247 research outputs found
Obstructions to Genericity in Study of Parametric Problems in Control Theory
We investigate systems of equations, involving parameters from the point of
view of both control theory and computer algebra. The equations might involve
linear operators such as partial (q-)differentiation, (q-)shift, (q-)difference
as well as more complicated ones, which act trivially on the parameters. Such a
system can be identified algebraically with a certain left module over a
non-commutative algebra, where the operators commute with the parameters. We
develop, implement and use in practice the algorithm for revealing all the
expressions in parameters, for which e.g. homological properties of a system
differ from the generic properties. We use Groebner bases and Groebner basics
in rings of solvable type as main tools. In particular, we demonstrate an
optimized algorithm for computing the left inverse of a matrix over a ring of
solvable type. We illustrate the article with interesting examples. In
particular, we provide a complete solution to the "two pendula, mounted on a
cart" problem from the classical book of Polderman and Willems, including the
case, where the friction at the joints is essential . To the best of our
knowledge, the latter example has not been solved before in a complete way.Comment: 20 page
Parallel and distributed Gr\"obner bases computation in JAS
This paper considers parallel Gr\"obner bases algorithms on distributed
memory parallel computers with multi-core compute nodes. We summarize three
different Gr\"obner bases implementations: shared memory parallel, pure
distributed memory parallel and distributed memory combined with shared memory
parallelism. The last algorithm, called distributed hybrid, uses only one
control communication channel between the master node and the worker nodes and
keeps polynomials in shared memory on a node. The polynomials are transported
asynchronous to the control-flow of the algorithm in a separate distributed
data structure. The implementation is generic and works for all implemented
(exact) fields. We present new performance measurements and discuss the
performance of the algorithms.Comment: 14 pages, 8 tables, 13 figure
A survey on signature-based Gr\"obner basis computations
This paper is a survey on the area of signature-based Gr\"obner basis
algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain
the general ideas behind the usage of signatures. We show how to classify the
various known variants by 3 different orderings. For this we give translations
between different notations and show that besides notations many approaches are
just the same. Moreover, we give a general description of how the idea of
signatures is quite natural when performing the reduction process using linear
algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table
Lie series for celestial mechanics, accelerators, satellite stabilization and optimization
Lie series applications to celestial mechanics, accelerators, satellite orbits, and optimizatio
- …