Abstract:
Two exact relations between mutlifractal exponents are shown to hold at the critical point of the Anderson localization transition. The first relation implies a symmetry of the multifractal spectrum linking the multifractal exponents with indices $q<1/2$ to those with $q>1/2$. The second relation connects the wave function multifractality to that of Wigner delay times in a system with a lead attached.

Abstract:
We relate the entropy of entanglement of ensembles of random vectors to their generalized fractal dimensions. Expanding the von Neumann entropy around its maximum we show that the first order only depends on the participation ratio, while higher orders involve other multifractal exponents. These results can be applied to entanglement behavior near the Anderson transition.

Abstract:
We study percolation as a critical phenomenon on a multifractal support. The scaling exponents of the the infinite cluster size ($\beta$ exponent) and the fractal dimension of the percolation cluster ($d_f$) are quantities that seem do not depend on local anisotropies. These two quantities have the same value as in the standard percolation in regular bidimensional lattices. On the other side, the scaling of the correlation length ($\nu$ exponent) unfolds new universality classes due to the local anisotropy of the critical percolation cluster. We use two critical exponents $\nu$ according to the percolation criterion for crossing the lattice in either direction or in both directions. Moreover $\nu$ is related to a parameter that characterizes the stretching of the blocks forming the tilling of the multifractal.

Abstract:
The harmonic measure (or diffusion field or electrostatic potential) near a percolation cluster in two dimensions is considered. Its moments, summed over the accessible external hull, exhibit a multifractal spectrum, which I calculate exactly. The generalized dimensions D(n) as well as the MF function f(alpha) are derived from generalized conformal invariance, and are shown to be identical to those of the harmonic measure on 2D random walks or self-avoiding walks. An exact application to the anomalous impedance of a rough percolative electrode is given. The numerical checks are excellent. Another set of exact and universal multifractal exponents is obtained for n independent self-avoiding walks anchored at the boundary of a percolation cluster. These exponents describe the multifractal scaling behavior of the average nth moment of the probabity for a SAW to escape from the random fractal boundary of a percolation cluster in two dimensions.

Abstract:
We study the use of the quantum wavelet transform to extract efficiently information about the multifractal exponents for multifractal quantum states. We show that, combined with quantum simulation algorithms, it enables to build quantum algorithms for multifractal exponents with a polynomial gain compared to classical simulations. Numerical results indicate that a rough estimate of fractality could be obtained exponentially fast. Our findings are relevant e.g. for quantum simulations of multifractal quantum maps and of the Anderson model at the metal-insulator transition.

Abstract:
A random field composed by Poisson distributed Brownian vortex filaments is constructed. The filament have a random thickness, length and intensity, governed by a measure $\gamma$. Under appropriate assumptions on $\gamma$ we compute the scaling law of the structure function and get the multifractal scaling as a particular case.

Abstract:
Many examples of signals and images cannot be modeled by locally bounded functions, so that the standard multifractal analysis, based on the H\"older exponent, is not feasible. We present a multifractal analysis based on another quantity, the p-exponent, which can take arbitrarily large negative values. We investigate some mathematical properties of this exponent, and show how it allows us to model the idea of "lacunarity" of a singularity at a point. We finally adapt the wavelet based multifractal analysis in this setting, and we give applications to a simple mathematical model of multifractal processes: Lacunary wavelet series.

Abstract:
In some clusters of galaxies, a diffuse non-thermal emission is present, not obviously associated with any individual galaxy. These sources have been identified as relics, mini-halos, and halos according to their properties and position with respect to the cluster center. Moreover in a few cases have been reported the existence of a diffuse radio emission not identified with a cluster, but with a large scale filamentary region. The aim of this work is to observe and discuss the diffuse radio emission present in the complex merging structure of galaxies ZwCl 2341.1+0000. We have obtained VLA observations at 1.4 GHz to derive a deep radio image of the diffuse emission. Low resolution VLA images show a diffuse radio emission associated to the complex merging region with a largest size = 2.2 Mpc. In addition to the previously reported peripheral radio emission, classified as a double relic, diffuse emission is detected along the optical filament of galaxies. The giant radio source discussed here shows that magnetic fields and relativistic particles are present also in filamentary structures. Possible alternate scenarios are: a giant radio halo in between two symmetric relics, or the merging of two clusters both hosting a central radio halo.

Abstract:
This paper is devoted to problem of detecting critical events at finiacial markets using methods of multifractal analysis. Namely, the local regularity of time-series is studied. As a result, one can find out a special behavior or signal of regularity before crashes. This spesial behaviour of local Hoelder exponents inherent in financial time series can be used in detecting critcal events or crashes at financial markets.

Abstract:
In this paper, we study many geometrical properties of contour loops to characterize the morphology of synthetic multifractal rough surfaces, which are generated by multiplicative hierarchical cascading processes. To this end, two different classes of multifractal rough surfaces are numerically simulated. As the first group, singular measure multifractal rough surfaces are generated by using the $p$ model. The smoothened multifractal rough surface then is simulated by convolving the first group with a so-called Hurst exponent, $H^*$ . The generalized multifractal dimension of isoheight lines (contours), $D(q)$, correlation exponent of contours, $x_l$, cumulative distributions of areas, $\xi$, and perimeters, $\eta$, are calculated for both synthetic multifractal rough surfaces. Our results show that for both mentioned classes, hyperscaling relations for contour loops are the same as that of monofractal systems. In contrast to singular measure multifractal rough surfaces, $H^*$ plays a leading role in smoothened multifractal rough surfaces. All computed geometrical exponents for the first class depend not only on its Hurst exponent but also on the set of $p$ values. But in spite of multifractal nature of smoothened surfaces (second class), the corresponding geometrical exponents are controlled by $H^*$, the same as what happens for monofractal rough surfaces.