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 ? ...

We characterize the functions defined on a weighted space, which are uniformly approximated by a sequence of positive linear operators and we obtain the range of the weights which can be used for uniform approximation. We, also, obtain an estimation of the remainder in terms of the usual modulus of continuity. We give particular results for the Szász-Mirakjan and Baskakov operators. Mathematics...