arXiv subjects

Viktor Levandovskyy

Publications and source records attributed to Viktor Levandovskyy.

26 records · Page 2Linked to original sources

arXiv2011

On Two-generated Non-commutative Algebras Subject to the Affine Relation

We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + ax + by + c for q in K^* and a, b, c in K. We prove, that among such algebras there are precisely five isomorphism classes. The representatives of these classes, which are ubiquitous operator algebras, are called model algebras. We derive explicit multiplication formulas for y^m*x^n in terms of standard monomials x^i*y^j for many algebras of the considered type. Such formulas are used in establishing formulas of binomial type and in implementing non-commutative multiplication in a computer algebra system. By using the formulas we also study centers and ring-theoretic properties of the non-commutative model algebras.

cs.SC↗

arXiv2010

A Symbolic Approach to Generation and Analysis of Finite Difference Schemes of Partial Differential Equations

In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent and discuss the applicability of them to nonlinear PDE's as well as to the case of variable coefficients. Moreover, we systematically use another symbolic technique, namely the cylindrical algebraic decomposition, in order to derive the conditions on the von Neumann stability of a difference s cheme for a linear PDE with constant coefficients. For stable schemes we demonst rate algorithmic and symbolic approach to handle both continuous and discrete di spersion. We present an implementation of tools for generation of schemes, which rely on Gröbner basis, in the system SINGULAR and present numerous e xamples, computed with our implementation. In the stability analysis, we use the system MATHEMATICA for cylindrical algebraic decomposition.

math-ph↗

arXiv2010

Constructive $D$-module Theory with \textsc{Singular}

We overview numerous algorithms in computational $D$-module theory together with the theoretical background as well as the implementation in the computer algebra system \textsc{Singular}. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein-Sato polynomials and also algorithms, recovering any kind of Bernstein-Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein-Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.

math.AG↗

arXiv2010

Exact linear modeling using Ore algebras

Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas the operators with varying coefficients were addressed only recently. This question can be tackled either using Gröbner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods to compute "most powerful unfalsified models" (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples.

math.OC↗

arXiv2010

Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases

In this paper we present two algorithms for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Gröbner bases. This can be viewed as the pre-processing for the computation of Jacobson normal form and also used for the computation of Smith normal form in the commutative case. We propose a general framework for handling, among other, operator algebras with rational coefficients. We employ special "polynomial" strategy in Ore localizations of non-commutative $G$-algebras and show its merits. In particular, for a given matrix $M$ we provide an algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds. The polynomial approach allows one to obtain more precise information, than the rational one e. g. about singularities of the system. Our implementation of polynomial strategy shows very impressive performance, compared with methods, which directly use fractions. In particular, we experience quite moderate swell of coefficients and obtain uncomplicated transformation matrices. This shows that this method is well suitable for solving nontrivial practical problems. We present an implementation of algorithms in SINGULAR:PLURAL and compare it with other available systems. We leave questions on the algorithmic complexity of this algorithm open, but we stress the practical applicability of the proposed method to a bigger class of non-commutative algebras.

math.RA↗

arXiv2010

Algorithms for Checking Rational Roots of $b$-functions and their Applications

Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose a family of algorithms called \texttt{checkRoot} for optimized check of whether a given rational number is a root of Bernstein-Sato polynomial and the computations of its multiplicity. This algorithms are used in the new approach to compute the whole global or local Bernstein-Sato polynomial and $b$-function of a holonomic ideal with respect to weights. They are applied in numerous situations, where there is a possibility to compute an upper bound for the polynomial. Namely, it can be achieved by means of embedded resolution, for topologically equivalent singularities or using the formula of A'Campo and spectral numbers. We also present approaches to the logarithmic comparison problem and the intersection homology D-module. Several applications are presented as well as solutions to some challenges which were intractable with the classical methods. One of the main applications consists of computing of a stratification of affine space with the local $b$-function being constant on each stratum. Notably, the algorithm we propose does not employ primary decomposition. Also we apply our results for the computation of Bernstein-Sato polynomials for varieties. The methods from this paper have been implemented in {\sc Singular:Plural} as libraries {\tt dmod.lib} and {\tt bfun.lib}. All the examples from the paper have been computed with this implementation.

math.AG↗

arXiv2010

Effective Methods for the Computation of Bernstein-Sato polynomials for Hypersurfaces and Affine Varieties

This paper is the widely extended version of the publication, appeared in Proceedings of ISSAC'2009 conference \citep*{ALM09}. We discuss more details on proofs, present new algorithms and examples. We present a general algorithm for computing an intersection of a left ideal of an associative algebra over a field with a subalgebra, generated by a single element. We show applications of this algorithm in different algebraic situations and describe our implementation in \textsc{Singular}. Among other, we use this algorithm in computational $D$-module theory for computing e. g. the Bernstein-Sato polynomial of a single polynomial with several approaches. We also present a new method, having no analogues yet, for the computation of the Bernstein-Sato polynomial of an affine variety. Also, we provide a new proof of the algorithm by Briançon-Maisonobe for the computation of the $s$-parametric annihilator of a polynomial. Moreover, we present new methods for the latter computation as well as optimized algorithms for the computation of Bernstein-Sato polynomial in various settings.

math.AG↗

arXiv2007

Obstructions to Genericity in Study of Parametric Problems in Control Theory

We investigate systems of equations, involving parameters from the point of view of both control theory and computer algebra. The equations might involve linear operators such as partial (q-)differentiation, (q-)shift, (q-)difference as well as more complicated ones, which act trivially on the parameters. Such a system can be identified algebraically with a certain left module over a non-commutative algebra, where the operators commute with the parameters. We develop, implement and use in practice the algorithm for revealing all the expressions in parameters, for which e.g. homological properties of a system differ from the generic properties. We use Groebner bases and Groebner basics in rings of solvable type as main tools. In particular, we demonstrate an optimized algorithm for computing the left inverse of a matrix over a ring of solvable type. We illustrate the article with interesting examples. In particular, we provide a complete solution to the "two pendula, mounted on a cart" problem from the classical book of Polderman and Willems, including the case, where the friction at the joints is essential . To the best of our knowledge, the latter example has not been solved before in a complete way.

math.OC↗