Description of research

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Chemical reaction systems
Feinberg modeled chemical reaction systems in the model of mass action kinetics as a dynamical system where the right hand side is given by sparse polynomials. Thus the system is given by a weighted directed graph and a weighted bipartite graph. Depending on the structure of the graphs the number of real positive solutions is interesting since this has consequences for the dynamics of the system. in some situations the system decouples, or is a binomial system solved with Smith normal form or Hermite normal form. Another situation where precisely one positive solution exists corresponds to a linear system on a toric variety generated by one polynomial. In the general situation the results by Sturmfels on the number of positive solutions apply. For some parameter region the number is given by the number of alternating cells of the mixed subdivision of the Newton polytopes. The boundary of the parameter region is given by the sparse resultant. For the chemical reaction system this means that parts of the directed graph are responsible for the positive solutions. This method has been generalized for counting the number of stable positive solutions. In the second paper the key idea of intersecting a deformed toric variety with a convex cone is emphasized which gives a new coordinate systems. These coordinates are called convex parameters by Clarke. Toric geometry gives good insight into the problem.
K. Gatermann, B. Huber:
A family of sparse polynomial systems arising in chemical reaction systems.
Journal of Symbolic Computation 33(3), 275-305, 2002. (ps-file available here )
K. Gatermann:
Counting stable solutions of sparse polynomial systems in chemistry.
Contemporary Math. Volume 286,
Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering,
Editors: E. Green, S. Hosten, R. Laubenbacher, V. Powers, 53--69, 2001. (ps-file available here )

Symbolic computations in symmetric bifurcation theory
- Bifurcation phenomena of higher codimension are studied with singularity theory. This involves algebraic objects such as tangent spaces which are modules of germs. The computation of the codimension of tangent spaces and the symmetry naturally involve Gröbner bases computations.
- Also traditional local bifurcation analysis involves symbolic computations since the start is with a generic equivariant vector field and derivatives, restriction to fixed point spaces are performed by Computer Algebra.
- Even more the method of reduction onto the orbit space transports the differential equations onto a different set of equations on a part of a real variety, which is structured by the isotropy lattice (stratification as semi-algebraic set). The best choice of coordinates on this orbit space reflecting the symmetry as best as possible is determined with the help of symbolic computations.
K. Gatermann:
Computer Algebra for equivariant dynamical systems.
Habilitationsschrift FU Berlin, 1999.
Lecture Notes in Mathematics 1728, Springer 2000.
K. Gatermann, R. Lauterbach:
Automatic classification of normal forms.
Nonlinear Analysis 34, 157-190, 1998. (ps-file available here )
A. Lari-Lavassani, W.F. Langford, K. Huseyin, K. Gatermann:
Steady-state mode interactions for D3 and D4-symmetric systems.
Dynamics of Continuous, Discrete and Impulsive Systems 6, 169--209, 1999. (ps-file available here )

Gröbner bases
A Maple implementation of the Buchberger algorithm including a Hilbert series driven version, truncation with respect to gradings, arbitrary matrix term orders, fields of algebraic extensions, sugar selection strategy etc. has been done. It is available by www . Gröbner bases are used for the solution of algebraic systems of equations. Moreover they are used for the computation of structual information of varieties and algebraic computations as in algorithmic invariant theory. For efficiency reasons the detection of Gröbner bases are useful since conversion algorithms from one term order to another are known. The detection of Gröbner bases for sparse polynomial systems with weighted matching of bipartite graphs is described in
K. Gatermann:
Computer Algebra for equivariant dynamical systems.
Habilitationsschrift FU Berlin, 1999.
Lecture Notes in Mathematics 1728, Springer 2000.

Algorithmic invariant theory
The algorithmic computation of fundamental invariants for finite groups by Sturmfels algorithm and its generalization to equivariants has been implemented in the Symmetry package in Maple.
Secondly, routines for completeness for invariants, equivariants, Reynold projection, Molien series and an algorithm for the computation of invariants of compact Lie groups are included as well.
K. Gatermann:
Computer Algebra for equivariant dynamical systems.
Habilitationsschrift FU Berlin, 1999. (ps-file available here )
K. Gatermann, F. Guyard:
Gröbner bases, invariant theory and equivariant dynamics.
Journal of Symbolic Computation 28, p. 275--302, 1999. (ps-file available here )
K. Gatermann: Semi-invariants, equivariants and algorithms.
Appl. Algebra Eng. Comm. Comput. 7, 105-124, 1996. (ps-file available here )

Equivariant dynamical systems
- theoretical investigation of mode interaction of steady states from the view of numerics (a.) secondary bifurcation points, b.) secondary Hopf bifurcation)
- recursive detectives for observation of symmetry of chaotic attractors
K. Gatermann, B. Werner,
Secondary Hopf Bifurcation Caused by Steady-state Steady-state Mode interaction. In J. Chaddam, M. Golubitsky, W. Langford, B. Wetton (Eds.), Pattern Formations: Symmetry Methods and Applications, p 209-224, Fields Institute Communications Volume 5, 1996. (SC 93-16, 1993).
K. Gatermann, B. Werner:
Group Theoretical Mode Interactions with Different Symmetries. International Journal on Bifurcation and Chaos 4, 177-191, 1994. (SC 93-3, 1993).
K. Gatermann:
A remark on the detection of symmetry of attractors.
In P. Chossat (Ed.), Dynamics, Bifurcation and Symmetry, New Trends and New Tools, pp. 123, Kluwer Academic Publishers, Dordrecht, 1994.
K. Gatermann:
Testing for S_n-Symmetry with a Recursive Detective.
In: H.W. Broer, S.A. von Gils, I. Hoveijn, F. Takens (Eds.), Nonlinear Dynamical Systems and Chaos, 61-78, Birkhäuser, Basel, Boston, Berlin, 1996. (SC 95-12, 1995).

SYMCON: symbolic-numerical solution of equivariant dynamical systems

Computer Algebra part: preparation of given system for numeric part of SYMCON , exploitation of permutation representation for symmetry reduced systems and blocks of Jacobian, computation of subgroup lattice, bifurcation subgroups, representatives of left cosets etc., documentation . It is based on the REDUCE-SYMMETRY package for linear representation theory for small finite groups.

Numerical part: numerical pathfollowing of steady states, computation of bifurcation points, Hopf points, stability, user friendly X interface

K. Gatermann: Computation of Bifurcation Graphs.
In: E. Allgower, K. Georg, R. Miranda (eds.), Exploiting Symmetry in Applied and Numerical Analysis: 1992 AMS-SIAM Summer Seminar in Applied Mathematics, July 26- August 1, 1992, AMS Lectures in Applied Mathematics V. 29, 187-201, Providence, Rhode Island, 1993. (SC 92-13, 1992).
K. Gatermann: Mixed symbolic-numeric solution of symmetrical nonlinear systems .
In St. Watt (Ed.), Proceedings of ISSAC-91 (Bonn, Germany, July 15-17, 1991), 431-432, ACM, New York, 1991.
K. Gatermann, A. Hohmann: Symbolic Exploitation of Symmetry in Numerical Pathfollowing.
Impact of Computing in Science and Engineering 3, 330-365, 1991.
(SC 90-11, 1990).
K. Gatermann, A. Hohmann:
Hexagonal Lattice Dome - Illustration of a Nontrivial Bifurcation Problem. Konrad-Zuse-Zentrum, Preprint SC 91-8, 1991.

Combinatorial-numerical algorithm for the solution of sparse polynomial systems
The theoretical basis is the upper bound by Bernshtein, Kushnirenko and Khovanski for the number of solutions. Numerical pathfollowing is combined with the computation of this bound by the dynamic subdivision algorithm.
J. Verschelde, K. Gatermann, R. Cools:
Mixed Volume Computation by Dynamic Lifting
applied to Polynomial System Solving.
Journal of Discrete & Computational Geometry 16, 69-112, 1996.
J. Verschelde, K. Gatermann:
Symmetric Newton Polytopes for Solving Sparse Polynomial Systems.
Adv. Appl. Math. 16 (1), 95-127, 1995. (SC 94-3, 1994).

Exact solution of symmetric algebraic systems
- finding factors before starting the Buchberger algorithm
- using invariants and equivariants
- using algebraic groups and comparison of methods
K. Gatermann:
Computer Algebra for equivariant dynamical systems.
Habilitationsschrift FU Berlin, 1999.
Lecture Notes in Mathematics 1728, Springer 2000.
K. Gatermann: Semi-invariants, equivariants and algorithms.
Appl. Algebra Eng. Comm. Comput. 7, 105-124, 1996.
H. Caprasse, J. Demaret, K. Gatermann, H. Melenk:
Power-law type solutions of fouth-order gravity for multidimensional Bianchi I universes. Int. J. Math. Phys. C. 2, 601-611, 1991. (SC 90-18, 1990).
K. Gatermann: Symbolic solution of polynomial equation systems with symmetry.
In Sh. Watanabe, M. Nagata (Eds.), Proceedings of ISSAC--90 (Tokyo, Japan, August 20-24, 1990), 112-119. ACM, New York, 1990. (SC 90-3, 1990).

Construction of cubature formulas
This is the topic of my dissertation. The knots of the formula are the common zeros of some sufficiently orthogonal polynomials which form a H-basis of the ideal. Since the domains are symmetric this approach has been combined with linear representation theory and invariant theory of finite groups.
K. Gatermann: Linear representations of finite groups and the ideal theoretical
construction of G-invariant cubature formulas.
In T.O. Espelid, A. Genz (eds.), Numerical Integration, 25-35,
Kluwer Academic Publishers, 1992.
K. Gatermann: Gruppentheoretische Konstruktion von symmetrischen Kubaturformeln.
Dissertation an der Universität Hamburg, 1990. (TR 90-1, 1990).
K. Gatermann:
The Construction of Symmetric Cubature Formulas for the Square and the Triangle.
Computing 40, 229--240, 1988.

Last Update: March 20. 2002 by Karin Gatermann
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