20,485 research outputs found
Galois differential algebras and categorical discretization of dynamical systems
A categorical theory for the discretization of a large class of dynamical
systems with variable coefficients is proposed. It is based on the existence of
covariant functors between the Rota category of Galois differential algebras
and suitable categories of abstract dynamical systems. The integrable maps
obtained share with their continuous counterparts a large class of solutions
and, in the linear case, the Picard-Vessiot group.Comment: 19 pages (examples added
Duality and separation theorems in idempotent semimodules
We consider subsemimodules and convex subsets of semimodules over semirings
with an idempotent addition. We introduce a nonlinear projection on
subsemimodules: the projection of a point is the maximal approximation from
below of the point in the subsemimodule. We use this projection to separate a
point from a convex set. We also show that the projection minimizes the
analogue of Hilbert's projective metric. We develop more generally a theory of
dual pairs for idempotent semimodules. We obtain as a corollary duality results
between the row and column spaces of matrices with entries in idempotent
semirings. We illustrate the results by showing polyhedra and half-spaces over
the max-plus semiring.Comment: 24 pages, 5 Postscript figures, revised (v2
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