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Smallest singular value of sparse random matrices

Alexander E. Litvak, Omar Rivasplata (2012)

Studia Mathematica

We extend probability estimates on the smallest singular value of random matrices with independent entries to a class of sparse random matrices. We show that one can relax a previously used condition of uniform boundedness of the variances from below. This allows us to consider matrices with null entries or, more generally, with entries having small variances. Our results do not assume identical distribution of the entries of a random matrix and help to clarify the role of the variances of the entries....

Some Borel measures associated with the generalized Collatz mapping

K. Matthews (1992)

Colloquium Mathematicae

This paper is a continuation of a recent paper [2], in which the authors studied some Markov matrices arising from a mapping T:ℤ → ℤ, which generalizes the famous 3x+1 mapping of Collatz. We extended T to a mapping of the polyadic numbers ℤ ^ and construct finitely many ergodic Borel measures on ℤ ^ which heuristically explain the limiting frequencies in congruence classes, observed for integer trajectories.

Some remarks about strong proximality of compact flows

A. Bouziad, J.-P. Troallic (2009)

Colloquium Mathematicae

This note aims at providing some information about the concept of a strongly proximal compact transformation semigroup. In the affine case, a unified approach to some known results is given. It is also pointed out that a compact flow (X,𝓢) is strongly proximal if (and only if) it is proximal and every point of X has an 𝓢-strongly proximal neighborhood in X. An essential ingredient, in the affine as well as in the nonaffine case, turns out to be the existence of a unique minimal subset.

Sparse recovery with pre-Gaussian random matrices

Simon Foucart, Ming-Jun Lai (2010)

Studia Mathematica

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 ℓ₁-minimization under the optimal condition m ≥ csln(eN/s). The main ingredient of the proof is a variation of a classical Restricted Isometry Property, where the inner norm becomes the ℓ₁-norm and the outer norm depends on probability distributions.

Spectral distribution of the free Jacobi process associated with one projection

Nizar Demni, Taoufik Hmidi (2014)

Colloquium Mathematicae

Given an orthogonal projection P and a free unitary Brownian motion Y = ( Y ₜ ) t ≥ 0 in a W*-non commutative probability space such that Y and P are *-free in Voiculescu’s sense, we study the spectral distribution νₜ of Jₜ = PYₜPYₜ*P in the compressed space. To this end, we focus on the spectral distribution μₜ of the unitary operator SYₜSYₜ*, S = 2P - 1, whose moments are related to those of Jₜ via a binomial-type expansion already obtained by Demni et al. [Indiana Univ. Math. J. 61 (2012)]. In this connection,...

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