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...

Several estimates for the convolution function C[f (x)] := x 1 f (y)f (x/y)(dy/y) and its iterates are obtained when f (x) is a suitable number-theoretic error term. We deal with the case of the asymptotic formula for T 0 |ζ(1/2 + it)| 2k dt (k = 1, 2), the general Dirichlet divisor problem, the problem of nonisomorphic Abelian groups of given order, and the Rankin-Selberg convolution. 1. Convo...

COVID-19 reached its fourth year of pandemic since 2020. The repeated waves infections have been driven by multiple factors such as pathological traits variants, diagnostic accuracy, and vaccination conditions. This study revisits analyzes the dynamic processes viral transmission to generate new scientific knowledge. A cascade model from one case another was developed, theoretically analyzed ho...

and Applied Analysis 3 It is worth mentioning that in the absence of the white noise, the existence of a global attractor for 1.3 1.4 was established in 40 . The main contribution of this paper is to extend the method of tail estimates to stochastic retarded LDSs and prove the existence of a random attractor for the infinite dimensional random dynamical system generated by stochastic retarded L...

A family of permutations F of [n] = {1, 2, . . . , n} is (ε, k)-min-wise independent if for every nonempty subset X of at most k elements of [n], and for any x ∈ X, the probability that in a random element π of F , π(x) is the minimum element of π(X), deviates from 1/|X| by at most ε/|X|. This notion can be defined for the uniform case, when the elements of F are picked according to a uniform d...

We consider the problem of designing (or augmenting) an electric power system such that it satisfies the N-k-ε survivability criterion while minimizing total cost. The survivability criterion requires that at least (1− ε) fraction of the total demand can still be met even if any k (or fewer) of the system components fail. We formulate this problem, taking into account both transmission and gene...

Let Xm,ε be the distribution over m bits (X1, . . . , Xm) where the Xi are independent and each Xi equals 1 with probability (1 + ε)/2 and 0 with probability (1 − ε)/2. We consider the smallest value ε∗ of ε such that the distributions Xm,ε and Xm,0 can be distinguished with constant advantage by a function f : {0, 1}m → S which is the product of k functions f1, f2, . . . , fk on disjoint input...