In the first part of this paper we prove that flow associated to a dispersive Burgers equation with non local term form $$\left| D \right| ^{\alpha -1} \partial _x u$$ , $$\alpha \in [1,+\infty [$$ is Lipschitz from bounded sets $$H^s_0({\mathbb {T}};{\mathbb {R}})$$ $$C^0([0,T],H^{s-(2-\alpha )^+}_0({\mathbb {R}}))$$ for $$T>0$$ and $$s>\big \lceil \frac{\alpha }{\alpha -1}\big \rceil -\frac{1...

Let S ⊂ R be contained in the unit ball. Let ∆(S) = {||a− b|| : a, b ∈ S}, the Euclidean distance set of S. Falconer conjectured that the ∆(S) has positive Lebesque measure if the Hausdorff dimension of S is greater than d 2 . He also produced an example, based on the integer lattice, showing that the exponent d 2 cannot be improved. In this paper we prove the Falconer distance conjecture for t...

It is an interesting, maybe surprising, fact that different dense subspaces of even “nice” topological spaces can have densities. So, our aim here to investigate the set densities all a space X we call double density spectrum and denote by dd(X). We improve result from [1] showing dd(X) always ω-closed (i.e., countably closed) if Hausdorff. manage give complete characterizations spectra Hausdor...

Abstract We establish a law of the iterated logarithm (LIL) for set real numbers whose n th partial quotient is bigger than α , where ( ) sequence such that ∑1/ finite. This shown to have Hausdorff dimension 1/2 in many cases and measure LIL absolutely continuous measure. The result obtained as an application strong invariance principle unbounded observables on limit sequential function system.

A subset S of a topological group G is said to be a suitable set if (a) it has the discrete topology, (b) it is a closed subset of G \ {l}, and (c) the subgroup generated by S is dense in G. K.H. Hoffmann and S.A. Morris proved that every locally compact group has a suitable set. In this paper it is proved that every metrizable topological group and every countable Hausdorff topological group h...

We consider various algorithmic solutions to image registration based on the alignment of a set of feature points. We present a number of enhancements to a branch-and-bound algorithm introduced by Mount, Netanyahu, and Le Moigne (Pattern Recognition, Vol. 32, 1999, pp. 17–38), which presented a registration algorithm based on the partial Hausdorff distance. Our enhancements include a new distan...

The relation between Hausdorff dimension of the singular spectrum of a Schrödinger operator and the decay of its potential has been extensively studied in many papers. In this work, we address similar questions from a different point of view. Our approach relies on the study of the socalled Krein systems. For Schrödinger operators, we show that some bounds on the singular spectrum, obtained rec...

The rate of growth of the partial quotients of an irrational number is studied relative to the rate of approximation of the number by its convergents. The focus is on the Hausdorff dimension of exceptional sets on which different growth rates are achieved. In this note we look at the rate of growth of the partial quotients ai of the irrational number x = [a1, a2, . . . ] = 1 a1 + 1 a2 + 1 a3 + ...