We revisit the Mittag-Leffler functions of a real variable t, with one, two and three order-parameters {α, β, γ}, as far as their Laplace transform pairs and complete monotonicity properties are concerned. These functions, subjected to the requirement to be completely monotone for t > 0, are shown to be suitable models for nonDebye relaxation phenomena in dielectrics including as particular cases the classical models referred to as Cole-Cole, Davidson-Cole and Havriliak-Negami. We show 3D plots of the response functions and of the corresponding spectral distributions, keeping fixed one of the three order-parameters. The colleagues Rudolf Gorenflo, Tibor K. Pogany andŽivorad Tomovski, when collaborating on use of the Prabhakar function, found a mistake in our use of the theorem by Gripenberg et al. in Section 2.3 of the first version. In this revised version we will properly apply the conditions of this theorem in order to improve our previous results. We take this occasion to correct a number of misprints and improve Section 2.3. Furthermore, we will better rearrange the original text and update the bibliography. The authors are thus very grateful to these colleagues for having pointed out the deficiencies of the previous analysis whose final results, however, remain still valid but less general, as it will be shown in the following. Foreword to the first revised version, June 2011This E-print reproduces the revised version of the paper published in EPJ- ST, Vol. 193 (2011), pp. 161-171. The revision concerns the proper use of the terms relaxation function and response function in the literature on dielectrics. In the published paper, starting from Eq.(1.1), the authors had referred the inverse Laplace transform of the complex susceptibility as the relaxation function. This is not correct because the inversion provides the so-called response function as pointed out to one of the authors (FM) by Prof. Karina Weron (KW) to whom the authors are very grateful. As a matter of fact the relationship between the response function and the relaxation function can better be clarified by their probabilistic interpretation investigated in several papers by KW. As a consequence, interpreting the relaxation function as a survival probability Ψ(t), the response function turns out to be the probability density function corresponding to the cumulative probability function Φ(t) = 1 − Ψ(t). Then, denoting by φ(t) the response function, we haveAs KW has pointed out, both functions have very different properties and describe different physical magnitudes; only in the Debye (pure exponential) 2 case the properties coincide. The relaxation function describes the decay of polarization whereas the response function its decay rate (the depolarization current). However, for physical realizability, both functions are required to be completely monotone with a proper spectral distribution so our analysis can be properly transferred from response functions to the corresponding relaxation functions, whereas the corresponding cumulative ...

Abstract. In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even sections of the Clifford bundle (over the RCST), thereby being certain particular sections of a new bundle named Spin-Clifford bundle (SCB). The conditions for the existence of the SCB are studied and are shown to be equivalent to the famous Geroch's theorem concerning to the existence of spinor structures in a Lorentzian spacetime. We introduce also the covariant and algebraic Dirac spinor fields and compare these with DHSF, showing that all the three kinds of spinor fields contain the same mathematical and physical information. We clarify also the notion of (Crumeyrolle's) amorphous spinors (Dirac-Kähler spinor fields are of this type), showing that they cannot be used to describe fermionic fields. We develop a rigorous theory for the covariant derivatives of Clifford fields (sections of the Clifford bundle (CB)) and of Dirac-Hestenes spinor fields. We show how to generalize the original Dirac-Hestenes equation in Minkowski spacetime for the case of a RCST. Our results are obtained from a variational principle formulated through the multiform derivative approach to Lagrangian field theory in the Clifford bundle.

Introducing products between multivectors of Cl(0,7) and octonions, resulting in an octonion, and leading to the non-associative standard octonionic product in a particular case, we generalize the octonionic X-product, associated with the transformation rules for bosonic and fermionic fields on the tangent bundle over the 7-sphere, and the XY-product. We also present the formalism necessary to construct Clifford algebra-parametrized octonions. Finally we introduce a method to construct generalized octonionic algebras, where their octonionic units are parametrized by arbitrary Clifford multivectors. The products between Clifford multivectors and octonions, leading to an octonion, are shown to share graded-associative, supersymmetric properties. We also investigate the generalization of Moufang identities, for each one of the products introduced. The X-product equals twice the parallelizing torsion, given by the torsion tensor, and is used to investigate the S7 Kac-Moody algebra. The X-product has also been used to obtain triality maps and G2 actions, and it leads naturally to remarkable geometric and topological properties, for instance the Hopf fibrations and twistor formalism in ten dimensions. The paramount importance of octonions in the search for unification is based, for instance, in the fact that by extending the division-algebra-valued superalgebras to octonions, in D=11 an octonionic generalized Poincare superalgebra can be constructed, the so-called octonionic M-algebra that describes the octonionic M-theory where the octonionic super-2-brane and the octonionic super-5-brane sectors are shown to be equivalent. Also, there are other vast generalizations and applications of the octonionic formalism such as the classification of quaternionic and octonionic spinors and the pseudo-octonionic formalism.Comment: 13 pages, to appear in Journal of Algebra 302 (1) (2006

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A novel method for the solution of the Schrödinger equation in the presence of exchange termsThe fractional Schrödinger equation is solved for the delta potential and the double delta potential for all energies. The solutions are given in terms of Fox's H -function. C