TY - GEN A1 - Gatermann, Karin T1 - Counting stable solutions of sparse polynomial systems in chemistry N2 - The polynomial differential system modelling the behavior of a chemical reaction is given by graphtheoretic structures. The concepts from toric geometry are applied to study the steady states and stable steady states. Deformed toric varieties give some insight and enable graph theoretic interpretations. The importance of the circuits in the directed graph are emphazised. The counting of positive solutions of a sparse polynomial system by B.\ Sturmfels is generalized to the counting of stable positive solutions in case of a polynomial differential equation. The generalization is based on a method by sparse resultants to detect whether a system may have a Hopf bifurcation. Special examples from chemistry are used to illustrate the theoretical results. T3 - ZIB-Report - 00-32 KW - mass action kinetics KW - toric variety KW - stability KW - circuit Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6007 ER - TY - GEN A1 - Gatermann, Karin T1 - Gruppentheoretische Konstruktion von symmetrischen Kubaturformeln. N2 - $G$-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. T3 - ZIB-Report - TR-90-01 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4652 ER - TY - GEN A1 - Caprasse, H. A1 - Demaret, J. A1 - Gatermann, Karin A1 - Melenk, Herbert T1 - Power-Law Type Solutions of Fourth-Order Gravity N2 - We study the power-law type solutions of the fourth order field equations derived from a generic quadratic Lagrangian density in the case of multidimensional Bianchi I cosmological models. All the solutions of the system of algebraic equations have been found, using computer algebra, from a search of the Groebner bases associated to it. While, in space dimension $ d = 3 $ , the Einsteinian Kasner metric is still the most general power-law type solution, for $ d > 3 $ , no solution, other than the Minkowski space-time, is common to the three systems of equations associated with the three contributions to the Lagrangian density. In the case of a pure Riemann-squared contribution (suggested by a recent calculation of the effective action for the heterotic string), the possibility exists to realize a splitting of the $ d $-dimensional space into a ( $ d - 3 $)-dimensional internal space and a physical 3- dimensional space, the latter expanding in time as a power bigger than 2 (about 4.5 when $ d = 9 $). T3 - ZIB-Report - SC-90-18 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-466 ER - TY - THES A1 - Gatermann, Karin T1 - Computer Algebra Methods for Equivariant Dynamical Systems N2 - 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. T3 - ZIB-Report - SC-99-26 KW - Groebner bases KW - algorithmic invariant theory KW - equivariant dynamics KW - orbit space reduction Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4140 ER - TY - GEN A1 - Gatermann, Karin A1 - Huber, Birkett T1 - A family of sparse polynomial systems arising in chemical reaction systems N2 - 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. T3 - ZIB-Report - SC-99-27 KW - mass action kinetics KW - sparse polynomial systems KW - positive solutions Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4150 ER - TY - GEN A1 - Gatermann, Karin T1 - Computation of Bifurcation Graphs. N2 - The numerical treatment of Equivariant parameter-dependent onlinear equation systems, and even more its automation requires the intensive use of group theory. This paper illustrates the group theoretic computations which are done in the preparation of the numerical computations. The bifurcation graph which gives the bifurcation subgroups is determined from the interrelationship of the irreducible representations of a group and its subgroups. The Jacobian is transformed to block diagonal structure using a modification of the transformation which transforms to block diagonal structure with respect to a supergroup. The principle of conjugacy is used everywhere to make symbolic and numerical computations even more efficient. Finally, when the symmetry reduced problems and blocks of Jacobian matrices are evaluated numerically, the fact that the given representation is a quasi-permutation representation is exploited automatically. T3 - ZIB-Report - SC-92-13 Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-837 ER - TY - GEN A1 - Gatermann, Karin A1 - Hohmann, Andreas T1 - Hexagonal Lattice Dome - Illustration of a Nontrivial Bifurcation Problem. N2 - The deformation of a hexagonal lattice dome under an external load is an example of a parameter dependent system which is equivariant under the symmetry group of a regular hexagon. In this paper the mixed symbolic-numerical algorithm SYMCON is applied to analyze its steady state solutions automatically showing their different symmetry and stability properties. T3 - ZIB-Report - SC-91-08 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-587 ER - TY - GEN A1 - Gatermann, Karin T1 - Symbolic solution of polynomial equation systems with symmetry. N2 - Systems of polynomial equations often have symmetry. The Buchberger algorithm which may be used for the solution ignores this symmetry. It is restricted to moderate problems unless factorizing polynomials are found leading to several smaller systems. Therefore two methods are presented which use the symmetry to find factorizing polynomials, decompose the ideal and thus decrease the complexitiy of the system a lot. In a first approach projections determine factorizing polynomials as input for the solution process, if the group contains reflections with respect to a hyperplane. Two different ways are described for the symmetric group Sm and the dihedral group Dm. While for Sm subsystems are ignored if they have the same zeros modulo G as another subsystem, for the dihedral group Dm polynomials with more than two factors are generated with the help of the theory of linear representations and restrictions are used as well. These decomposition algorithms are independent of the finally used solution technique. We used the REDUCE package Groebner to solve examples from CAPRASSE, DEMARET and NOONBURG which illustrate the efficiency of our REDUCE program. A short introduction to the theory of linear representations is given. In a second approach problems of another class are transformed such that more factors are found during the computation; these transformations are based on the theory of linear representations. Examples illustrate these approaches. The range of solvable problems is enlarged significantly. T3 - ZIB-Report - SC-90-03 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-311 ER - TY - GEN A1 - Gatermann, Karin A1 - Hohmann, Andreas T1 - Symbolic Exploitation of Symmetry in Numerical Path-following. N2 - 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. T3 - ZIB-Report - SC-90-11 KW - pathfollowing KW - mixed symbolic-numeric algorithm KW - parameter dependent KW - nonlinear systems KW - linear representations Y1 - 1990 N1 - No preprint available; the article appeared in: IMPACT Comput. Sci. Eng. 3 (1991) pp. 330-365 ER - TY - GEN A1 - Gatermann, Karin A1 - Lauterbach, Reiner T1 - Automatic classification of normal forms N2 - The aim of this paper is to demonstrate a specific application of Computer Algebra to bifurcation theory with symmetry. The classification of different bifurcation phenomena in case of several parameters is automated, based on a classification of Gröbner bases of possible tangent spaces. The computations are performed in new coordinates of fundamental invariants and fundamental equivariants, with the induced weighted ordering. In order to justify the approach the theory of intrinsic modules is applied. Results for the groups $D_3, Z_2,$ and $ Z_2\times Z_2$ demonstrate that the algorithm works independent of the group and that new results are obtained. T3 - ZIB-Report - SC-95-03 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1692 ER - TY - GEN A1 - Verschelde, Jan A1 - Gatermann, Karin T1 - Symmetric Newton Polytopes for Solving Sparse Polynomial Systems. N2 - The aim of this paper is to compute all isolated solutions to symmetric polynomial systems. Recently, it has been proved that modelling the sparse structure of the system by its Newton polytopes leads to a computational breakthrough in solving the system. In this paper, it will be shown how the Lifting Algorithm, proposed by Huber and Sturmfels, can be applied to symmetric Newton polytopes. This symmetric version of the Lifting Algorithm enables the efficient construction of the symmetric subdivision, giving rise to a symmetric homotopy, so that only the generating solutions have to be computed. Efficiency is obtained by combination with the product homotopy. Applications illustrate the practical significance of the presented approach. T3 - ZIB-Report - SC-94-03 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1331 ER - TY - GEN A1 - Gatermann, Karin A1 - Werner, Bodo T1 - Group Theoretical Mode Interactions with Different Symmetries. N2 - 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. T3 - ZIB-Report - SC-93-03 Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-994 ER - TY - GEN A1 - Gatermann, Karin T1 - Semi-invariants, equivariants and algorithms. N2 - The results from invariant theory and the results for semi-invariants and equivariants are summarized in a way suitable for the combination with Gröbner basis computation. An algorithm for the determination of fundamental equivariants using projections and a Poincar\'{e} 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 T3 - ZIB-Report - SC-94-11 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1411 ER - TY - GEN A1 - Gatermann, Karin A1 - Werner, Bodo T1 - Secondary Hopf Bifurcation Caused by Steady-State Steady-State Mode Interaction. N2 - In two-parameter systems with symmetry two steady state bifurcation points of different symmetry types coalesce generically within one point. Under certain group theoretic conditions involving the action of the symmetry group on the kernels, we show that secondary Hopf bifurcation is borne by the mode interaction. We explain this phenomenon by using linear representation theory. For motivation an example with $D_3$-symmetry is investigated where the main properties causing the Hopf bifurcation are summarized. T3 - ZIB-Report - SC-93-16 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1122 ER - TY - GEN A1 - Gatermann, Karin T1 - Testing for Sn-Symmetry with a Recursive Detective N2 - 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. T3 - ZIB-Report - SC-95-12 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1786 ER - TY - GEN A1 - Gatermann, Karin A1 - Guyard, Frederic T1 - Gröbner bases, invariant theory and equivariant dynamics N2 - This paper is about algorithmic invariant theory as it is required within equivariant dynamical systems. The question of generic bifurcation equations requires the knowledge of fundamental invariants and equivariants. We discuss computations which are related to this for finite groups and semisimple Lie groups. We consider questions such as the completeness of invariants and equivariants. Efficient computations are gained by the Hilbert series driven Buchberger algorithm. Applications such as orbit space reduction are presented. T3 - ZIB-Report - SC-96-37 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2470 ER -