Green function for the diffusion limit of one-dimensional continuous time random walks

It is shown how the fractional probability density diffusion equation for the diffusion limit of one-dimensional continuous time random walks may be derived from a generalized Markovian Chapman-Kolmogorov equation. The non-Markovian behaviour is incorporated into the Markovian Chapman-Kolmogorov equation by postulating a Levy like distribution of waiting times as a kernel. The Chapman-Kolmogorov equation so generalised then takes on the form of a convolution integral. The dependence on the initial conditions typical of a non-Markovian process is treated by adding a time dependent term involving the survival probability to the convolution integral. In the diffusion limit these two assumptions about the past history of the process are sufficient to reproduce anomalous diffusion and relaxation behaviour of the Cole-Cole type. The Green function in the diffusion limit is calculated using the fact that the characteristic function is the Mittag-Leffler function. Fourier inversion of the characteristic function yields the Green function in terms of a Wright function. The moments of the distribution function are evaluated from the Mittag-Leffler function using the properties of characteristic functions and a relation between the powers of the second moment and higher order even moments is derived. (C) 2004 Elsevier B.V. All rights reserved.

High harmonic generation in the relativistic limit

The generation of extremely bright coherent X-ray pulses in the femtosecond and attosecond regime is currently one of the most exciting frontiers of physics - allowing, for the first time, measurements with unprecedented temporal resolution(1-6). Harmonics from laser - solid target interactions have been identified as a means of achieving fields as high as the Schwinger limit(2,7) (E = 1.3 x 10(16) V m(-1)) and as a highly promising route to high-efficiency attosecond (10(-18) s) pulses(8) owing to their intrinsically phase-locked nature. The key steps to attain these goals are achieving high conversion efficiencies and a slow decay of harmonic efficiency to high orders by driving harmonic production to the relativistic limit(1). Here we present the first experimental demonstration of high harmonic generation in the relativistic limit, obtained on the Vulcan Petawatt laser(9). High conversion efficiencies (eta> 10(-6) per harmonic) and bright emission (> 10(22) photons s(-1) mm(-2) mrad(-2) (0.1% bandwidth)) are observed at wavelengths <4 nm ( the 'water-window' region of particular interest for bio-microscopy).

Polynomials on Banach spaces with unconditional bases

We study the classes of homogeneous polynomials on a Banach space with unconditional Schauder basis that have unconditionally convergent monomial expansions relative to this basis. We extend some results of Matos, and we show that the homogeneous polynomials with unconditionally convergent expansions coincide with the polynomials that are regular with respect to the Banach lattices structure of the domain.

Multiplicative spectra of Banach spaces

The multiplicative spectrum of a complex Banach space X is the class K(X) of all (automatically compact and Hausdorff) topological spaces appearing as spectra of Banach algebras (X,*) for all possible continuous multiplications on X turning X into a commutative associative complex algebra with the unity. The properties of the multiplicative spectrum are studied. In particular, we show that K(X^n) consists of countable compact spaces with at most n non-isolated points for any separable hereditarily indecomposable Banach space X. We prove that K(C[0,1]) coincides with the class of all metrizable compact spaces.