Catálogo de publicaciones - libros

A -additive measure over a space of paths is constructed to give the solution to the Fokker-Planck equation associated with a stochastic differential equation with coefficient function of polynomial growth by making use of nonstandard analysis.

We survey recent results on existence of optimal controls for stochastic Navier-Stokes equations in 2 and 3 dimensions using Loeb space methods.

Consider the equation: together with periodic boundary conditions and initial condition (, 0) = (). This corresponds a Navier-Stokes problem where the incompressibility condition has been dropped. The major difficulty in existence proofs for this simplified problem is the unbounded advection term, ( · ∇).

We present a proof of local-in-time existence of a smooth solution based on a discretization by a suitable Euler scheme. It will be shown that this solution exists in an interval [0, ), where ≤ 1/, with depending only on and the values of the Lipschitz constants of and at time 0. The argument given is based directly on local estimates of the solutions of the discretized problem.

Abraham Robinson discovered a rigorous approach to calculus with infinitesimals in 1960 and published it in []. This solved a 300 year old problem dating to Leibniz and Newton. Extending the ordered field of (Dedekind) “real” numbers to include infinitesimals is not difficult algebraically, but calculus depends on approximations with transcendental functions. Robinson used mathematical logic to show how to extend all real functions in a way that preserves their propertires in a precise sense. These properties can be used to develop calculus with infinitesimals. Infinitesimal numbers have always fit basic intuitive approximation when certain quantities arc “small enough,” but Leibniz, Euler, and many others could not make the approach free of contradiction. Section 1 of this article uses some intuitive approximations to derive a few fundamental results of analysis. We use approximate equality, x ≈ y, only in an intuitive sense that “x; is sufficiently close to ”.

This paper is a follow-up of K. Hrbacek’s article showing how his approach can be pedagogically helpful when introducing analysis at pre-university level.

Conceptual difficulties arise in elementary pedagogical approaches. In most cases it remains difficult to explain at pre-university level how the derivative is calculated at nonstandard values or how an internal function is defined. Hrbacek provides a modified version of IST [] (rather Péraire’s RIST) which seems to reduce all these difficulties. This system is briefly presented here in its pedagogical form with an application to the derivative. It must be understood as a state-of-the-art report.