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...

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...

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...

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...

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.

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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...

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...

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...

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...

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...

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...

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 $F(X) = \sum_{i=0}^d a_i X^{d-i}$...

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...

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 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...

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...

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...

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...

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,...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

What is the connection of sparse interpolation from computer algebra with exponential analysis from signal processing, Padé approximation in approximation theory, and tensor decomposition in multilinear algebra? We explain that these seemingly unrelated topics are actually deeply intertwined. While several of these connections deserve to be investigated more thoroughly, new results have been...

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...

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...

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...

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 ...