Let A be a subset of a finite field F := Z/qZ for some prime q. If |F | Î´ < |A| < |F | 1âˆ’Î´ for some Î´ > 0, then we prove the estimate |A + A| + |A Â· A| â‰¥ c(Î´)|A| 1+Îµ for some Îµ = Îµ(Î´) > 0. This is aâ€¦ (More)

Given a monotone graph property P consider p P the proba bility that a random graph with edge probability p will have P The function d p P dp is the key to understanding the threshold behavior of theâ€¦ (More)

is globally wellposed in time. More precisely, we obtain a unique solution u = uÏ† âˆˆ CH1([0,âˆž[) such that for all time, u(t) depends continuously on the data Ï† (in fact, the dependence is even realâ€¦ (More)

University Press. All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage andâ€¦ (More)

where Q = (0, 2Ï€). The notations L,W , etc. refer to L(T),W (T), etc. or to 2Ï€-periodic functions in Lploc(R),W 1,p loc (R), etc. We denote by L p # the space of functions in L satisfying (1.2).â€¦ (More)

We give a new explicit construction of n Ã— N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some Îµ > 0, large N and any n satisfying N1âˆ’Îµ â‰¤ n â‰¤ N , we construct RIP matricesâ€¦ (More)

A subset A of f0; 1g n is called monotone provided if x 2 A; x 0 2 f0; 1g n ; x i x 0 i for i = 1; : : : ; n then x 0 2 A. For 0 p 1, deene p the product measure on f0; 1g n with weights 1 ? p at 0â€¦ (More)