Broadband frequency-selective fading channels usually have the inherent sparse nature. By exploiting the sparsity, adaptive sparse channel estimation (ASCE) methods, e.g., reweighted L1-norm least mean square (RL1-LMS), could bring a performance gain if additive noise satisfying Gaussian assumption. In real communication environments, however, channel estimation performance is often deteriorate...

The focus of this survey paper is on the distribution function FNβ(t) for the largest eigenvalue in the finite N Gaussian Orthogonal Ensemble (GOE, β = 1), the Gaussian Unitary Ensemble (GUE, β = 2), and the Gaussian Symplectic Ensemble (GSE, β = 4) in the edge scaling limit of N → ∞. These limiting distribution functions are expressible in terms of a particular Painlevé II function. Comparison...

The Short Integer Solution (SIS) and Learning With Errors (LWE) problems are the foundations for countless applications in latticebased cryptography, and are provably as hard as approximate lattice problems in the worst case. An important question from both a practical and theoretical perspective is how small their parameters can be made, while preserving their hardness. We prove two main resul...

Given an involutive negator N and a leftcontinuous t-norm T whose contour line C0 is continuous on ]0, 1], we build a rotationinvariant t-norm from a rescaled version of T and its left, right and front rotation. Depending on the involutive negator N and the set of zero divisors of T , some reshaping of the rescaled version of T may occur during the rotation process. The rescaled version of T it...

The paper is concerned with best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial is taken in L2 with the Gegenbauer weight corresponding to a parameter α , while the derivative is measured in L2 with the Gegenbauer weight for a parameter β . Under the assumption that β −α is an intege...

For an m × N underdetermined system of linear equations with independent pre-Gaussian random coefficients satisfying simple moment conditions, it is proved that the s-sparse solutions of the system can be found by `1-minimization under the optimal condition m ≥ c s ln(eN/s). The main ingredient of the proof is a variation of a classical Restricted Isometry Property, where the inner norm becomes...

There’s a different way to go about the problem of finding all ways of writing n = x + y. Recall that in the Gaussian integers Z[i] we have an automorphism α 7→ ᾱ. Thus the function Nα = αᾱ = x + y is multiplicative. Thus our question is just to find all ways of writing n as a norm from the Gaussian integers. To answer this question we need to know a little about the structure of Z[i]. (Note th...

this formula defines a Schwartz function, and hence the solution u = uf ∈ S also, and the mapping f 7→ uf is continuous in the sense that if fn is a sequence of Schwartz functions such that ‖fn − f‖α,β → 0 for every Schwartz semi-norm ‖ · ‖α,β , then also ‖un − u‖α,β → 0 for every Schwartz semi-norm, where un = ufn , u = uf . In fact the formula above extends to define a distributional solution...

We present a simple numerical method for constructing the optimal (generalized) averaged Gaussian quadrature formulas which are the optimal stratified extensions of Gauss quadrature formulas. These extensions exist in many cases in which real positive Kronrod formulas do not exist. For the Jacobi weight functions w(x) ≡ w(α,β)(x) = (1− x)α(1 + x)β (α, β > −1) we give a necessary and sufficient ...

This paper proposes solution to the following non-convex optimization problem: min || x || p subject to || y Ax || q Such an optimization problem arises in a rapidly advancing branch of signal processing called ‘Compressed Sensing’ (CS). The problem of CS is to reconstruct a k-sparse vector xnX1, from noisy measurements y = Ax+ , where AmXn (m<n) is the measurement matrix and mX1 is additive no...