9TH WORKSHOP ON COMPUTER ALGEBRA
DUBNA 2005
May 1718, 2005, Dubna
The 9th twoday workshop on Computer Algebra will be held on May 1718, 2005. This worshop is a traditional joint meeting of Moscow computer algebra seminar with the seminar on computational and applied mathematics of the Laboratory of Information Technologies of the Joint Institute for Nuclear Research. It is the continuation of a series of workshops which started in 1997 by Compuer Science Faculty of MSU, Skobeltsyn Institute of Nuclear Physics of MSU and Joint Institute for Nuclear Research. The workshop is conceived with the purpose of presenting topics of current interest and providing a stimulating environment for scientific discussion on new developments in computer algebra.
The duration of reports is 20 mins including discussion. Invited 30 mins reports will take place also. Participants who plan to visit Dubna for one day only should inform Alla Bogolubskaya about preferable date of the visit.
Current hotel prices now are 745 rub/night  a single room.
The transport timetable is at the site: http://www.dubna.ru/transport.php
Most interesting reports will be published (after a review treatment) in "Programmirovanie" Journal. The full version of papers (better in Russian) should be passed to Sergei Abramov in during of the seminar (NOT LATER!) in printed form (as a paper) AND as a file in PostScript or PDF format.
First Name: ______________________
Second Name: _____________________
Family Name: _____________________
Affiliation: ____________________
Position: ________________________
Abstract (including the title and a list of authors but without a list of references).
LIST OF ABSTRACTS
Symbolic Solutions of Inhomogeneous Linear Ordinary Differential Equations  
Anna Ryabenko (Dorodnicyn Computing Centre RAS)  
For a given l.o.d.e. Ly(x)=f(x) with polynomial coefficients and a rational right hand side f(x), we determine a set of points x=a such that there can be a power series solution y(x)=\sum_{n=0}^\infty c_n (xa)^n with hypergeometric coefficients, i.e. c_{n+1}/c_n is a rational in n for large enought n.


Inverse Problem for Discrete Elliptic Equation with Prescribed Symmetry Conditions  
S.I.Serdyukova (JINR, Dubna)  
We developed a numerical and analytical algorithm for reconstructing the twodimensional discrete elliptic equation of a part of spectrum and prescribed symmetry conditions. The right problem is solved in rectangle M \times N with zeroboundary conditions. If given symmetry conditions are satisfied the eigenfunctions can be prolonged from the rectangle on the whole plain with reserving continuity of the first derivatives. The problem is reduced to reconstruction of a symmetric fivediagonal matrix. We proved that when prescribed symmetry conditions are satisfied the considered block threediagonal matrix and all its blocks are persymmery. Such matrix has L<MN different elements and can be reconstructed of L given eigenvalues. Matrix elements and lacking MNL eigenvalues are determined by solving a polynomial system constituted of the Vieta relations for 4 multipliers of the characteristic polynomial of the matrix. Numerical experiments were produced. The polynomial systems were derived and solved by using CAS REDUCE.


On Regular and Logarithmic Solutions of Ordinary Linear Differential Systems  
Sergei A. Abramov (Dorodnicyn Computing Center RAS), Manuel Bronstein (INRIA, France), Denis E. Khmelnov (Dorodnicyn Computing Center RAS)  
We present an approach to find all the regular solutions of systems of linear
ordinary differential equations using the EG'desingularization algorithm as an
auxiliary tool. A similar approach to find all the solutions with entries in
C(z)[log(z)] is presented as well, together with a new hybrid method for constructing the denominator of rational and logarithmic solutions.


NORMAL FORMS AND INTEGRABILITY OF ODE SYSTEMS  
A.D. Bruno (Keldysh Institute of Applied Mathematics), V.F. Edneral (Moscow State University after Lomonosov)  
We consider a special case of the EulerPoisson system, describing the motion of a rigid body with a fixed point. It is the autonomous ODE system of sixth order with one parameter. For the stationary points, we compute the resonant normal form for two low level resonances of the system using a program based on the MATHEMATICA package. Our results show that in cases of the existence of an additional first integral of the system its normal form is degenerate. So we assume that the integrability of a system can be checked through its normal form.


On Factorization of Some Classes of Nonlinear Differential Equations  
L.M.Berkovich (Samara State University)  
The current paper considers some classes of nonlinear differential equations, that allows exact linearization and also different methods of factorization.


On EulerImshenetskyDarboux Transformation of Secondorder Linear Differential Equations  
L.M.Berkovich, S.A.Evlakhov (Samara State University) 

On the basis of EulerImshenetskyDarboux the sequence of the equations, integrable in terms of an input equation, constructs.


Testing the Membership in Differential Ideals Generated by a Polynomial Composition  
M.V. Kondratieva, A.I. Zobnin (MSU, Dept. of Mechanics and Mathematics)  
The membership problem posed for finitely generated differential ideals in ordinary differential polynomial rings is still open. It is solved just in several important cases, including the case of radical ideals, the case of isobaric ideals and the case of known finite or parametric differential standard basis. The authors suggested the process that builds such a basis with respect to certain orderings and always terminates if the latter is finite. The method that reduces testing the membership in saturated ideals generated by a polynomial composition to simpler solvable problems is described.


THE ALGORITHMS OF SYMBOLIC CALCULATIONS IN STUDYING SOME PROBLEMS OF COSMIC DYNAMICS  
`  A.Prokopenya (Brest Technical University, Belarus)  
Studying some models of cosmic dynamics, for example, elliptic restricted manybody problem or satellite motion along the elliptic orbit in the central field, we have to solve a number of problems where bulky symbolic calculations are involved. Obtaining the equations of motion, a search of their stationary solutions and studying their stability are just some examples of such problems. In the present paper we discuss the algorithms of symbolic calculations we use for solving the problems. The examples of their implementation in codes of the system Mathematica are shown.


The Proof of Nonexistence of Elliptic Solutions of the Cubic Complex GinzburgLandau Equation  
S.Yu. Vernov (Skobeltsyn Institute of Nuclear Physics of MSU)  
The cubic complex onedimensional GinzburgLandau equation is considered. Using the Hone's method, based on the use of the Laurentseries solutions and the residue theorem, we have proved that this equation has neither elliptic standing wave nor elliptic travelling wave solutions. This result amplifies Hone's result, that the GinzburgLandau equation has no elliptic travelling wave solutions. The proof has been reduced to solve a system of algebraic equations, that allows to automatize calculations.


Parallel Integration  
M. Bronstein (INRIA, http://wwwsop.inria.fr/cafe/Manuel.Bronstein/pmint/ ) 



Parallel integration, is an attempt to avoid the recursive nature of the Risch integration algorithm by considering the function to integrate as a multivariate rational function in the various terms appearing in it. We present new theorems that refine the classical Liouville theorem on elementary integration in the case of purely transcendental integrands. By predicting precisely the new functions that can appear in the integral, they yield a very simple parallel integration method that is applicable also to nonelementary and nonmonomial functions, such as the Lambert W and the Bessel functions. The method is so simple that an experimental Maple program takes less than 100 lines of code.


Solution Spaces of HSystems and the OreSato Theorem  
S.A. Abramov (Dorodnicyn Computing Centre RAS), M. Petkovsek (University of Ljubljana, Slovenia)  
An $H$system is a system of firstorder linear homogeneous difference equations for a single unknown function (sequence) $T(n_1,\dots,n_d)$. The coefficients of an $H$system are in $\cs [n_1,\dots ,n_d]$. We consider two spaces of solutions of $H$systems: the space $V_1$ of solutions defined everywhere on $\zs ^d$, and the space $V_2$ of solutions that are defined at all nonsingular points of $\zs ^d$. We investigate the dimensions of the spaces $V_1$, $V_2$, and prove for the case $d=1$ that if the equation has singularities then $1 \le \dim V_1 < \dim V_2 < \infty$, and for any integers $s, t$ such that $1 \le s < t$ there exists an equation with $\dim V_1 = s$ and $\dim V_2 = t$ (the case where there is no singularity is trivial: $\dim V_1=\dim V_2 =1$). Then we show that in the case $d>1$ the possibilities are even richer: for any $s,t\in \zs_+ \cup \{\infty\}$ there exists an $H$system with $\dim V_1 =s$ and $\dim V_2=t$. Additionally we revisit the SatoOre theorem
and show that, contrary to some interpretations
in the literature, this theorem does not imply that
any solution of an $H$system is of the form


Gosper's Algorithm, Accurate Summation, and Discrete NewtonLeibniz Formula  
S.A. Abramov (Dorodnicyn Computing Centre RAS), M. Petkovsek (University of Ljubljana, Slovenia)  
Sufficient conditions are given for validity of the discrete NewtonLeibniz formula when the indefinite sum is obtained either by Gosper's algorithm or by Accurate Summation algorithm. It is shown that sometimes a polynomial can be factored from the summand in such a way that the safe summation range is increased.


Applications of BK Method of LPDOs Factorization  
Shemyakova Katya (1.RISC, Austria; 2. MSU, Dept. of Mechanics and Mathematics)  
The problem of factorization of LPDOs is wellknown and a lot of
deep results are already obtained (Tsarev and its bib).
However, mostly the results were not constructive and play the
role of pure existence theorems.


Optimal Algorithm to Compute Essential Values of Hypergeometric Type Functions  
Alexey Bogolubsky (MSU, Dept. of Mechanics and Mathematics), Sergey Skorokhodov (Dorodnicyn Computing Centre RAS)  
The new method is proposed to compute the values of big set of functions at their singular points. Method is based on using of complete asymptotical expansions by Hurwitz functions \zeta(s, \alpha) for efficient summation of Dirichlettype series. Application of the method for computation of practically important function p_F_p1 ([a]; [b]; 1) is being considered in detail; the comparison with existing methods and their program realizations has been carried out. One of the real challenges is to make efficient choice of \alpha parameter (which is influencing convergence speed greatly); to do it, one must solve strongly nonlinear optimization problem. Computer analytical calculations play key role on both stages of developing and usage of the method. Besides, stochastic approach is also applied efficiently: genetic alghorithm is used to solve one of the subproblems. The whole of the methods developed allows to achieve significant perfomance increase in comparison with traditional techniques.


Analysis of the Solution Component Stability of Linear Differential or Recurrent Equations System with Maple  
G.V. Plotnikov (Dept. of Computational Mathematics and Cybernetics, MSU)  
The complex of the MAPLEprocedures based on AbramovBronstein's algorithm is described. This complex allows to consider components of the solution of the linear differential or recurrent equations system (to examine only a part of unknowns). Application of this algorithm in stability of solution components is also analyzed. Cases of constant and variable coefficients are examined.


Experimental Study of Efficiency of Involutive Criteria  
V.P. Gerdt, D.A. Yanovich (JINR, Dubna)  
In this paper we present results of computer experiments to study effectiveness of involutive criteria for avoiding some useless prolongations in construction of polynomial Janet bases. These bases are typical representative of involutive bases. Our experimental study shows that the role of criteria in the involutive approach is definitely weaker than that in Buchberger's algorithm.


Symbolic Algorithm for Decomposition of Evolution Operator for TimeDependent Schroedinger Equation  
V.P. Gerdt, A.A. Gusev, V.A. Rostovtsev, T.V. Tupikova, S.I. Vinitsky (JINR, Dubna), Y. Uwano (Hakodate University, Japan)  
An symbolic algorithm based on unitary evolution operator decomposition to generate a multilayer implicit schemes for solving the time dependent Schroedinger equation is elaborated. In framework of the algorithm a gauge transformation for extraction of symmetric operators for construction chonomical evolution schemes is realised also. The efficiency of the generated schemes is demonstrated on some integrable models.


CONSTRUCTIVITY OF INVOLUTIVE DIVISIONS  
Alexander Semenov (MSU, Dept. of Mechanics and Mathematics)  
Constructivity of involutive divisions
plays a significant role in the theory of
involutive bases, as this property is sufficient
for existence of the minimal involutive
basis of an arbitrary polynomial ideal


Analysis Methods of Multiple Eigenvalues of Spheroidal Wave Functions  
Skorokhodov S.L. (Dorodnicyn Computing Centre RAS), Khristoforov D.V. (MSU, Dept. of Mechanics and Mathematics)  
Spheroidal wave functions are eigenfunctions of special differential operator
of second order. They play an important role in many applied problems of diffraction,
astrophysics, etc.


Discrete Relations On Abstract Simplicial Complexes  
V. V. Kornyak (JINR, Dubna)  
A system of discrete relations is, on the one hand,
generalization of a cellular automaton, or, on the other hand,
a settheoretic analog of a system of polynomial equations.
Methods of investigation of systems of discrete relations
based on settheoretic and topological constructions
are considered. The proposed approach is implemented as a C
program. The results of application of the approach to


Symbolicnumeric program for solving twodimensional Schroedinger equation  
I.N.Belyaeva, N.A. Chekanov (Belgorod State University, Belgorod), A.A. Gusev, V.A. Rostovtsev (JINR, Dubna), Yu.A.Ukolov (Belgorod State University, Belgorod), Y. Uwano (Hakodate University, Hakodate, Japan) and S.I. Vinitsky (JINR, Dubna)  


The symbolicnumeric program for solving of the twodimensional Schroedinger equation. The rresponding algorithm of this program using a conventional pseudocode is described too. As example, the energy spectrum and wave functions for generalized HenonHeiles Hamiltonian were obtained.


Weakadmissible Monomial Orderings and the Universal Groebner Base  
N.N.Vasiliev (PDMI RAS, St.Petersburg )  
We consider non admissible monomial orderings which are only adjusted to usual division of monomials. We suggest some ways to define Groebner bases for such kind of orderings. We prove also the finiteness of universal base of an ideal defined for all weak admissible orderings.
