In previous lectures we defined a randomness extractor as follow, Definition 1 Function Ext : {0, 1}n × {0, 1}d → {0, 1}m is (k, ε)extractor if for any distribution W ∈ {0, 1}n, with H∞(W ) ≥ k SD(Ext(W ; I), Um) ≤ ε. ♦ If (I,Ext(W ; I) ≈ε (I, Um) we say Ext is a strong extractor. We saw that if H = {hi : {0, 1}n → {0, 1}m|∀i ∈ {0, 1}d} is a universal hash family and m ≤ k − 2 log(1/ε) then Ext...

We describe a deterministic algorithm that, for constant k, given a k-DNF or kCNF formula φ and a parameter ε, runs in time linear in the size of φ and polynomial in 1/ε and returns an estimate of the fraction of satisfying assignments for φ up to an additive error ε. For k-DNF, a multiplicative approximation is also achievable in time polynomial in 1/ε and linear in the size of φ. Previous alg...

In this paper we prove higher integrability results for vector fields B,E, (B,E) ∈ L2− (Ω, R) × L2−ε(Ω, R), ε small, such that div B = 0, curl E = 0 satisfying a “reverse” inequality of the type |B| + |E| ≤ ( K + 1 K ) 〈B,E〉+ |F | with K ≥ 1 and F ∈ L(Ω, R), r > 2 − ε. Applications to the theory of quasiconformal mappings and partial differential equations are given. In particular, we prove reg...

and Applied Analysis 3 Proposition 3. For any constant R ≥ 0 there exists a positive constant K = K(R) such that, for any initial data u 1 (0), u 2 (0) with ‖u i (0)‖ Hη,ε ≤ R, i = 1, 2 one has 󵄩󵄩󵄩󵄩 S ε,η (t)u 1 (0) − S ε,η (t)u 2 (0) 󵄩󵄩󵄩󵄩Hη,ε ≤ e (K 2 /ε 2 )t󵄩󵄩󵄩󵄩u1 (0) − u2 (0) 󵄩󵄩󵄩Hη,ε , (20) where S ε,η (t) is the solution semigroup of the problem (1). Proof. Let u 1 , u 2 , two solutions of ...

Let k be a an algebraically closed field of arbitrary characteristic, and we let h : A(k(t)) −→ R≥0 be the usual Weil height for the n-dimensional affine space corresponding to the function field k(t) (extended to its algebraic closure). We prove that for any affine variety V ⊂ A defined over k(t), there exists a positive real number ε := ε(V ) such that if P ∈ V (k(t)) and h(P ) < ε, then P ∈ ...

We propose an algorithm to sample and mesh a k-submanifold M of positive reach embedded in R. The algorithm first constructs a crude sample of M. It then refines the sample according to a prescribed parameter ε, and builds a mesh that approximates M. Differently from most algorithms that have been developped for meshing surfaces of R, the refinement phase does not rely on a subdivision of R (su...

This paper investigates the following natural greedy procedure for clustering in the bi-criterion setting: iteratively grow a set of centers, in each round adding the center from a candidate set that maximally decreases clustering cost. In the case of k-medians and k-means, the key results are as follows. • When the method considers all data points as candidate centers, then selecting O(k log(1...

Let F : X → X be a C 2 loc map in a Banach space X, and A be its Frèchet derivative at the element w := w ε , which solves the problem (*) ˙ w = −A −1 ε (F (w) + εw), w(0) = w 0 , where A ε := A + εI. Assume that A −1 ε ≤ cε −k , 0 < k ≤ 1, 0 < ε > ε 0. Then (*) has a unique global solution, w(t), there exists w(∞), and (* *) F (w(∞)) + εw(∞) = 0. Thus the DSM (Dynamical Systems Method) is just...