Results 1  10
of
13
Tropical matrix duality and Green’s D relation
 Journal of the London Mathematical Society
"... iv ..."
(Show Context)
Rational semimodules over the maxplus semiring and geometric . . .
, 2002
"... We introduce rational semimodules over semirings whose addition is idempotent, like the maxplus semiring. We say that a subsemimodule of the free semimodule S n over a semiring S is rational if it has a generating family that is a rational subset of S n, S n being thought of as a monoid under the ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
We introduce rational semimodules over semirings whose addition is idempotent, like the maxplus semiring. We say that a subsemimodule of the free semimodule S n over a semiring S is rational if it has a generating family that is a rational subset of S n, S n being thought of as a monoid under the entrywise product. We show that for various semirings of maxplus type whose elements are integers, rational semimodules are stable under the natural algebraic operations (union, product, direct and inverse image, intersection, projection, etc). Rational semimodules are a tool to extend the geometric approach of linear control to discrete event systems. In particular, we show that the reachable and observable spaces of maxplus linear dynamical systems
Multiplicative structure of 2 × 2 tropical matrices
, 907
"... We study the algebraic structure of the semigroup of all 2 × 2 tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green’s relations and the idempotents and maximal subgroups of this semigroup. 1 ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
(Show Context)
We study the algebraic structure of the semigroup of all 2 × 2 tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green’s relations and the idempotents and maximal subgroups of this semigroup. 1
SEMIGROUP IDENTITIES IN THE MONOID OF TWOBYTWO TROPICAL MATRICES
, 2009
"... We show that the monoid M2 ( ) of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
We show that the monoid M2 ( ) of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity
Backward Reachability of Autonomous MaxPlusLinear Systems ⋆
"... Abstract: This work discusses the backward reachability of autonomous MaxPlusLinear (MPL) systems, a class of continuousspace discreteevent models that are relevant for applications dealing with synchronization and scheduling. Given an MPL system and a continuous set of final states, we charact ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract: This work discusses the backward reachability of autonomous MaxPlusLinear (MPL) systems, a class of continuousspace discreteevent models that are relevant for applications dealing with synchronization and scheduling. Given an MPL system and a continuous set of final states, we characterize and compute its "backward reach tube" and "backward reach sets," namely the set of states that can reach the final set within a given event interval or at a fixed event step, respectively. We show that, in both cases, the computation can be done exactly via manipulations of differencebound matrices. Furthermore, we illustrate the application of the backward reachability computations over safety and transient analysis of MPL systems.
Computational Techniques for Reachability Analysis of MaxPlusLinear Systems ⋆
"... Abstract This work discusses a computational approach to reachability analysis of MaxPlusLinear (MPL) systems, a class of discreteevent systems widely used in synchronization and scheduling applications. Given a set of initial states, we characterize and compute its "reach tube," namely ..."
Abstract
 Add to MetaCart
Abstract This work discusses a computational approach to reachability analysis of MaxPlusLinear (MPL) systems, a class of discreteevent systems widely used in synchronization and scheduling applications. Given a set of initial states, we characterize and compute its "reach tube," namely the collection of set of reachable states (regarded stepwise as "reach sets"). By an alternative characterization of the MPL dynamics, we show that the exact computation of the reach sets can be performed quickly and compactly by manipulations of differencebound matrices, and further derive worstcase bounds on the complexity of these operations. The approach is also extended to backward reachability analysis. The concepts and results are elucidated by a running example, and we further illustrate the performance of the approach by a numerical benchmark: the technique comfortably handles twentydimensional MPL systems (i.e., with twenty continuous state variables), and as such it outperforms the stateoftheart alternative approaches in the literature.
Improved Matrix Pair . . .
, 2006
"... We improve undecidability bounds for problems involving two integer matrices. We prove ..."
Abstract
 Add to MetaCart
We improve undecidability bounds for problems involving two integer matrices. We prove
unknown title
, 2006
"... Algèbres maxplus et mathématiques de la décision/Maxplus algebras and mathematics of decision ..."
Abstract
 Add to MetaCart
Algèbres maxplus et mathématiques de la décision/Maxplus algebras and mathematics of decision