[Show abstract][Hide abstract] ABSTRACT: In this paper we show how to compute algorithmically the full set of
algebraically independent constraints for singular mechanical and
field-theoretical models with polynomial Lagrangians. If a model under
consideration is not singular as a whole but has domains of dynamical (field)
variables where its Lagrangian becomes singular, then our approach allows to
detect such domains and compute the relevant constraints. In doing so, we
assume that the Lagrangian of a model is a differential polynomial and apply
the differential Thomas decomposition algorithm to the Euler-Lagrange
equations.
[Show abstract][Hide abstract] ABSTRACT: The purpose of this paper is to develop constructive versions of Stafford’s theorems on the module structure of Weyl algebras A
n
(k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford’s theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A
n
(k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series.
[Show abstract][Hide abstract] ABSTRACT: The MAPLE package "Janet" comes in two parts, the first dealing with polynomials and commutative algebra, the second with linear PDEs. Here the first part, called "Involutive", is described. Amongst others it contains a MAPLE and a C++ implementation of the involutive technique for polynomial modules as an alternative for conventional Gröbner basis techniques.
[Show abstract][Hide abstract] ABSTRACT: This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized.
Multidimensional Systems and Signal Processing 04/2014; 26(2). DOI:10.1007/s11045-014-0280-9 · 1.62 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper presents an algorithmic method to study structural properties of
nonlinear control systems in dependence of parameters. The result consists of a
description of parameter configurations which cause different control-theoretic
behaviour of the system (in terms of observability, flatness, etc.). The
constructive symbolic method is based on the differential Thomas decomposition
into disjoint simple systems, in particular its elimination properties.
[Show abstract][Hide abstract] ABSTRACT: To compute difference Groebner bases of ideals generated by linear
polynomials we adopt to difference polynomial rings the involutive algorithm
based on Janet-like division. The algorithm has been implemented in Maple in
the form of the package LDA (Linear Difference Algebra) and we describe the
main features of the package. Its applications are illustrated by generation of
finite difference approximations to linear partial differential equations and
by reduction of Feynman integrals. We also present the algorithm for an ideal
generated by a finite set of nonlinear difference polynomials. If the algorithm
terminates, then it constructs a Groebner basis of the ideal.
[Show abstract][Hide abstract] ABSTRACT: Using new results on the general Monge parametrization (see [25] and the references therein) recently obtained in [21], i.e., on the possibility to extend the con-cept of image representation to non-controllable multidimen-sional linear systems, we show that we can transform some quadratic variational problems (e.g., optimal control prob-lems) with differential constraints into free variational ones directly solvable by means of the standard Euler-Lagrange equations. This result generalizes for non-controllable multi-dimensional linear systems the results obtained in [11], [19] for controllable ones. In particular, in the 1-D case, this result allows us to avoid the controllability condition commonly used in the behavioural approach literature for the study of optimal control problems with a finite horizon and replace it by the stabilizability condition for the ones with an infinite horizon.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we consider systems of algebraic and non-linear partial
differential equations and inequations. We decompose these systems into
so-called simple subsystems and thereby partition the set of solutions. For
algebraic systems, simplicity means triangularity, square-freeness and
non-vanishing initials. Differential simplicity extends algebraic simplicity
with involutivity. We build upon the constructive ideas of J. M. Thomas and
develop them into a new algorithm for disjoint decomposition. The given paper
is a revised version of a previous paper and includes the proofs of correctness
and termination of our decomposition algorithm. In addition, we illustrate the
algorithm with further instructive examples and describe its Maple
implementation together with an experimental comparison to some other
triangular decomposition algorithms.
[Show abstract][Hide abstract] ABSTRACT: Some changes of the traditional scheme for finding rational solutions of linear differential, difference and q-difference homogeneous equations with rational coefficients are proposed. In many cases these changes allow one to predict the absence ...
ACM Communications in Computer Algebra 01/2011; DOI:10.1145/1940475.1940480
[Show abstract][Hide abstract] ABSTRACT: This paper provides methods to decide whether a given analytic function of several complex variables is a linear combination of nitely many given analytic functions with coecients of the following special form: Each one of these coe- cients is a composition of an unknown analytic function of less arguments than the function to be expressed, with xed analytic functions. Methods which compute suitable coecient functions in the armative case are presented as well.
Mathematics in Computer Science 09/2010; 4(2-3):231-242. DOI:10.1007/s11786-010-0053-2
[Show abstract][Hide abstract] ABSTRACT: In this paper we consider disjoint decomposition of algebraic and non-linear
partial differential systems of equations and inequations into so-called simple
subsystems. We exploit Thomas decomposition ideas and develop them into a new
algorithm. For algebraic systems simplicity means triangularity, squarefreeness
and non-vanishing initials. For differential systems the algorithm provides not
only algebraic simplicity but also involutivity. The algorithm has been
implemented in Maple.
[Show abstract][Hide abstract] ABSTRACT: In this paper we consider finite difference approximations for numerical solving of systems of partial differential equations of the form f1 = · · · = fp = 0, where F := {f1, ..., fp} is a set of linear partial differential polynomials over the field of rational functions with rational coefficients. For orthogonal and uniform solution grids we strengthen the generally accepted concept of equation-wise consistency (e-consistency) of the difference equations f1 = · · · = fp = 0 as approximation of the differential ones. Instead, we introduce a notion of consistency of the set of all linear consequences of the difference polynomial set f := {f, ..., fp} with the linear subset of the differential ideal 〈F〉. The last consistency, which we call s-consistency (strong consistency), admits algorithmic verification via a Gröbner basis of the difference ideal 〈f〉. Some related illustrative examples of finite difference approximations, including those which are e-consistent and s-inconsistent, are given.
Symbolic and Algebraic Computation, International Symposium, ISSAC 2010, Munich, Germany, July 25-28, 2010, Proceedings; 01/2010
[Show abstract][Hide abstract] ABSTRACT: The paper remarks on the average number of cycles of the elements of a finite permutation group in general and computes this
number for the alternating groups and wreath products using the cycle number indicator.
Archiv der Mathematik 11/2009; 93(5):445-449. DOI:10.1007/s00013-009-0045-0 · 0.39 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper explains the relevance of partitioning the set of standard monomials into cones for constructing a Noether normalization for an ideal in a polynomial ring. Such a decomposition of the complement of the corresponding initial ideal in the set of all monomials - also known as a Stanley decomposition - is constructed in the context of Janet bases, in order to come up with sparse coordinate changes which achieve Noether normal position for the given ideal.
[Show abstract][Hide abstract] ABSTRACT: In this work we announce the package to compute connection and C-connection matrices. is based on our abstract homological algebra package . We emphasize that the notion of braids is irrelevant for the definition and for the computation of such matrices. We introduce the notion of triangles that suffices to state the definition of (C-) connection matrices. The notion of octahedra, which is equivalent to that of braids is also introduced.
[Show abstract][Hide abstract] ABSTRACT: Computing the relations for the coefficients satisfied by the
characteristic polynomial of the Kronecker product of a general
$n \times n$ matrix by a general $m \times m$ matrix leads to
an elimination problem that is already difficult for small
values of $n$ and $m$. In this article we focus
on the problems for $(n, m) \in \{ (2,3), (2,4), (3,3)$ and
use these problems for developing and testing a new elimination
technique called elimination by degree steering.
[Show abstract][Hide abstract] ABSTRACT: A well-known result due to J.T. Stafford asserts that a stably free left module M over the Weyl algebras D=An(k) or Bn(k)–where k is a field of characteristic 0–with is free. The purpose of this paper is to present a new constructive proof of this result as well as an effective algorithm for the computation of bases of M. This algorithm, based on the new constructive proofs [Hillebrand, A., Schmale, W., 2001. Towards an effective version of a theorem of Stafford. J. Symbolic Comput. 32, 699–716; Leykin, A., 2004. Algorithmic proofs of two theorems of Stafford. J. Symbolic Comput. 38, 1535–1550] of J.T. Stafford’s result on the number of generators of left ideals of D, performs Gaussian elimination on the formal adjoint of the presentation matrix of M. We show that J.T. Stafford’s result is a particular case of a more general one asserting that a stably free left D-module M with is free, where denotes the stable rank of a ring D. This result is constructive if the stability of unimodular vectors with entries in D can be tested. Finally, an algorithm which computes the left projective dimension of a general left D-module M defined by means of a finite free resolution is presented. It allows us to check whether or not the left D-module M is stably free.
[Show abstract][Hide abstract] ABSTRACT: In the seventies, the study of transfer matrices of time-invariant linear systems of Ordinary Differential Equations (ODEs) led to the development of the polynomial approach Kailath, Thomas [”Linear systems”, Prentice-Hall Information and System Sciences Series. Englewood Cliffs, N.J.: Prentice-Hall, Inc. XXI, 682 p. (1980; Zbl 0454.93001)], V. Kucera [Discrete linear control. The polynomial equation approach, John Wiley & Sons. Published in co-edition with Prague: Academia, Publishing house of the Czechoslovak Academy of Sciences. 206 p. (1979; Zbl 0432.93001)], H.H. Rosenbrock [”State-space and multivariable theory”, London, Thomas Nelson & Sons, Ltd. VIII, 257 p. £5.00 (1970; Zbl 0246.93010)]. In particular, the univariate polynomial matrices play a central role in this approach (e.g., Hermite, Smith and Popov forms, invariant factors, primeness, Bézout/Diophantine equations).
[Show abstract][Hide abstract] ABSTRACT: The central notion of this work is that of a functor between categories of
finitely presented modules over so-called computable rings, i.e. rings R where
one can algorithmically solve inhomogeneous linear equations with coefficients
in R. The paper describes a way allowing one to realize such functors, e.g.
Hom, tensor product, Ext, Tor, as a mathematical object in a computer algebra
system. Once this is achieved, one can compose and derive functors and even
iterate this process without the need of any specific knowledge of these
functors. These ideas are realized in the ring independent package homalg. It
is designed to extend any computer algebra software implementing the
arithmetics of a computable ring R, as soon as the latter contains algorithms
to solve inhomogeneous linear equations with coefficients in R. Beside
explaining how this suffices, the paper describes the nature of the extensions
provided by homalg.
Journal of Algebra and Its Applications 01/2007; 7(3). DOI:10.1142/S0219498808002813 · 0.45 Impact Factor