It is shown that the initial value problem (IVP) of the generalized KdV equation @tu + @x(a(u)) + @ 3 xu = 0; u(x; 0) = (x) is well posed in the classical Sobolev space H(R) with s > 3=4, which thus establishes a nonlinear map K from H(R) to C([ T; T ];H(R)). Then it is proved that (i) if a = a(x) is a C function on R to R, then K is in nitely many times Frechet di erentiable; (ii) if a = a(x) ...

probability 1. (Note that we do not assume any relation of dependence or indepenaence between X(r) and X(s) for m $ n.) 1 See M. Frechet, Recherches theoriques modernes, Vol. 1, Paris, 1937, p. 27. 2 See H. Cram&r, Mathematical methods of statistics, Princeton, 1946, p. 93. Compare F. P. Cantelli, Considerazioni sulla legge uniforme dei grandi numeri ecc., Giornale dell 'Istituto Italiano degli...

We show that in two dimensions or higher, the Mordukhovich-loffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Frechet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with ...

Consider two (n−1)-dimensional manifolds, S and S in R. We say that they are normalcompatible when the closest projection of each one onto the other is a homeomorphism. We give a tight condition under which S and S are normal-compatible. It involves the minimum feature size of S and of S and the Hausdorff distance between them. Furthermore, when S and S are normal-compatible, their Frechet dist...

Several generalized Wallman type spaces are considered as well as various lattices of subsets therein. In particular, regularity of these lattices and consequences are investigated. Also considered are necessary and sufficient conditions for these lattices to be Lindel6f as well as replete, prime complete, and fully replete.

We prove the existence of designs of small size in a number of contexts. In particular our techniques can be applied to prove the existence of ndesigns on S of size Od(n d log(n)d−1).

A separable Banach space X contains l1 isomorphically if and only if X has a bounded fundamental total wc∗0-stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total wc∗0-biorthogonal system.