Daniel Robertz

RWTH Aachen University, Aachen, North Rhine-Westphalia, Germany

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Publications (36)4.11 Total impact

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    Markus Lange-Hegermann, Daniel Robertz
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    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.
    12/2012;
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    Vladimir P. Gerdt, Daniel Robertz
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    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.
    06/2012;
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    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.
    08/2011;
  • ACM Communications in Computer Algebra 01/2011;
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    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.
    08/2010;
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    Computer Algebra in Scientific Computing - 12th International Workshop, CASC 2010, Tsakhkadzor, Armenia, September 6-12, 2010. Proceedings; 01/2010
  • Vladimir P. Gerdt, Daniel Robertz
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    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
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    Wilhelm Plesken, Daniel Robertz
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    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 01/2010; 4:231-242.
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    Daniel Robertz
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    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.
    J. Symb. Comput. 01/2009; 44:1359-1373.
  • W. Plesken, D. Robertz
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    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.
    Experimental Mathematics 01/2008; 17(2008). · 0.73 Impact Factor
  • Frédéric Chyzak, Alban Quadrat, Daniel Robertz
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    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 [20, 22, 44]. 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).
    04/2007: pages 233-264;
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    Mohamed Barakat, Daniel Robertz
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    ABSTRACT: In this work we announce the Maple package conley to compute connection and C-connection matrices. conley is based on our abstract homological algebra package homalg. 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.
    01/2007;
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    Mohamed Barakat, Daniel Robertz
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    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.
    01/2007;
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    Wilhelm Plesken, Daniel Robertz
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    ABSTRACT: New elimination methods are applied to compute polynomial relations for the coefficients of the characteristic polynomial of certain families of matrices such as tensor squares.
    Computer Algebra in Scientific Computing, 10th International Workshop, CASC 2007, Bonn, Germany, September 16-20, 2007, Proceedings; 01/2007
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    A. Quadrat, D. Robertz
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    ABSTRACT: It is well-known that a time-varying controllable ordinary differential linear system is flat outside some singularities. In this paper, we prove that every time-varying controllable linear system is a projection of a flat system. We give an explicit description of a flat system which projects onto a given controllable one. This phenomenon is similar to a classical one largely studied in algebraic geometry and called the blowing-up of a singularity. These results simplify the ones obtained in [6] and generalize them to MIMO multidimensional systems. Finally, we prove that every controllable multi-input ordinary differential linear system with polynomial coefficients is flat.
    Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on; 01/2006
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    Mohamed Barakat, Daniel Robertz
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    ABSTRACT: Homological algebra is a natural extension of the theory of modules over rings. The category of modules and their homomorphisms is replaced by the category of chain complexes and their chain maps. A module is represented by any of its resolutions. The module is then recovered as the only non-trivial homology of the resolution. The notions of derived functors and their homologies, connecting homomorphism and the resulting long exact homology sequences play a central role in homological algebra. The MAPLE-package homalg (1, 2) provides a way to deal with these powerful no- tions. The package is abstract in the sense that it is independent of any specific ring arithmetic. If one specifies a ring, not necessarily commutative, in which one can solve the ideal membership problem and compute syzygies, the above homological algebra constructions over that ring become accessible using homalg. As the name of this package suggests, our intention has been to make as much as possible of the basic homological machinery available in a computer algebra system without the need to specify the ring of operators from the beginning.
    Proceedings of the X meeting on computational algebra and its applications; 01/2006
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    Mohamed Barakat, Daniel Robertz
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    ABSTRACT: Methods from homological algebra (16) play a more and more important role in the study of multidimensional linear systems (15, 14, 5). The use of modules allows an algebraic treatment of linear systems which is independent of their presentations by systems of equations. The type of linear system (ordinary/partial differential equations, time-delay systems, discrete systems...) is encoded in the (non-commutative) ring of (differential, shift, ...) operators over which the modules are defined. In this framework, homological algebra gives very general information about the structural properties of linear systems. Homological algebra is a natural extension of the theory of modules over rings. The category of modules and their homomorphisms is replaced by the category of chain complexes and their chain maps. A module is represented by any of its resolutions. The module is then recovered as the only non-trivial homology of the resolution. The notions of derived functors and their homologies, connecting homomorphism and the resulting long exact homology sequences play a central role in homological algebra. The MAPLE-package homalg (1, 2) provides a way to deal with these powerful notions. The package is abstract in the sense that it is independent of any specific ring arithmetic. If one specifies a ring in which one can solve the ideal membership problem and compute syzygies, the above homological algebra constructions over that ring become accessible using homalg. In this paper we introduce the package homalg and present several applications of homalg to the study of multidimensional linear systems using available MAPLE-packages which provide the ring arithmetics, e.g. OreModules (4, 6) and Janet (3, 13).
    Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems; 01/2006
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    Alban Quadrat, Daniel Robertz
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    ABSTRACT: The purpose of this paper is to give a con-structive algorithm for the computation of bases of finitely presented free modules over the Weyl algebras of differen-tial operators with polynomial or rational coefficients. In particular, we show how to use these results in order to recognize when a multidimensional linear system defined by partial differential equations with polynomial or rational coefficients is flat and, if so, to compute flat outputs and the injective image representations of the system. These new results are based on recent constructive proofs of a famous result in non-commutative algebra due to J. T. Stafford [27]. The different algorithms have been implemented in the package STAFFORD [25] based on OREMODULES [2]. These results allow us to achieve the general solution of the so-called Monge problem for multidimensional linear systems defined by partial differential equations with polynomial or rational coefficients. Finally, we constructively answer an open question posed by Datta [5] on the possibility to generalize the results of [13] to multi-input multi-output polynomial time-varying controllable linear systems. We show that every controllable ordinary differential linear system with at least two inputs and polynomial coefficients is flat. Keywords— Flat multidimensional linear systems, injective image representation, constructive computation of bases of free modules, Stafford's results, non-commutative algebra.
    01/2006;
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    W. Plesken, D. Robertz
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    ABSTRACT: Characteristic zero representations of the (2,3,7)-triangle group in degrees up to seven are constructed by using Janet's algorithm for solving polynomial equations. These are used to find families of Hurwitz groups, i.e. finite epimorphic images. For some varieties of representations it is investigated whether additional relations can be uniformly imposed and still result in subvarieties of representations. The methods are of more general interest. Some remarks about the interaction of positive characteristics and characteristic zero are made.
    Journal of Algebra. 01/2006;
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    Vladimir P. Gerdt, Daniel Robertz
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    ABSTRACT: A Maple package for computing Gröbner bases of linear difference ideals is described. The underlying algorithm is based on Janet and Janet-like monomial divisions associated with finite difference operators. The package can be used, for example, for automatic generation of difference schemes for linear partial differential equations and for reduction of multiloop Feynman integrals. These two possible applications are illustrated by simple examples of the Laplace equation and a one-loop scalar integral of propagator type.
    Nuclear Instruments and Methods in Physics Research Section A Accelerators Spectrometers Detectors and Associated Equipment 09/2005; · 1.14 Impact Factor

Publication Stats

214 Citations
4.11 Total Impact Points

Institutions

  • 2010–2011
    • RWTH Aachen University
      • Lehrstuhl B für Mathematik
      Aachen, North Rhine-Westphalia, Germany
  • 2006
    • National Institute for Research in Computer Science and Control
      Le Chesney, Île-de-France, France
  • 2005
    • Joint Institute for Nuclear Research
      • Laboratory of Information Technologies
      Dubna, Moskovskaya, Russia