Substitution in Nash functions

Global Problems on Nash Functions

This is a survey on the history of and the solutions to the basic global problems on Nash functions, which have been only recently solved, namely: separation, extension, global equations, Artin-Mazur description and idempotency, also noetherianness. We discuss all of them in the various possible contexts, from manifolds over the reals to real spectra of arbitrary commutative rings. Nash functio...

Nash Manifolds and Schwartz Functions on Them

These are the lecture notes for my talk on December 1, 2014 at the BIRS workshop “Motivic Integration, Orbital Integrals, and Zeta-Functions”. 1. Semi-algebraic sets and the Seidenberg-Tarski theorem In this section we follow [BCR]. Definition 1.1. A subset A ⊂ R is called a semi-algebraic set if it can be presented as a finite union of sets defined by a finite number of polynomial equalities a...

Nash Equilibria in Discrete Routing Games with Convex Latency Functions

We study Nash equilibria in a discrete routing game that combines features of the two most famous models for non-cooperative routing, the KP model [16] and the Wardrop model [27]. In our model, users share parallel links. A user strategy can be any probability distribution over the set of links. Each user tries to minimize its expected latency, where the latency on a link is described by an arb...

Extremal functions for the sharp L− Nash inequality

The Continuous Newton's Method, Inverse Functions, and Nash-Moser

The conventional Newton’s method for finding a zero of a function F : R → R, assuming that (F ′(y))−1 exists for at least some y in R, is the familiar iteration: pick z0 in R n and define zk+1 = zk − (F ′(zk))−1F (zk) (k = 0, 1, 2, . . . ), hoping that z1, z2, . . . converges to a zero of F . What can stop this process from finding a zero of F ? For one thing, there might not be a zero of F . F...