Nondifferentiable Dynamic: Two Examples

Dynamic algebras: Examples, constructions, applications

Dynamic algebras combine the classes of Boolean (B ∨ ′ 0) and regular (R ∪ ; ∗) algebras into a single finitely axiomatized variety (B R 3) resembling an R-module with “scalar” multiplication 3. The basic result is that ∗ is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynam...

Nondifferentiable Optimization via Approximation*

Optimization problems with nondifferentiable cost functionals, particularly minimax problems, have received considerable attention recently since they arise naturally in a variety of contexts. Optimality conditions for such problems have been derived by several authors while a number of computational methods have been proposed for their solution (the reader is referred to [1] for:a fairly compl...

Stably ergodic approximation: two examples

It has been conjectured that the stably ergodic diieomorphisms are open and dense in the space of volume-preserving, partially hyperbolic diieomorphisms of a compact manifold. In this paper we deal with two recalcitrant examples; the standard map cross Anosov and the ergodic automorphisms of the four torus. In both cases we show that they may be approximated by stably ergodic diieomorphisms whi...

Two Examples of Affine Manifolds

An affine manifold is a manifold with a distinguished system of affine coordinates, namely, an open covering by charts which map homeomorphically onto open sets in an affine space E such that on overlapping charts the homeomorphisms differ by an affine automorphism of E. Some, but certainly not all, affine manifolds arise as quotients Ω/Γ of a domain in E by a discrete group Γ of affine transfo...

Approximate Dynamic Programming by Practical Examples