A (k, ε)-biased sample space is a distribution over {0, 1}n that ε-fools every nonempty linear test of size at most k. Since they were introduced by Naor and Naor (SIAM J. Computing, 1993), these sample spaces have become a central notion in theoretical computer science with a variety of applications. When constructing such spaces, one usually attempts to minimize the seed length as a function ...

Abstract We prove a Nekhoroshev type theorem for the nonlinear Schrödinger equation iut = −∆u+ V ⋆ u+ ∂ūg(u, ū) , x ∈ T, where V is a typical smooth potential and g is analytic in both variables. More precisely we prove that if the initial datum is analytic in a strip of width ρ > 0 with a bound on this strip equals to ε then, if ε is small enough, the solution of the nonlinear Schrödinger equa...

Let 1 ≤ t ≤ 7 be an integer and let F be a k-uniform hypergraph on n vertices. Suppose that |A∩B∩C∩D| ≥ t holds for all A,B,C,D ∈ F . Then we have |F | ≤ (n−t k−t ) if | k n − 2 |< ε holds for some ε > 0 and all n > n0(ε). We apply this result to get EKR type inequalities for “intersecting and union families” and “intersecting Sperner families.”

This example is studied using Floquet theory. A classic example is k(t) = 1 + ε cosωt, where one finds regions of instability in the ε, ω plane for which there are solutions which grow exponentially in time. The instability regions are open and with closures touching ε = 0 at critical frequencies (see [A], [MW]). For any t, the constant coefficient problems with k frozen at k(t) is conservative...

Degenerations, contractions and deformations of various algebraic structures play an important role in mathematics and physics. There are many different definitions and special cases of these notions. We try to give a general definition which unifies these notions and shows the connections among them. Here we focus on contractions of Lie algebras and algebraic groups. 1. Contractions, degenerat...

New high-order accurate finite difference schemes based on defect correction are considered for an initial boundary-value problem on an interval for singularly perturbed parabolic PDEs with convection; the highest space derivative in the equation is multiplied by the perturbation parameter ε, ε ∈ (0, 1]. Solutions of the well-known classical numerical schemes for such problems do not converge ε...

We provide a complete description of the giant component of the Erdős-Rényi random graph G(n, p) as soon as it emerges from the scaling window, i.e., for p = (1 + ε)/n where εn → ∞ and ε = o(1). Our description is particularly simple for ε = o(n), where the giant component C1 is contiguous with the following model (i.e., every graph property that holds with high probability for this model also ...

A matrix A ∈ Cq×N satisfies the restricted isometry property of order k with constant ε if it preserves the l2 norm of all k-sparse vectors up to a factor of 1± ε. We prove that a matrix A obtained by randomly sampling q = O(k · log k · logN) rows from an N × N Fourier matrix satisfies the restricted isometry property of order k with a fixed ε with high probability. This improves on Rudelson an...

We briefly survey some recent progress on list decoding algorithms for binary codes. The results discussed include: – Algorithms to list decode binary Reed-Muller codes of any order up to the minimum distance, generalizing the classical GoldreichLevin algorithm for RM codes of order 1 (Hadamard codes). These algorithms are “local” and run in time polynomial in the message length. – Construction...

In metallic wires, electron-electron interactions lead to quasi-elastic scattering, which modifies the electron phase-coherence time, and to inelastic scattering, which redistributes energy between electrons. Several recent experiments have shown that these properties can be sampledependent 1,2. We have found furthermore that the low-temperature saturation of the phasecoherence time τφ of quasi...