and F4 algorithm that take into consideration the pecularities of computing of the Groebner basis over F₂. Besides, some additional optimizations were applied to the Buchberger algorithm, which could be applied in the general case. And we studied the problem of strategy of choosing the critical pairs at each step, and the result is that the usual _„Normal strategy" is not so good in case of F₂ with field equations added. We compare the efficiency of the algorithm with one of the fastest known implemented algorithms - Faugere's library FGb 1.34 for Maple. We use some standard benchmark tests.

We discuss one approach for constructing a parallel computer algebra. The base of this approach is a "temporal" tree algorithm, which is represented by the weighted tree. Data is passed through the edges of a graph, computational procedures are allocated in the vertexes of the graph and the weights of the edges denote the order of the priority of data which passes through these edges.

There are proposed two recursive block algorithms for matrix inversion, which have the computational complexity the same as the matrix multiplication has. These algorithms do not require the procedure of the choice of the leader element or leader block during the computational process. Such algorithms are effective for parallel computations.

Our aim is to add a prover to a computer algebra library. Our prover is based on many-sorted term rewriting, unfailing completion, and inductive reasoning. The completion procedure includes also superpositions with Boolean terms, which enables proofs in the predicate logic. The completion procedure can be considered as a generalization of the Groebner basis method known in algebra. The inductive inference cooperates in a natural way with completion. The prover distributes the resource between the branches of the search tree according to heuristics. There is also an intriguing point in the strategy: the step of the search for probable lemmata. We intend to link this prover to our computer algebra library, and both parts are implemented in the Haskell functional language.

Key words: automatic equational prover, term rewriting, inductive reasoning.

We consider non-transitive head-or-tail game: two players agree on some integer n; then both of them select a binary n-word (“head”=0, “tail”=1) and begin flipping a coin. The winner is the player whose word appears first as a block of n consecutive outcomes. The solution of the problem that was suggested by Conway requires construction of a special matrix. We suggest recursive algorithm for the fast construction of the Conway matrix. Results are visualized as a square, where numbers of rows and columns code binary words in a usual way; color depends on the probability of the event “row wins”.

The Rosenfeld-Groebner algorithm decomposes a radical differential ideal into characterizable components. One can bound orders of derivatives occurring while the algorithm is performed by applying a special differential reduction algorithm. We claim that such a bound would then be true for any differential reduction algorithm.

The Groebner bases techniques is applied to the analysis of the so-called Yang-Mills mechanics, which is the degenerate Hamiltonian model derived from the Yang-Mills field theory supposing a spatial homogeneity of the fields. Due to the large number of variables and constraints calculations modulo a set of constraints (the polynomial functions of coordinates and momenta) usage of the standard build in Maple and Mathematica packages turns out to be insufficient. To overcome this problem the special homogeneous Groebner basis has been constructed in Mathematica codes. This basis is build by increasing the degree of polynomials at every subsequent step. The algorithm and implementation of the basis for the deducing and classifying of the complete set of constraints of the Yang-Mills mechanics are presented.

The problem of finding of the two rational functions f(x), g(x) for given rational function f(x) such that f(x)=g(x+1)-g(x)+r(x), the denominator of r(x) has minimal degree, and the denominator of g(x) has minimal degree for given conditions, is considered. An algorithm for solving that problem that does not require complete factorization of polynomials is proposed. The algorithm is implemented in computer algebra system Maple.

A modern computer algebra system gives an opportunity of exact experimental computations including formula obtaining. It makes possible receiving solutions of a problem under concideration for a set of small size inputs . The results can provide a base to set up a conjecture on the general solution of the problem. After this the computer algebra system can help to check the conjecture.

In the work we consider important dynamic properties of involutive divisions - constructivity and monotonicity, which were firstly introduced by V.P. Gerdt and Yu.A. Blinkov. The problem of description of all constructive and monotone involutive divisions is still unsolved and non-trivial. In the contribution to the workshop we give a small overview of the previously obtained results and consider some questions on the Janet division - the most used in the set of all involutive divisions. And example of an involutive division which is superior to all variants of Janet divisions on a particular set is given.

In the report a problem of diagnostics for optical surfaces is considered. The light from distorted surface pass thought the Hartmann diaphragm and rays of light displace from positions the position according to undistorted surface. To determine a degree of distortion of the optical surface we use an expansion by Zernike polynomials that allow to synthesize its correction. To provide a stable procedure of mathematical synthesis of surface correction we use Tikhonov stabilizing functional calculated with the help of matrices of partial derivatives of Zernike polynomials. We realized the algorithm of building Zernike polynomials and matrix of their partial derivatives in module of GNU Public License's Computer Algebra System Axiom. This module permits to construct basic functions and to deal with them afterwards.

In the report a program package for a computer building of equations for motion of holonomic multibody mechanical systems in a symbolic form is presented. The package consists of Mathematica-code programs, is intended for obtaining of governing dynamical equations for systems analized with a tree configuration and generates Fortran-subroutines for calculation of kinematic and dynamic characteristics of mechanical systems.

We study the Orr-Sommerfeld equation for the Couette flow in a channel. A new efficient method for computation eigenvalues $\lambda_n$ was elaborated for the large Reynolds numbers R >> 1. Using the system Maple and numerical evaluations we find, that the eigenvalues $\lambda_n$ have denumerable number of branch points $R_k > 1$ at which the eigenvalues $\lambda_n$ with two different numbers $n_1$ and $n_2$ form the double eigenvalues.

An efficient method for computation eigenvalues $\lambda_n$ for the Coulomb spheroidal wave equation was elaborated in a case of complex parameters $b$ and $c$. Using the system Maple and numerical evaluations we find, that some points $b_s$ and $c_s$ are branch points of the second order for eigenvalues $\lambda_{n}(b, c)$ with different numbers $n_1$ and $n_2$. A large number of these singular points was evaluated.

The computation of the dominant positive roots of univariate polynomials is a key step for real root isolation algortihms. In this talk we propose a new approach for computing dominant real roots, and compare it with other existing methods. For roots greater than unity we obtain better results than the criteria of Lagrange and Longchamp. We also discuss a new version of our bounds for polynomials with an even number of sign changes.

In is well known that Shimura differential operator is defined explicitly in the case of dimension 2. But in the case of dimension 3 the explicit expression for it is unknown. A Mathematica 5.2 package was developed. It allows to simplify the calculation of Shimura differential operator in the case of dimension 3. Several functions from this package will be observed. Also their construction and aplication will be disscused.

In this talk we describe various connections between monomial orders, multidimensional Young diagrams and combinatorics of Groebner Fan. Algorithms for enumeration of admissible, weak-admissible and convex monomial orders will be presented. These algorithms are based on connection between enumeration combinatorics of Young diagrams and combinatorics of monomial orderings. We shall describe also some results and hypothesis about the shape of Groebner fan for random polynomial ideals.

The Conte-Musette method for the search of both elliptic and degenerated elliptic solutions has been modified for the search of only elliptic solutions to systems of differential equations. A key idea of this a priory restriction is to simplify calculations by means of the use of a few Laurent series solutions instead of one and the use of the residue theorem. The application of this approach to the quintic complex one-dimensional Ginzburg-Landau equation (CGLE5) allows us to find elliptic solutions in the wave form. Restrictions on coefficients, which are necessary conditions for the existence of elliptic solutions for the CGLE5, have been found as well. We demonstrate that to find elliptic solutions the analysis of a system of differential equations is preferable to the analysis of the equivalent single differential equation.

Symbolic-numerical algorithms for solving boundary problems are reviewed using a set of the quantum mechanics tasks.

In this talk we present our recent results of our implementations of Buhberger algorithm and involutive algorithm for computation of Groebner bases over F₂. Special attention is payed to the efficiency of these implementaions, particularly to the vectorization used to storage the inner data of the monomial class. We compare the efficiency of our implementations with a few other computer algrebra systems: Singular 3.0.2, CoCoA 4.5 and FGb 1.34 for Maple. As benchmarks we use the standart ones and some other benchmarks.

Differential standard bases are differential counterparts of Groebner bases. A finite or parametric standard basis of a differential ideal allows us to test the membership. However, such bases are often infinite. Finiteness criteria for differential standard bases can be stated in terms of generators and properties of admissible orderings. We also discuss possible generalizations of differential standard bases (so called differential H-bases). The results of the report have been obtained after special computation with computer algebra system Maple.

Several years ago we presented programm complex for parallel computations of Groebner and involutive bases that works on shared-memory computers. But unfortunately number of processors that we can use simultaniously is low (about 2-8 by now) because of hardware restrictions. In this paper programm for distributed computation exploiting same strategy and principles but working on network of heterogeneous machines will be presented. We will evaluate the peak/average load on processor and network bandwidth and point several approaches to optimize resource using.