@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{Gatermann1990, author = {Gatermann, Karin}, title = {Symbolic solution of polynomial equation systems with symmetry.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-311}, number = {SC-90-03}, year = {1990}, abstract = {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.}, language = {en} } @misc{GatermannHohmann1990, author = {Gatermann, Karin and Hohmann, Andreas}, title = {Symbolic Exploitation of Symmetry in Numerical Path-following.}, number = {SC-90-11}, year = {1990}, abstract = {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.}, language = {en} } @misc{Gatermann1990, author = {Gatermann, Karin}, title = {Gruppentheoretische Konstruktion von symmetrischen Kubaturformeln.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4652}, number = {TR-90-01}, year = {1990}, abstract = {\$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 gr{\"o}bner bases.}, language = {de} }