We prove that the moduli of U-convexity, introduced by Gao (1995), of the ultrapower X̃ of a Banach space X and of X itself coincide whenever X is super-reflexive. As a consequence, some known results have been proved and improved. More precisely, we prove that uX(1) > 0 implies that both X and the dual space X∗ of X have uniform normal structure and hence the “worth” property in Corollary 7 of ...

Let α ≥ 1 and let (X, d, μ) be an α-homogeneous metric measure space with conformal Assouad dimension equal to α. Then there exists a weak tangent of (X, d, μ) with uniformly big 1-modulus.

Let T (x; r) denote the total occupation measure of the ball of radius r centered at x for Brownian motion in IR 3. We prove that sup jxjj1 T (x; r)=(r 2 jlog rj) ! 16== 2 a.s. as r ! 0, thus solving a problem posed by Taylor in 1974. Furthermore, for any a 2 (0; 16== 2), the Hausdorr dimension of the set of \thick points" x for which limsup r!0 T (x; r)=(r 2 jlog rj) = a, is almost surely 2 ? ...

In this paper we consider the Interband Light Absorption Coefficient (ILAC), in a symmetric form, in the case of random operators on the d-dimensional lattice. We show that the symmetrized version of ILAC is either continuous or has a component which has the same modulus of continuity as the density of states.

It is known that both Watson-Nadaraya and Gasser-Muller types of regression estimators have some disadvantages. A smooth version of local polynomial regression estimators are proposed to remedy the disadvantages. The mean squared error and mean integrated squared errors are computed explicitly. It turns out that by suitably selecting a kernel and a bandwidth, the proposed estimator has at least...

The theory of Eisenstein series is fundamental for the spectral theory of automorphic forms. It was first developed by Selberg, and was completed by Langlands ([Lan76]; see also [MW95]). There are several known proofs for the meromorphic continuation of Eisenstein series (apart from very special cases of Eisenstein series which can be expressed in terms of Tate integrals). In all these proofs i...

In Part I, we construct a class of examples of initial velocities for which the unique solution to the Euler equations in the plane has an associated flow map that lies in no Hölder space of positive exponent for any positive time. In Part II, we explore inverse problems that arise in attempting to construct an example of an initial velocity producing an arbitrarily poor modulus of continuity o...

This article provides a detailed analysis of the behavior of suprema and moduli of continuity for a large class of random fields which generalize Gaussian processes, sub-Gaussian processes, and random fields that are in the nth chaos of a Wiener process. An upper bound of Dudley type on the tail of the random field’s supremum is derived using a generic chaining argument; it implies similar resu...

We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form ∂t−∆+ b ·∇ resp. −∆+ b ·∇ with a divergence-free drift b. We prove the Liouville theorem and Harnack inequality when b ∈ L∞(BMO) resp. b ∈ BMO−1 and provide a counterexample demonstra...