In this paper, we present some new results on the thinnest coverings that can be obtained in Hamming or Euclidean spaces if spheres and ellipsoids are covered with balls of some radius ε. In particular, we tighten the bounds currently known for the ε-entropy of Hamming spheres of an arbitrary radius r. New bounds for the ε-entropy of Hamming balls are also derived. If both parameters ε and r ar...

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper an approximation ρ̂ that is based on ellipsoid norms, that can be computed by...

In this paper, we present some new results on the thinnest coverings that can be obtained in Hamming or Euclidean spaces if spheres and ellipsoids are covered with balls of some radius ε. In particular, we tighten the bounds currently known for the ε-entropy of Hamming spheres of an arbitrary radius r. New bounds for the ε-entropy of Hamming balls are also derived. If both parameters ε and r ar...

We establish bounds on the spectral radii for a large class of sparse random matrices, which includes the adjacency matrices of inhomogeneous Erdős-Rényi graphs. For the Erdős-Rényi graph G(n, d/n), our results imply that the smallest and second-largest eigenvalues of the adjacency matrix converge to the edges of the support of the asymptotic eigenvalue distribution provided that d log n. Toget...

We study the wall-crossing behavior of the index of BPS states forD4−D2−D0 brane systems on a Calabi-Yau 3-fold at large radius and point out that not only is the “BPS index at large radius” chamber-dependent, but that the changes in the index can be large in the sense that they dominate single-centered black hole entropy. We discuss implications for the weak coupling OSV conjecture. We also an...

We prove explicit bounds on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set S ⊂ Rk defined by a quantifier-free formula involving s polynomials in Z[X1, . . . ,Xk] having degrees at most d, and whose coefficients have bitsizes at most τ . Our bound is an explicit function of s, d, k and τ , and does not contain ...

Given two independent Poisson point processes Φ(1),Φ(2) in Rd, the AB Poisson Boolean model is the graph with points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centred at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolat...

Dynamics and morphology of hole growth in a film of power hardening viscoplastic solid [ yield stress approximately (strain-rate)(n)] is investigated. At short times the growth is exponential and depends on the initial hole size. At long times, for n>1 / 3, the growth is again exponential but with a different exponent. However, for n<1 / 3 the hole growth slows and the hole radius approaches an...

By applying the Edwards self-consistent field theory, we calculate the polymer density and free energy excesses caused by the presence of nanocolloids in the excluded volume polymer system. Using the obtained results, we have calculated the depletion potential U as a function of the separation between colloids, colloid radius, polymer volume fraction, and polymer gyration radius. Upon analyzing...

We introduce and discuss shape based models for finding the best interpolation data when reconstructing missing regions in images by means of solving the Laplace equation. The shape analysis is done in the framework of Γ-convergence, from two different points of view. First, we propose a continuous PDE model and get pointwise information on the ”importance” of each pixel by a topological asympt...