@misc{GatermannGuyard1996, author = {Gatermann, Karin and Guyard, Frederic}, title = {Gr{\"o}bner bases, invariant theory and equivariant dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2470}, number = {SC-96-37}, year = {1996}, abstract = {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.}, language = {en} } @misc{CaprasseDemaretGatermannetal.1990, author = {Caprasse, H. and Demaret, J. and Gatermann, Karin and Melenk, Herbert}, title = {Power-Law Type Solutions of Fourth-Order Gravity}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-466}, number = {SC-90-18}, year = {1990}, abstract = {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 \$).}, language = {en} } @misc{GatermannLauterbach1995, author = {Gatermann, Karin and Lauterbach, Reiner}, title = {Automatic classification of normal forms}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1692}, number = {SC-95-03}, year = {1995}, abstract = {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 Gr{\"o}bner 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.}, language = {en} } @misc{VerscheldeGatermann1994, author = {Verschelde, Jan and Gatermann, Karin}, title = {Symmetric Newton Polytopes for Solving Sparse Polynomial Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1331}, number = {SC-94-03}, year = {1994}, abstract = {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.}, language = {en} } @misc{GatermannWerner1993, author = {Gatermann, Karin and Werner, Bodo}, title = {Secondary Hopf Bifurcation Caused by Steady-State Steady-State Mode Interaction.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1122}, number = {SC-93-16}, year = {1993}, abstract = {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.}, language = {en} } @misc{Gatermann1994, author = {Gatermann, Karin}, title = {Semi-invariants, equivariants and algorithms.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1411}, number = {SC-94-11}, year = {1994}, abstract = {The results from invariant theory and the results for semi-invariants and equivariants are summarized in a way suitable for the combination with Gr{\"o}bner 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}, language = {en} } @misc{Gatermann1995, author = {Gatermann, Karin}, title = {Testing for Sn-Symmetry with a Recursive Detective}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1786}, number = {SC-95-12}, year = {1995}, abstract = {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.}, language = {en} } @misc{Gatermann1992, author = {Gatermann, Karin}, title = {Computation of Bifurcation Graphs.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-837}, number = {SC-92-13}, year = {1992}, abstract = {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.}, language = {en} } @misc{GatermannWerner1992, author = {Gatermann, Karin and Werner, Bodo}, title = {Group Theoretical Mode Interactions with Different Symmetries.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-994}, number = {SC-93-03}, year = {1992}, abstract = {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.}, language = {en} } @misc{GatermannHohmann1991, author = {Gatermann, Karin and Hohmann, Andreas}, title = {Hexagonal Lattice Dome - Illustration of a Nontrivial Bifurcation Problem.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-587}, number = {SC-91-08}, year = {1991}, abstract = {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.}, language = {en} }