We investigate graph-directed iterated function systems in mixed Euclidean and p-adic spaces. Hausdorff measure and Hausdorff dimension in such spaces are defined, and an upper bound for the Hausdorff dimension is obtained. The relation between the Haar measure and the Hausdorff measure is clarified. Finally, we discus an example in R×Q2 and calculate upper and lower bounds for its Hausdorff di...

Given 2 ≤ k ≤ n, the minimal (n− 1)-dimensional Gaussian measure of the union of the boundaries of k disjoint sets of equal Gaussian measure in Rn whose union is Rn is of order √ log k. A similar results holds also for partitions of the sphere Sn−1 into k sets of equal Haar measure.

For the action of an algebraic cellular automaton on a Markov subgroup, we show that the Cesàro mean of the iterates of a Markov measure converges to the Haar measure. This is proven by using the combinatorics of the binomial coefficients on the regenerative construction of the Markov measure.

In this paper we characterise, in terms of the upper Dini derivative, when the Clarke subdiierential mapping of a real-valued locally Lipschitz function is a minimal weak cusco. We then use this characterisation to deduce some new results concerning Lips-chitz functions with minimal subdiierential mappings. In the papers 7] and 1], the authors investigate a class of locally Lipschitz functions ...

In recent years, there has been greater attempt to find numerical solutions of differential equations using wavelet's methods. The following method is based on vector forms of Haar-wavelet functions. In this paper, we will introduce one dimensional Haar-wavelet functions and the Haar-wavelet operational matrices of the fractional order integration. Also the Haar-wavelet operational matrices of ...

LetMn be a random n n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of Mn to converge to a Gaussian limit as n ! 1. By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenva...

Let M be a monoid (e.g. N, Z, or Z), and A an abelian group. A is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F : A −→ A that commutes with all shift maps. Let μ be a (possibly nonstationary) probability measure on A; we develop sufficient conditions on μ and F so that the sequence {Fμ}N=1 weak*-converges to the Haar measure on A, in density (and...