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  • 1
    Electronic Resource
    Electronic Resource
    The construction of symmetric cubature formulas for the square and the triangle (1988)
    Springer
    Computing 40 (1988), S. 229-240 
    Details
    ISSN: 1436-5057
    Keywords: 65D32 ; 65H10 ; Symmetric cubature formulas
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die Gewichte und Knoten einer symmetrischen Kubaturformel werden durch Lösung eines nichtlinearen Gleichungsystems bestimmt. die Anzahl der Gleichungen und ihre Struktur werden für symmetrische Kubaturformeln für das Quadrat und das Dreieck untersucht. Eine neue Kubaturformel vom Grad 7 mit 12 Knoten wird für das Dreieck angegeben.
    Notes: Abstract The weights and nodes of a symmetric cubature formula are determined by solving a system of nonlinear equations. The number of equations and their structure are investigated for symmetric cubature formulas for the square and the triangle. A new cubature formula of degree 7 with 12 nodes is given for the triangle.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02251251
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    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    Semi-Invariants, equivariants and algorithms (1996)
    Springer
    Applicable algebra in engineering, communication and computing 7 (1996), S. 105-124 
    Details
    ISSN: 1432-0622
    Keywords: Invariant theory ; Linear representations of finite groups ; Gröbner bases ; Computation of fundamental equivariants ; Solution of systems of polynomial equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics , Technology
    Notes: Abstract The results from Invariant Theory and the results for semi-invariants and equivariants are summarized in a suitable way for combining with Gröbner basis computation. An algorithm for the determination of fundamental equivariants using projections and a Poincaré series is described. Secondly, an algorithm is given for the representation of an equivariant in terms of the fundamental equivariants. Several ways for the exact determination of zeros of equivariant systems are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01191379
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    Paper (German National Licenses)
    Fulltext
  • 3
    Electronic Resource
    Electronic Resource
    Semi-Invariants, Equivariants and Algorithms (1996)
    Springer
    Applicable algebra in engineering, communication and computing 7 (1996), S. 105-124 
    Details
    ISSN: 1432-0622
    Keywords: Keywords: Invariant theory ; Linear representations of finite groups ; Gröbner bases ; Computation of fundamental equivariants ; Solution of systems of polynomial equations.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics , Technology
    Notes: Abstract.  The results from Invariant Theory and the results for semi-invariants and equivariants are summarized in a suitable way for combining with Grobner basis computation. An algorithm for the determination of fundamental equivariants using projections and a Poincaré series is described. Secondly, an algorithm is given for the representation of an equivariant in terms of the fundamental equivariants. Several ways for the exact determination of zeros of equivariant systems are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01191379
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    Paper (German National Licenses)
    Fulltext
  • 4
    Book
    Book
    Computer algebra methods for equivariant dynamical systems; 1728 (2000)
    Berlin u.a. :Springer,
    Details
    Title: Computer algebra methods for equivariant dynamical systems; 1728
    Author: Gatermann, Karin
    Publisher: Berlin u.a. :Springer,
    Year of publication: 2000
    Pages: 153 S.
    Series Statement: Lecture notes in mathematics 1728
    Type of Medium: Book
    URL: http://www.zbmath.org/?q=an:0944.65131
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    Zuse Institute Berlin get availability status
  • 5
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    Group Theoretical Mode Interactions with Different Symmetries. (1992)
    Details
    Publication Date: 2014-02-26
    Description: In two-parameter systems two symmetry breaking bifurcation points of different types coalesce generically within one point. This causes secondary bifurcation points to exist. The aim of this paper is to understand this phenomenon with group theory and the innerconnectivity of irreducible representations of supergroup and subgroups. Colored pictures of examples are included.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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    OPUS
  • 6
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    Testing for Sn-Symmetry with a Recursive Detective (1995)
    Details
    Publication Date: 2014-02-26
    Description: In the recent years symmetric chaos has been studied intensively. One knows which symmetries are admissible as the symmetry of an attractor and which transitions are possible. The numeric has been developed using equivariant functions for detection of symmetry and augmented systems for determination of transition points. In this paper we look at this from a sophisticated group theoretic point of view and from the view of scientific computing, i.e. efficient evaluation of detectives is an important point. The constructed detectives are based on Young's seminormal form for $S_n$. An application completes the paper.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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    OPUS
  • 7
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    Computer Algebra Methods for Equivariant Dynamical Systems (1999)
    Details
    Publication Date: 2014-02-27
    Description: An introductory chapter on Groebner bases is given which also includes new results on the detection of Groebner bases for sparse polynomial systems. Algorithms for the computation of invariants and equivariants for finite groups, compact Lie groups and algebraic groups are presented and efficient implementation and time comparision are discussed. This chapter also inlcudes improvements of the computation of Noether normalisation and Stanley decomposition. These results are applied in symmetric bifurcation theory and equivariant dynamics. As preparation of the investigation of the orbit space reduction three methods are compared for solving symmetric polynomial systems exactly. The method of orbit space reduction is improved by using the Cohen-Macaulayness of the invariant ring and nested Noether normalization. Finally this is applied for a case of mode interaction in the Taylor-Couette problem.
    Keywords: ddc:000
    Language: English
    Type: doctoralthesis , doc-type:doctoralThesis
    Format: application/postscript
    Format: application/pdf
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    OPUS
  • 8
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    A family of sparse polynomial systems arising in chemical reaction systems (1999)
    Details
    Publication Date: 2014-02-26
    Description: A class of sparse polynomial systems is investigated which is defined by a weighted directed graph and a weighted bipartite graph. They arise in the model of mass action kinetics for chemical reaction systems. In this application the number of real positive solutions within a certain affine subspace is of particular interest. We show that the simplest cases are equivalent to binomial systems while in general the solution structure is highly determined by the properties of the two graphs. First we recall results by Feinberg and give rigorous proofs. Secondly, we explain how the graphs determine the Newton polytopes of the system of sparse polynomials and thus determine the solution structure. The results on positive solutions from real algebraic geometry are applied to this particular situation. Examples illustrate the theoretical results.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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    OPUS
  • 9
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    Gruppentheoretische Konstruktion von symmetrischen Kubaturformeln. (1990)
    Details
    Publication Date: 2014-02-26
    Description: $G$-invariant cubature formulas for numerical integration over n-dimensional, $G$- invariant integration regions are computed symbolically. The nodes are the common zeros of some $d$-orthogonal polynomials which build an $H$-basis of an ideal. Approaches for these polynomials depending on parameters are made with the help of the theory of linear representations of a group $G$. This theory is also used for the effective computation of necessary conditions which determines the parameters. Another approach uses invariant theory and gröbner bases.
    Keywords: ddc:000
    Language: German
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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    OPUS
  • 10
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    Symbolic Exploitation of Symmetry in Numerical Path-following. (1990)
    Details
    Publication Date: 2014-02-26
    Description: Parameter-dependent systems of nonlinear equations with symmetry are treated by a combination of symbolic and numerical computations. In the symbolic part of the algorithm the complete analysis of the symmetry occurs, and it is here where symmetrical normal forms, symmetry reduced systems, and block diagonal Jacobians are computed. Given a particular problem, the symbolic algorithm can create and compute through the list of possible bifurcations thereby forming a so-called tree of decisions correlated to the different types of symmetry breaking bifurcation points. The remaining part of the algorithm deals with the numerical pathfollowing based on the implicit reparametrisation as suggested and worked out by Deuflhard/Fiedler/Kunkel. The symmetry preserving bifurcation points are computed using recently developed augmented systems incorporating the use of symmetry. {\bf Keywords:} pathfollowing, mixed symbolic-numeric algorithm, parameter-dependent, nonlinear systems, linear representations.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
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