3rd International Conference Computer Algebra


3rd International Conference "Computer Algebra", Moscow, June 17-21, 2019

The 3rd international conference “Computer Algebra” will be held in Moscow from June 17 to June 21, 2019. The conference will be co-organized by Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control” of Russian Academy of Sciences (CCAS) and Peoples’ Friendship University of Russia (PFUR).

Location and Time

Conference meetings will be held in two locations: in CCAS building (40 Vavilova street, Moscow) on June 17,18 and 21, 2019 and in PFUR building (3 Ordzhonikidze street, Moscow) on June 19 and 20, 2019.

Main Topics

The two main topics of the conference are the following:

  1. Open questions in computer algebra. In particular, the focus is on problems which as yet are still unsolved, but for which it is reasonable to expect that with concentrated research efforts they could be worked out in the near future.
  2. Systematization (classification) of computer-algebraic tools. In particular, the goal is to propose principles for selection of appropriate computer-algebraic tools for solving concrete mathematical and applied problems.

Besides these two particular topics, the more traditional topics such as symbolic algorithms, mathematical software, and application aspects are discussed at the conference as well.

Key words: computer algebra, symbolic algorithms, implementation, software aspects, applied aspects, open questions, computer-algebraic tools.

Social Program

Best city tours on double-decker buses

  • Antonio Jiménez-Pastor
  • Dmitry Sergeevich Kulyabov
  • Victor Edneral
    • 09:30 09:50
      1-0: Registration
    • 09:50 10:00
      1-0: Opening
    • 10:00 11:25
      1-1: Plenary Speakers

      Chair: V.P.Gerdt

      • 10:00
        The Difference Ring Approach for Symbolic Summation 40m

        A major breakthrough in symbolic summation was Michael Karr's algorithm (1981) that solves the following problem. Given a summand $f(k)$ in a $\Pi\Sigma$-difference field $\mathbb F$, decide constructively if there exists a $g(k)$ in $\mathbb F$ such that $f(k)=g(k+1)-g(k)$
        holds. Given such a solution $g(k)$ and summing the telescoping equation over a valid summation range yields $\sum_{k=a}^bf(k)=g(b+1)-g(a)$.
        From the point of view of applications one is usually given only the summand $f(k)$ in terms of indefinite nested sums and products, and not a difference field $\mathbb F$ in which $f(k)$ can be represented properly.
        In this talk I will elaborate how indefinite nested sums defined over hypergeometric products can be always represented in $R\Pi\Sigma$-difference rings, an enhanced definition of Karr's $\Pi\Sigma$-fields. As a bonus we will obtain an algebraic machinery that can rewrite an expression in terms of indefinite nested sums and products to an equivalent expression such that among the arising sums and products (excepts elements such as $\alpha^k$ where $\alpha$ is a root of unity) no algebraic relations exist.
        Exploiting in addition the available difference ring/field algorithms for Zeilberger's creative telescoping and recurrence solving (utilizing as special cases Abramov's recurrence solver for rational functions and Petkovsek's Hyper algorithm), I will demonstrate how non-trivial summation problems in combinatorics and particle physics can be tackled with the help of the summation package Sigma.

        Speaker: Carsten Schneider (Research Institute for Symbolic Computation (RISC))
      • 10:45
        Geometric integrators in mechanics - the need for computer algebra tools 40m

        In this contribution we will describe some objects of the generalized geometry that appear naturally in the qualitative analysis of mechanical systems. In particular we will discuss the Dirac structures within the framework of the systems with constraints as well as of the port-Hamiltonian systems.

        From the mathematical point of view, Dirac structures generalize simultaneously symplectic and Poisson structures. As for mechanics, the idea is to design numerical methods that preserve these structures and thus guarantee good physical behaviour in the simulation.

        Then, we will present a framework which is even more general -- the one of differential graded manifolds (also called Q-manifolds), and discuss some possible ways of using them for the ``structure preserving integrators'' in mechanics.

        For all of the mentioned constructions we will explain the problems that arise in generic situations -- most of them are open, but we think they are suitable for handling with various computer algebra approaches.

        Speaker: Vladimir Salnikov (CNRS / La Rochelle Université)
    • 11:30 12:00
      1-1: Coffee break; discussions

      Chair: V.P.Gerdt

      • 11:30
        Coffee break; discussions 30m
    • 12:00 12:25

      Chair: V.P.Gerdt

      • 12:00
        Non-local singularities on families of periodic solutions to ODEs 25m

        We consider degenerate solutions on families of periodic solutions to ODEs. The degeneracy can mean any property of the solution that isolates the solution from the generic case. This can be a bifurcation or a topological peculiarity on the family that causes a failure of a numerical algorithm that was used for generic cases. We suggest a method of computation of such degeneracies with variational equations of higher order with the same accuracy as ordinary solutions.

        Speaker: Prof. Victor Varin (KIAM Moscow Russia)
    • 12:30 14:00
      1-1: Lunch break

      Chair: V.P.Gerdt

      • 12:30
        Lunch break 1h 30m
    • 14:00 15:25
      1-2: Plenary Speakers

      Chair: M.Petkovšek

      • 14:00
        Normal form of the periodically perturbed Hamiltonian system 40m

        Near a stationary solution we consider the Hamiltonian system with such perturbation, that the unperturbed Hamiltonian function is autonomous and the perturbation of the Hamiltonian function is periodic in time. First we remind the normal form of the autonomous Hamiltonian function. Second we describe the normal form of the periodic perturbation of the Hamiltonian function. It can always be reduced to the time independent Hamiltonian. It allows to compute the local families of periodic solutions of the initial system. We also discuss problems of the computer algebra arising in these computations.

        Speaker: Alexander Bruno (Keldysh Institute of Applied Mathematics)
      • 14:45
        Making Many More Matrix Multiplication Methods 40m

        Strassen's algorithm for matrix multiplication is based on the observation that the product of two $2 \times 2$ matrices can be computed with only 7 multiplications. It is known that there is no way to do it with fewer multiplications, and that Strassen's method is essentially the only way to do it with 7 multiplications. For $3 \times 3$-matrices, the situation is less clear. More than 40 years ago, Laderman gave a method that uses 23 multiplications, and still nobody knows whether there is a way to do it with fewer multiplications. Laderman's method is not unique. Several authors have later presented isolated additional methods that work for arbitrary coefficient rings, and there is even a family of methods with three free parameters but restricted to coefficient rings containing $\mathbb{Q}$. Using SAT solvers and Gr\"obner bases technology, we have found more than 10000 pairwise inequivalent new matrix multiplication methods with 23 multiplications. We were able to cluster them together into families with up to 16 parameters which apply without any restriction on the coefficient domain. In conclusion, the set of methods with 23 multiplications appears to be much larger than it seemed to be until now, and it is quite possible that the methods we found are still just the tip of the iceberg. Although our results have no immediate implications on the complexity of matrix multiplication, we do hope that the vast amount of new methods will eventually contribute to a better understanding of why these methods exist at all, and whether 23 is best possible for $3\times 3$.

        Speaker: Manuel Kauers
    • 15:30 15:55

      Chair: M.Petkovšek

      • 15:30
        Algebraically mimetic finite difference approximations on regular orthogonal grids oriented to solution of PDE 25m

        In the given talk we discuss the algorithmic aspects of differential and difference algebra related to construction, on uniform and orthogonal grids, of finite difference approximations to polynomially - nonlinear partial differential equations and to systems of such equations. As this takes place, we impose on a finite difference approximation the condition of strong consistency with the differential equations. If the condition holds, then we call the approximation algebraically mimetic. This condition strengthens the commonly accepted requirement of consistency of differential ideal associated with the input differential the approximation, which we call weak consistency, and means approximability of any consequence of the input differential equation(s) by an element in the perfect difference ideal generated by the polynomials occurring in the finite difference approximation. In the case of linear equations one can use difference \Gr bases for the elimination and differential Groebner bases for the verification of strong consistency. If the input differential equations are nonlinear, then instead of differential Groebner bases, which can be infinite, one can use the differential Thomas decomposition. Nonlinear difference Groebner bases can also be infinite, and there are needs in design of mathematical methods, algorithms and software for construction of a difference analogue of the differential Thomas decomposition.

        Speaker: Prof. Vladimir Gerdt
    • 16:00 16:30
      1-2: Coffee break; discussions

      Chair: M.Petkovšek

      • 16:00
        Coffee break; discussions 30m
    • 16:30 17:55

      Chair: M.Petkovšek

      • 16:30
        Linearizing differential equations: Riccati solutions as Dn-finite functions 25m

        D-finite (or holonomic) functions are a class of formal power series that satisfy
        linear differential equation with polynomial coefficients. The finite representation of these functions (using the differential equation and some initial conditions) boosted the development of algorithms working symbolically over them. This has been recently extended to the DD-finite class (functions satisfying linear differential equations with D-finite coefficients) and implemented some closure properties. It was also proved that DD-finite functions (and also their generalization to the D$^n$-finite functions) are differentially algebraic. In this document we show how solutions to non-linear differential equations (starting with the Riccati differential equation) are always D$^n$-finite
        functions for some $n$ and proposed some ideas to set the difference between D$^n$-finite functions and differentially algebraic functions.

        Speaker: Antonio Jiménez-Pastor (Doctoral Program Computational Mathematics (JKU))
      • 17:00
        Symmetric Matrices Whose Entries Are Linear Functions 25m

        There exists a large set of real symmetric matrices whose entries are linear functions in several variables such that each matrix in the set is definite at some point, that is, after substitution some numbers instead of variables. In particular, this property holds for almost all such matrices of order two or three, whose entries depend on two or three variables, respectively. The same property holds for almost all matrices whose entries depend on a larger number of variables, when the number exceeds the order of the matrix. For example, for each matrix of second partial derivatives of a multivariate third degree polynomial, all matrix entries are linear functions. Some examples have considered in details. At last the determinant of such matrices is considered. For almost every symmetric matrix whose entries are linear functions, the determinant of the matrix is positive at some point and it is negative at another point.

        Speaker: Alexandr Seliverstov (Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute))
      • 17:30
        On a Provable Program for Generic Arithmetic of Fractions 25m

        There are described the design principles for certified programs for arithmetic of fractions over any domain with the greatest common divisor function. This is a small part of the library DoCon-A of certified programs for a computer algebra library designed by the author. In this system, programs include definitions for the corresponding mathematical notions and proofs for the main properties of the implemented methods. These proofs are checked by the compiler. It is used a purely functional programming language Agda, which also supports the feature of dependent types. It is described the technique for providing formal machine-checked proofs for a certain optimized method to sum fractions.

        Speaker: Dr Sergei Meshveliani (Program systems institute of Russian Academy of sciences, Pereslavl-Zalessky)
    • 10:00 11:25
      2-1: Plenary Speakers

      Chair: A.Prokopenya

      • 10:00
        Experience in using the Fast Automatic Differentiation to solve inverse coefficient problems 40m

        The Fast Automatic Differentiation methodology (FAD-methodology) is a modern approach to help you effectively solve optimal control problems of complex dynamic systems. Using this methodology, the authors had been resolved several interesting theoretically and practically important problems. This paper demonstrates the use of FAD-methodology to solve inverse coefficient problems. The importance of these problems due to the need to create new materials, some characteristics of which are not known in advance, and you can identify them by solving inverse problems.
        The inverse problem considered is devoted to the determination of temperature-dependent thermal conductivity of the substance according to the results of a pilot monitoring of temperature field in the investigated substance and heat flux on the surface of an object. The consideration is based on the initial boundary value problem for the non-stationary heat equation. The inverse coefficient problem is reduced to the variational problem: it is required to find such dependence of the thermal conductivity coefficient on the temperature, at which the temperature field and heat fluxes at the boundary of the object, obtained as a result of solving the direct problem, differ little from the data obtained experimentally. The algorithm for the numerical solution of the posed inverse problem based on the modern FAD-methodology was proposed. The examples of solving the inverse coefficient problem confirm the efficiency of the proposed algorithm.

        Speaker: Mr Yury Evtushenko (Dorodnicyn Computing Centre, Federal Research Center "Computer Science and Control" of Russian Academy of Sciences)
      • 10:45
        Inverse Zeilberger’s problem 40m

        Given a sum of the form $s_n = \sum_{k=0}^n F(n,k)$ where $F(n,k)$ is proper hypergeometric, Zeilberger's algorithm returns a linear recurrence with rational coefficients satisfied by $s_n$. Here we consider the inverse problem: Given a linear recurrence operator $L$ with polynomial coefficients, find solutions that have the form of a definite sum. As a small first step towards its solution, we present an algorithm which, given $L$ and a product of binomial coefficients of the form $F(n,k) = \prod_{i=1}^m \binom{a_i n + b_i}{k}$ with $a_i \in \N \setminus \{0\}$, returns a linear recurrence operator $L'$ with rational coefficients such that $L(\sum_{k=0}^\infty F(n,k) h_k) = 0$ if and only if $L' h = 0$.

        Speaker: Mr Marko Petkovšek (University of Ljubljana)
    • 11:30 12:00
      2-1: Coffee break; discussions

      Chair: A.Prokopenya

    • 12:00 12:25

      Chair: A.Prokopenya

      • 12:00
        Laurent solutions of linear ordinary differential equations with coefficients in the form of truncated power series 25m

        Linear ordinary differential equations with formal power series coefficients represented in a truncated form are considered. We discuss the information on solutions belonging to the field of Laurent formal series which can be obtained from this representation of a given equation.
        We emphasize that we are interested in such information about solutions
        which is invariant with respect to possible prolongations of the
        truncated series representing the coefficients of the equation.

        Speakers: Sergei Abramov, Denis Khmelnov, Anna Ryabenko
    • 12:30 14:00
      2-1: Lunch

      Chair: A.Prokopenya

    • 10:00 11:25
      3-1: Plenary Speakers

      Chair: Th.Cluzeau

      • 10:00
        Interpolating Reduced Data Based on Piecewise-Cubic Splines 40m

        The problem of fitting a sequence of reduced data with piecewise-cubics $\hat\gamma$ is discussed. The interpolation points are in arbitrary Euclidean space with the respective interpolation knots unavailable. The unknown knots are estimated by the so-called exponential parameterization which depends on a single parameter $\lambda\in[0,1]$. We review the existing results determining the asymptotics in approximating $\gamma$ with the interpolant $\hat\gamma$.
        With the aid of computer algebra and analytic arguments this paper addresses the following issues:
        a) the sharpness of the convergence rates in estimation $\gamma-\hat\gamma\circ\phi$ is demonstrated,
        b) the necessity of more-or-less uniformity of $Q_m$ and of the regularity of $\gamma$ in proving the latter is justified to be essential,
        c) the sufficient conditions for $\phi$ to form a genuine parameterization are formulated in terms of algebraic inequalities which can be visualized in Mathematica.

        Speaker: Prof. Ryszard Kozera (Warsaw University of Life Sciences -SGGW)
      • 10:45
        Hilbert series of noncommutative algebras and context-free languages 40m

        The Hilbert series is one of the most important algebraic invariants of infinite-dimensional graded associative algebra. The noncommutative Groebner basis machine reduces the problem of finding Hilbert series to the case of monomial algebra.

        We apply both noncommutative and commutative Groebner bases theory as well as the theory of formal languages to provide a new method for symbolic computation of Hilbert series of graded associative algebras. Whereas in general the problem of computation oh such a Hilbert series is known to be algorithmically unsolvable, we have describe a general class of algebras (called {\em homologically unambiguous}) with unambiguous context-free set of relations for which our method give effective algorithms. Unlike previously known methods, our algorithm is applicable to algebras with irrational Hilbert series and produces an algebraic equation which defines the series. The examples include infinitely presented monomials algebras as well as finitely presented algebras with irrational Hilbert series such that the associated monomial algebras are homologically unambiguous.

        Speaker: Dmitri Piontkovskiy (National Research University Higher School of Economics)
    • 11:30 12:00
      3-1: Coffee break; discussions

      Chair: Th.Cluzeau

      • 11:30
        Coffee break; discussions 30m
    • 12:00 12:25

      Chair: Th.Cluzeau

      • 12:00
        On symbolic computation of the solution to the inverse problem of the calculus of variations 25m

        We study the problem of finding a Lagrangian function for the given system of $n$ second order ordinary differential equations (the inverse problem of the calculus of variations).
        As it was discovered by J. Douglas (1941), the inverse problem of the calculus of variations can be reduced to the problem of finding nontrivial solution for the overdetermined system of first order partial differential equations with $n(n+1)/2$ unknown functions, but solving this PDE system requires huge analytical calculations and classification of solutions was obtained by J. Douglas only for the case $n=2$.
        We discuss algorithmic aspects of solving the inverse problem of the calculus of variations for the case $n=3$ and their implementation in computer algebra systems Maple and SymPy. We present symbolic calculation of Lagrangian for the equations of motion of a symmetric rigid body with a fixed point as a demonstration of the algorithm.

        Speaker: Mr Sergey Shorokhov (RUDN University)
    • 12:30 14:00
      3-1: Lunch break

      Chair: Th.Cluzeau

      • 12:30
        Lunch break 1h 30m
    • 14:00 14:40
      3-2: Plenary Speakers

      Chair: D.Ştefănescu

      • 14:00
        An additive Ore-Sato theorem on compatible rational functions 40m

        The Ore--Sato theorem describes a multiplicative structure of multivariate hypergeometric terms. This theorem was first proved by Ore in 1930 in the bivariate case and then extended to the multivariate case by Sato in the 1960s via homological method. In 2002, Abramov and Petkovsek gave an elementary proof of the Ore--Sato theorem and proved a slightly corrected version of a conjecture of Wilf and Zeilberger on multivariate hypergeometric terms. Other different proofs of this theorem were also presented by Payne in 1997 and Hou in 2001 in their PhD dissertations. The continuous analogue of the Ore--Sato theorem was first obtained by Christopher in 1997 for bivariate compatible rational functions and later extended by Zoładek to the multivariate case in 1998. More recently, this result has been extended further to the multivariate continuous-discrete case by Chen et al. in 2011. The mixed extension of the Ore--Sato theorem has been used to prove the Wilf--Zeilberger conjecture on mixed hypergeometric terms by Chen and Koutschan in 2018. We will present an additive version of the Ore--Sato theorem which explores all possible rational Wilf--Zeilberger pairs.

        Speaker: Dr Shaoshi Chen (KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
    • 14:45 20:00

      Chair: D.Ştefănescu

      • 14:45
        Computing Primitive Idempotents in Centralizer Ring of Permutation Representation of Wreath Product 25m

        An approach to compute the complete set of primitive orthogonal idempotents in the centralizer ring of the permutation representation of a wreath product is described. This set determines the decomposition of the representation into irreducible components. In quantum mechanics, these idempotents manifest themselves as projection operators into irreducible invariant subspaces of the Hilbert space of a multipartite quantum system. The C implementation of the approach is able to construct irreducible decompositions of high-dimensional and high-rank representations of wreath products.

        Speaker: Vladimir Kornyak (Laboratory of Information Technologies, Joint Institute For Nuclear Research)
      • 15:15
        Application of Computer Algebra Methods for Investigation of the Stationary Motions of the System of Two Connected Bodies Moving along a Circular Orbit 25m

        Computer algebra and numeric methods are used to investigate properties of a nonlinear algebraic system that determines the equilibrium orientations for a system of two bodies connected by a spherical hinge that moves along a circular orbit under the action of gravitational torque.
        The main attention is paid to study the conditions of existence of the equilibrium orientations for the system of two bodies for special cases, when one of the principal axes of inertia both the first and second body coincides with the normal to the orbital plane, with radius vector or tangent to the orbit.
        To determine the equilibrium orientations for the system of two bodies, the system of 12 stationary algebraic equations is decomposed into 9 subsystems. The computer algebra method based on the algorithm for the construction of a Gröbner basis applied to solve the stationary motion system of algebraic equations. Depending on the parameters of the problem, the number of equilibria is found by numerical analysis of the real roots of the algebraic equations from the Gröbner basis constructed.

        Speaker: Dr Sergey Gutnik (Moscow State Institute of International Relations (MGIMO University))
      • 15:45
        An operator matrix having a given space of solutions 25m

        We consider differential full rank operator matrices over a differential field $K$ of characteristic zero. The constant field of $K$ is assumed to be algebraically closed. Together with each operator $A$ we consider its solution space $V_A$; the components of solutions are supposed to belong to the universal Picard-Vessiot extension $\Lambda$ of $K$. We prove that for any given finite set $F \subset \Lambda^m$ there exists a matrix whose solution space is generated by $F$ (the entries of that matrix are, in general, in $\Lambda$, not necessarily in $K$).

        Speakers: Sergei Abramov, Prof. Moulay Barkatou (XLIM UMR 7252 CNRS)
      • 16:15
        Close-Packed Dimers on a Cylinder Boundary. Derivation of Generating Functions in the Maple System 25m

        The problem of counting the number of perfect matchings on the cylinders
        $C_m \times P_n$ is considered for the case when the matching contains a given
        edge on the boundary of the cylinder. For fixed values of
        $3 \leq m \leq 14$ a set of recurrence relations and associated
        generating functions is constructed. Application of the obtained results to the
        dimer problem allowed us to deduce closed form expressions for the probabilities
        of occupation a given edge of the lattice with a dimer when $n \to \infty$. A set
        of corrections to the limit values of probabilities due to the finite order of
        the graph is calculated.

        Speaker: Dr Sergey Perepechko (Petrozavodsk State University)
      • 16:45
        Canonical representation of monomials with indices and contractions 25m

        The article deals with the problem of reducing monomials with indices to canonical form. This problem is met in many fields of mathematics and physics and is one of the key algorithmic problems of computer algebra. The most commonly used approach to the problem is based on a double coset. However, this approach has some limitations. For example, it cannot be used for expressions in which have linear relationships with more then two terms. The article provides a definition of the canonical form for monomials from indexed factors, taking into account summation indices and linear relations for them. Linear relations can include two and more terms with different order of indices. Also a special colored graph associated with monomials is introdiced and formulate the problem of calculating the canonical form of a monomial in terms of the canonical numbering of such graphs. The final part of the paper presents an algorithm for reducing monomials to canonical form, based on the calculation of the set of canonical numbering of edges of a color graph conjugated to the monomial. The algorithm does not use the construction of adjacency classes for the group renaming summation indices. The computational complexity of the proposed algorithm does not depend on the number of summation indices, but on the number of automorphisms of the color graph and is small for expressions that occur in practice.

        Speaker: Alexander Kryukov (M.V.Lomonosov Moscow State University)
      • 17:15
        Discussions 45m
      • 18:00
        Dinner (restaurant of Computing centre) 2h
    • 10:00 11:25
      4-1: Plenary Speakers

      Chair: D.S.Kulyabov

      • 10:00
        An efficient algorithm for the simultaneous triangularization of a finite set of matrices 40m

        In the study of linear differential systems, one can be interested in deciding whether a set of $m$ given square matrices $A_1,\ldots,A_m$ are simultaneously triangularizable or not. If the answer is yes, then we sometimes need to compute effectively an invertible matrix $P$ such that, for all $i \in \{1,\ldots,m\}$, the matrix $P^{-1}\,A_i\,P$ is upper triangular.

        The classical approach consists in using Lie algebra theory to test whether the matrix Lie algebra spanned by the $A_i$'s is solvable (e.g., using the so-called derived series) and if so, find a basis in which all matrices of the Lie algebra are upper triangular using a constructive version of Lie's theorem on solvable algebras for computing common eigenvectors.

        In this presentation, we will rather consider the following result due to McCoy: matrices $A_1,\ldots,A_m$ are simultaneously triangularizable if and only if, for every scalar polynomial $p(x_1,\ldots,x_m)$ in the (non-commutative) variables $x_1,\ldots,x_m$, each of the matrices $p(A_1,\ldots,A_m)[A_i,A_j]=p(A_1,\ldots,A_m) (A_i\,A_j-A_j\,A_i)$ ($i,j=1,\ldots,m$) is nilpotent. We shall show that the proof of this result can be turned into an efficient algorithm for computing particular common eigenvectors of $A_1,\ldots,A_m$. As a consequence, this yields an efficient algorithm for the simultaneous triangularization problem. Note that this new approach does not require the construction of the Lie algebra spanned by the matrices $A_i$'s. The algorithm has been implemented in Maple and we will show comparisons to the implementation of the ``Lie algebra method'' included in the DifferentialGeometry/LieAlgebras package of Maple.

        Speaker: Thomas Cluzeau (University of Limoges)
      • 10:45
        On Evaluation of Bessel Functions of the First Kind via Prony-like Methods 40m

        Bessel functions are widely applied in mathematics, physics and engineering. In this work, we apply two variants of Prony's method on Bessel functions of first kind of integer order and obtain approximations as sums of sinusoidal functions. Experiments show that the Prony-like methods yield approximations of high accuracy.

        Speaker: Min Wu (East China Normal University)
    • 11:30 14:00

      Chair: D.S.Kulyabov

      • 11:30
        Coffee break; discussions 30m
      • 12:00
        Bifurcations of periodic solutions to Hamiltonian system with discrete symmetries 25m

        We consider an autonomous Hamiltonian system with two degrees of freedom, which system of canonical equations is t-invariant under Klein four-group of linear canonical automorphisms of the extended phase space of the system. Three types of bifurcations of a family of doubly symmetric periodic solutions: saddle-node bifurcation, pitch-fork bifurcation and period multiplying bifurcation, are investigated by means of simplified monodromy matrix of periodic solution. For last two types of bifurcations different scenarios are studied. All mentioned above kinds of bifurcations were found and investigated for the case of doubly symmetric periodic solutions of the Hill problem. Most computations were provided with CAS Maple.

        Speaker: Alexander Batkhin (Keldysh Institute of Applied Mathematics of RAS)
      • 12:30
        Lunch break 1h 30m
    • 14:00 14:40
      4-2: Plenary Speakers

      Chair: M.Barkatou

      • 14:00
        Computational aspects of irreducible polynomials 40m

        We discuss some results on testing the irreducibility of polynomials and give some methods for constructing irreducible polynomials. We first survey classical methods of factorization of polynomials over the integers and irreducibility criteria that are based on properties of Newton's polygon.
        Finally we give irreducibility criteria for univariate polynomials $F(X) = \sum_{i=0}^d a_i X^{d-i}$ over a discrete valuation domain in terms of Newton's polygon. We give applications to bivariate polynomials.

        Speaker: Doru Stefanescu (University of Bucharest)
    • 14:45 16:40

      Chair: M.Barkatou

      • 14:45
        An algorithm for solving a family of diophantine equations of degree four which satisfy Runge's condition 25m

        In modern computer algebra systems (such as Mathematica, Maple, SageMath, etc.) there is a small number of implemented algorithms for solving diophantine equations in integers. In this paper we suggest an algorithmic implementation of elementary version of Runge's method for a family of diophantine equations of degree four. Although the fact that Runge's method has been known for more than 100 years, its implementation in computer algebra systems (CAS) is very limited (due to very large theoretical upper bounds for solutions which is useless for practical computer implementation via brute force method).

        The elementary version of Runge's method for diophantine equations of small degree is based on a convenient parametrization which provides enumerating possible integer solutions. In some cases this leads to the practically working solving algorithms for the diophantine equations of degree three and four.

        The software implementation of the proposed solving method (in its optimized version) is planned in PARI/GP CAS.

        Speaker: Alexey Kytmanov (Siberian Federal University)
      • 15:15
        Symbolic Computation in Studying the Dynamics of Generalized Atwood's Machine 25m

        A generalized model of the Atwood machine when a pulley of finite radius is replaced by two small separate pulleys and one body can swing in a vertical plane is considered. Doing necessary symbolic calculations, we have obtained equations of motion of the system and analyzed their solutions in case of small oscillations. We have shown that in case of small difference of masses of the bodies there exists a state of dynamical equilibrium of the system when the oscillating body acts like a pendulum whose length undergoes small oscillation. The corresponding periodic solutions of the equations of motion are obtained in the form of power series in terms of a small parameter. Comparison of the obtained results with the corresponding numerical solution of the equations of motion demonstrates its validity. All the relevant calculations are performed with the computer algebra system Wolfram Mathematica.

        Speaker: Prof. Alexander Prokopenya (Warsaw University of Life Sciences)
      • 15:45
        Coffee break; discussions 30m
      • 16:15
        Linearizability of differential equations: theory and algorithms 25m

        Solving nonlinear differential equations is one of the fundamental and practically important research challenges in mathematics. However, the problem of their algorithmic linearizability so far remained unsolved. In this contribution, we propose a solution of this problem for a wide class of nonlinear ordinary differential equation of arbitrary order. Criteria for linearizability of scalar ordinary differential equations is based on algorithmic calculus of determining Lie symmetry equations. Two main points are analysis of dimension of determining system and construction of derived algebra for abstract Lie symmetry algebra. New recent generalization for PDE will be also discussed.

        Speaker: Dmitry Lyakhov (King Abdullah University of Science and Technology)
    • 10:00 14:30

      Chair: L.A.Sevastianov

      • 10:00
        Algorithm for Calculating Interpolation Hermite Polynomials in $d$-dimensional hypercube in the Analytical Form 25m

        A new symbolic algorithm implemented in Maple for constructing Hermitian finite elements in the standard $d$-dimensional hypercube is presented. The basis functions of finite elements are polynomials, determined from a specially constructed set of values of the polynomials themselves and their first partial derivatives at the corners of the hypercube. Such a choice of values allows one to construct a piecewise polynomial basis continuous on the boundaries of finite elements together with the derivatives up to the first order. In the case of a $d$-dimensional cube, it is shown that the basis functions are determined by a product of $d$ interpolation Hermite polynomials depending on each of the $d$ variables with continuous first derivatives on the boundaries of finite elements. It can be used to solve elliptic boundary value problems by means of the finite element method. The efficiency and accuracy order of the finite element scheme, algorithm and program are demonstrated by the example of an exactly solvable Helmholtz problem for a three-dimensional cube.

        Speaker: Dr Alexander Gusev (JINR)
      • 10:30
        CAS pseudo-random numbers generators statistical test suits 25m

        For a long time the implementation of pseudo-random number sequence generators in standard programming language libraries and mathematical packages was of poor quality. The situation has started to improve relatively recently. Even nowadays a large number of libraries and poorly supported mathematical packages utilize the old algorithms of pseudo-random numbers generation. We describe four actual sets of statistical tests that can be used to test the generator that is used in a particular software system. The emphasis is on the use of command-line utilities, to avoid low-level C or C++ programming.

        Speaker: Мигран Нельсонович Геворкян
      • 11:00
        Coffee break; discussions 30m
      • 11:30
        Effective Schrödinger equation for one-dimensional potential box with rapidly oscillating width 25m

        The method of the effective Schrödinger equation is applied to the one-dimensional motion of a particle in a potential box with a potential function and high-frequency time-dependent boundary conditions. It was assumed that the oscillation amplitudes are small in comparison with the potential box width.The resulting system can be considered as a particle in a static potential box, located in the field of some effective potential, different from the initial one. In this work, we consider a particular case where the width of the box oscillates, while its center does not move, with the subsequent construction of the effective Schrödinger equation. Oscillations of width lead to additional terms in the effective equation proportional not only to the second derivative of the potential, but to the first derivative as well. This means that effects from fast external driving can be observed only in cases where the potentials have regions with large values of $V''(x)$ and $V'(x)$. Another fundamental difference is that in additional terms the first and second derivatives are multiplied by $x$ and $x^2$. In the case of width oscillations, compared to the oscillations of the box as a whole (this case was previously calculated elsewhere), the changes are even more significant and striking. In particular, the potential barrier may turn into a potential well (and vice versa).

        Speaker: Nikolay Tretyakov (Russian Presidential Academy of National Economy and Public Administration)
      • 12:00
        Symbolic-numeric implementation of the four potential method for calculating normal modes: an example of square electromagnetic waveguide with rectangular insert 25m

        In this paper, the Maple computer algebra system is used to construct a symbolic-numeric implementation of the method for calculating normal modes of square closed waveguides in a vector formulation. The method earlier proposed by Malykh et al. [M.D. Malykh, L.A. Sevastianov, A.A. Tiutiunnik, N.E. Nikolaev. On the representation of electromagnetic fields in closed waveguides using four scalar potentials // Journal of Electromagnetic Waves and Applications, 32 (7), 886-898 (2018)] will be referred to as the method of four potentials. The Maple system is used at all stages of treating the system of differential equations for four potentials: the generation of the Galerkin basis, the substitution of approximate solution into the system under study, the formulation of a computational problem, and its approximate solution. Thanks to the symbolic-numeric implementation of the method, it is possible to carry out calculations for a large number of basis functions of the Galerkin decomposition with reasonable computation time and then to investigate the convergence of the method and verify it, which is done in the present paper, too.

        Speaker: Dmitriy Divakov (RUDN University)
      • 12:30
        MAPLE program for calculating transition probabilities of hydrogen-like atoms in quantum mechanics with non-negative distribution function 25m

        The program is proposed for a realization of the symbolic algorithm based on the quantum mechanics with non-negative probability distribution function (QDF) and for calculations of transition probabilities for hydrogen-like atoms. Transition probabilities are calculated and compared with experimental data. Calculations of radial functions were used in calculations of oscillators strengths and transitional probabilities. These transition probabilities are calculated by the Galerkin method with the Sturm functions of the hydrogen atom as coordinate functions. Matrix elements of physical operators are required when the accurate theoretical determination of atomic energy levels, orbitals and radiative transition data need to be obtained for open-shell atoms and ions. It turns out that QDF seems to be equivalent to the traditional quantum mechanics in regard to predictions of experimental values. However, the existence of a phase-space probabilistic quantum theory may be an important advance towards the explanation and interpretation of quantum mechanics. Computer algebra methods seem to be absolutely necessary and indispensable for such calculations.

        Speaker: Nikolay Tretyakov (Russian Presidential Academy of National Economy and Public Administration)
      • 13:00
        Lunch 1h 30m
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