On the size of the algebraic difference of two random Cantor sets

In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference set of two independent copies. We prove that this is the case for the so called Mandelbrot percolation. On the other hand the same is not always true if we apply a slightly more general construction of random Cantor sets. We also present a complete solution for the deterministic case.Comment: This replacement corrects an important omission in the proof of Theorem 1(a

On Disjoint hypercubes in Fibonacci cubes

The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma\_n$, is the subgraph of $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. We study the maximum number of disjoint subgraphs in $\Gamma\_n$ isomorphic to $Q\_k$, and denote this number by $q\_k(n)$. We prove several recursive results for $q\_k(n)$, in particular we prove that $q\_{k}(n) = q\_{k-1}(n-2) + q\_{k}(n-3)$. We also prove a closed formula in which $q\_k(n)$ is given in terms of Fibonacci numbers, and finally we give the generating function for the sequence $\{q\_{k}(n)\}\_{n=0}^{ \infty}$

Forecasting the CATS benchmark with the Double Vector Quantization method

The Double Vector Quantization method, a long-term forecasting method based on the SOM algorithm, has been used to predict the 100 missing values of the CATS competition data set. An analysis of the proposed time series is provided to estimate the dimension of the auto-regressive part of this nonlinear auto-regressive forecasting method. Based on this analysis experimental results using the Double Vector Quantization (DVQ) method are presented and discussed. As one of the features of the DVQ method is its ability to predict scalars as well as vectors of values, the number of iterative predictions needed to reach the prediction horizon is further observed. The method stability for the long term allows obtaining reliable values for a rather long-term forecasting horizon.Comment: Accepted for publication in Neurocomputing, Elsevie

Quantum Communication with Quantum Dot Spins

Single electron spins in quantum dots are attractive for quantum communication because of their expected long coherence times. We propose a method to create entanglement between two remote spins based on the coincident detection of two photons emitted by the dots. Local nodes of several qubits can be realized using the dipole-dipole interaction between trions in neighboring dots and spectral addressing, allowing the realization of quantum repeater protocols. We have performed a detailed feasibility study of our proposal based on tight-binding calculations of quantum dot properties.Comment: 4 pages, 2 figures, new and improved version, explicit performance estimate

A statistical mechanics framework for the large-scale structure of turbulent von K{\'a}rm{\'a}n flows

In the present paper, recent experimental results on large scale coherent steady states observed in experimental von K{\'a}rm{\'a}n flows are revisited from a statistical mechanics perspective. The latter is rooted on two levels of description. We first argue that the coherent steady states may be described as the equilibrium states of well-chosen lattice models, that can be used to define global properties of von K{\'a}rm{\'a}n flows, such as their temperatures. The equilibrium description is then enlarged, in order to reinterpret a series of results about the stability of those steady states, their susceptibility to symmetry breaking, in the light of a deep analogy with the statistical theory of Ferromagnetism. We call this analogy "Ferro-Turbulence