A double sequence x = {xk,l} of points in R is slowly oscillating if for any given ε > 0, there exist α = α(ε) > 0, δ = δ(ε) > 0, and N = N(ε) such that |xk,l−xs,t| < ε whenever k, l ≥ N(ε) and k ≤ s ≤ (1 + α)k, l ≤ t ≤ (1 + δ)l. We study continuity type properties of factorable double functions defined on a double subset A × A of R into R, and obtain interesting results related to uniform cont...

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...

The interplay between multiscale homogenization and dimension reduction for nonlinear elastic thin plates is analyzed in the case in which the scaling of the energy corresponds to Kirchhoff’s nonlinear bending theory for plates. Different limit models are deduced depending on the relative ratio between the thickness parameter h and the two homogenization scales ε and ε.

In this paper we deal with singular perturbations of nonlinear problems depending on a small parameter ε > 0. First we consider the abstract theory of singular perturbations of variational inequalities involving some nonlinear operators, defined in Banach spaces, and describe the asymptotic behaviour of these solutions when ε → 0. Then these abstract results are applied to some boundary value p...

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 ∈ ...