We evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets. This is applied in two different directions. The first is the determination of whether or not several classes of linear functionals satisfy the uniform Central Limit Theorem or the uniform Law of Large Numbers. The second direction, which was the main motivation of this study, is the ...

In this paper we present two approaches to estimate the Hausdorff dimension of an invariant compact set of a dynamical system: the method of characteristic exponents (estimates of KaplanYorke type) and the method of Lyapunov functions. In the first approach, using Lyapunov's first method we exploit charachteristic exponents for obtaining such estimate. A close relationship with uniform asymptot...

Abstract: Let X be a building of uniform thickness q+1. L-Betti numbers of X are reinterpreted as von-Neumann dimensions of weighted L-cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness q. The weighted cohomology makes sense for all real positive values of q, and is computed for small q. If the Davis comp...

In this paper we develop a solution to the discrete-time robust output feedback control problem for Linear Time-Varying (LTV) systems. The solution is developed along the strategy set up in [1] and the main ingredient in its derivation is the extension of the well-known bounded real lemma in a (discrete) timevarying context, developed in [2]. This approach contributes to the conceptual simplici...

For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik–Chervonenkis (VC) dimension or (ii) for every ε > 0 there is a finite partition π such the essential π -boundary of each set has measure at most ε. Immediate corollaries include the fact that a separable family with finite VC dimension has finite bracketing numbers, and satisfies uni...

Gaussian particles provide a flexible framework for modelling and simulating three-dimensional star-shaped random sets. In our framework, the radial function of the particle arises from a kernel smoothing, and is associated with an isotropic random field on the sphere. If the kernel is a von Mises–Fisher density, or uniform on a spherical cap, the correlation function of the associated random f...

We give a computer free proof of the Deligne, Cohen and deMan formulas for the dimensions of the irreducible g-modules appearing in g, k ≤ 4, where g ranges over the exceptional complex simple Lie algebras. We give additional dimension formulas for the exceptional series, as well as uniform dimension formulas for other representations distinguished by Freudenthal along the rows of his magic cha...

Suppose G is a hyperbolic group whose boundary ∂∞G has topological dimension k. If ∂∞G is quasi-symmetrically homeomorphic to an Ahlfors kregular metric space, then, modulo a finite normal subgroup, G is isomorphic to a uniform lattice in the isometry group Isom(Hk+1) of hyperbolic (k+1)-space.

A thorough account is given of the derivation of uniform semiclassical approximations to the particle and kinetic energy densities of N noninteracting bounded fermions in one dimension. The employed methodology allows the inclusion of non-perturbative effects via an infinite resummation of the Poisson summation formula.

We study generalized Horn-Kapranov rational parametrizations of inhomogeneous sparse discriminants from both a theoretical and an algorithmic perspective. We show that all these discriminantal parametrizations are birational, we prove some results on the corresponding implicit equations, and we propose a combinatorial algorithm to compute their degree in the uniform case of (co)dimension 3.