Newly developed high peak power lasers have opened the possibilities of driving coherent light sources operating with laser plasma accelerated beams and wave undulators. We speculate on the combination of these two concepts and show that the merging
of the underlying technologies could lead to new and interesting possibilities to achieve truly compact, coherent radiator devices.
We develop a heuristic model embedding Kleiber and Murray laws to describe mass growth, metastasis and vascularization in cancer. We analyze the relevant dynamics using different evolution equations (Verhulst, Gompertz and others). Their extension to
reaction diffusion equation of the Fisher type is then used to describe the relevant metastatic spreading in space. Regarding this last point, we suggest that cancer diffusion may be regulated by Levy flights mechanisms and discuss the possibility that the associated reaction diffusion equations are of the fractional type, with the fractional coefficient being determined by the fractal nature of the capillary evolution.
The symbolic method is used to get explicit formulae for the products or powers of Bessel functions and for the relevant integrals.
Recent experiments have shown that photoluminescence decay of silicon nanocrystals can be described by the stretched exponential function. We show here that the associated decay probability rate is the one-sided Levy stable distribution which describ
es well the experimental data. The relevance of these conclusions to the underlying stochastic processes is discussed in terms of Levy processes.
Symbolic methods of umbral nature play an important and increasing role in the theory of special functions and in related fields like combinatorics. We discuss an application of these methods to the theory of lacunary generating functions for the Lag
uerre polynomials for which we give a number of new closed form expressions. We present furthermore the different possibilities offered by the method we have developed, with particular emphasis on their link to a new family of special functions and with previous formulations, associated with the theory of quasi monomials.
We comment on the exponential parameterization of the quark mixing matrix, by stressing that it naturally incorporates the Cabibbo structure and the hierarchical features of the Wolfenstein form. We extend our results to the neutrino mixing and intro
duce an exponential generator of the tribimaximal matrix.
The Humbert-Bessel are multi-index functions with various applications in electromagnetism. New families of functions sharing some similarities with Bessel functions are often introduced in the mathematical literature, but at a closer analysis they a
re not new, in the strict sense of the word, and are shown to be expressible in terms of already discussed forms. This is indeed the case of the re-modified Bessel functions, whose properties have been analyzed within the context of coincidence problems in probability theory. In this paper we show that these functions are particular cases of the Humbert-Bessel ones.
We review some aspects of the theory of spherical Bessel functions and Struve functions by means of an operational procedure essentially of umbral nature, capable of providing the straightforward evaluation of their definite integrals and of successi
ve derivatives. The method we propose allows indeed the formal reduction of these family of functions to elementary ones of Gaussian type. We study the problem in general terms and present a formalism capable of providing a unifying point of view including Anger and Weber functions too. The link to the multi-index Bessel functions is also briefly discussed.
We present a number of identities involving standard and associated Laguerre polynomials. They include double-, and triple-lacunary, ordinary and exponential generating functions of certain classes of Laguerre polynomials.
The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermor
e, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics.
