We study numerically the roughness exponent zeta of an in-plane fracture front slowly propagating along a heterogeneous interface embedded in an elastic body, using a model based on the evolution of a process zone rather than a fracture line. We find zeta=0.60+/-0.05. For the first time, simulation results are in close agreement with experimental results. We then show that the roughness exponen...

Explicit formulae for Weber-Schafheitlin type integrals with exponent 1 are derived. The results of these integrals are distributions on R+. Received 07 April 2008; Revised 30 June 2008

In this note, we provide a non trivial example of differential equation driven by a fractional Brownian motion with Hurst parameter 1/3 < H < 1/2, whose solution admits a smooth density with respect to Lebesgue’s measure. The result is obtained through the use of an explicit representation of the solution when the vector fields of the equation are nilpotent, plus a Norris type lemma in the roug...

in this paper we give an elementary argument about the atoms and coatoms of the latticeof all subgroups of a group. it is proved that an abelian group of finite exponent is strongly coatomic.

Let B and e B be two independent, d-dimensional fractional Brownian motions with Hurst parameter H ∈ (0, 1) . Assume d ≥ 2. We prove that the intersection local time of B and e B I(BH , e BH) = Z

Numerical results for the directed polymer model in 1+4 dimensions in various types of disorder are presented. The results are obtained for a system size that is considerably larger than considered previously. For the extreme "strong" disorder case (min-max system), associated with the directed percolation model, the expected value of the meandering exponent, ζ=0.5, is clearly revealed, with ve...

Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L2 and two constants c1, c2 > 0 such that for all x, y ∈ G we have ‖ f (x)− f (y)‖2 ≥ c1d(x, y)− c2. Tessara [16] proved that the Hilbert compression exponent of...

A positive recurrent, aperiodic Markov chain is said to be long range dependent (LRD) when the indicator function of a particular state is LRD. This happens if and only if the return time distribution for that state has infinite variance. We investigate the question of whether other instantaneous functions of the Markov chain also inherit this property. We provide conditions under which the fun...

Dynamical chaos is studied in the Hamiltonian motion of ions confined in a Penning trap and forming so-called microplasmas. The dynamical chaos of the ion motion is characterized by the maximum Lyapunov exponent. Results are reported on the dependence of this exponent on the energy of the system, on the number of ions, as well as on the geometry of the trap. Different dynamical regimes are char...

Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L2 and a constant c > 0 such that for all x, y ∈ G we have ‖ f (x) − f (y)‖2 ≥ cd(x, y). In [2] it was shown that the Hilbert compression exponent of the wreath ...