We consider the non uniform AdS black string equations in arbitrary number of dimension in a perturbative approach up to order 2. We restrict the study in order 2 to the backreacting modes, since they provide the first relevant corrections on the thermodynamical quantities of the solutions. Our results suggests the existence of a new stable phase for AdS non uniform black strings, namely long n...

VC-dimension is the measure of model complexity (capacity) used in VC-theory. The knowledge of the VC-dimension of an estima-2 tor is necessary for rigorous complexity control using analytic VC generalization bounds. Unfortunately, it is not possible to obtain the analytic estimates of the VC-dimension in most cases. Hence, it has been recently proposed to measure the VC-dimension of an estimat...

Uniform goethite nanotubes were synthesized from the reaction of hydrazine with Fe(III)-oleate complex immobilized in reverse micelles. The nanotubes have interesting parallelogram cross section with uniform edge dimension of as small as 7 nm. The edge dimensions and lengths of the nanotubes were easily controlled by varying the reaction conditions.

In this paper the geometric dimension of an oriented matroid is introduced. It is the minimal euclidian dimension where its separoid (to be defined) can be realized as a family of convex sets. We show that in the uniform case, it is enough to know this invariant to decide if the oriented matroid is linear .

We prove that the uniform spanning forests of Z and Z have qualitatively different connectivity properties whenever ` > d ≥ 4. In particular, we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex, which we call the component graph. We show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and...

We prove upper and lower bounds on the VC dimension of sparse univariate polynomi-als over reals, and apply these results to prove uniform learnability of sparse polynomials and rational functions. As another application we solve an open problem of Vapnik ((Vap-nik 82]) on uniform approximation of the general regression functions, a central problem of computational statistics (cf. Vapnik 82]), ...