41 research outputs found
Extended Grassmann and Clifford algebras
This paper is intended to investigate Grassmann and Clifford algebras over
Peano spaces, introducing their respective associated extended algebras, and to
explore these concepts also from the counterspace viewpoint. The exterior
(regressive) algebra is shown to share the exterior (progressive) algebra in
the direct sum of chiral and achiral subspaces. The duality between scalars and
volume elements, respectively under the progressive and the regressive products
is shown to have chirality, in the case when the dimension n of the Peano space
is even. In other words, the counterspace volume element is shown to be a
scalar or a pseudoscalar, depending on the dimension of the vector space to be
respectively odd or even. The de Rham cochain associated with the differential
operator is constituted by a sequence of exterior algebra homogeneous subspaces
subsequently chiral and achiral. Thus we prove that the exterior algebra over
the space and the exterior algebra constructed on the counterspace are only
pseudoduals each other, when we introduce chirality. The extended Clifford
algebra is introduced in the light of the periodicity theorem of Clifford
algebras context, wherein the Clifford and extended Clifford algebras Cl(p,q)
can be embedded in Cl(p+1,q+1), which is shown to be exactly the extended
Clifford algebra. Clifford algebras are constructed over the counterspace, and
the duality between progressive and regressive products is presented using the
dual Hodge star operator. The differential and codifferential operators are
also defined for the extended exterior algebras from the regressive product
viewpoint, and it is shown they uniquely tumble right out progressive and
regressive exterior products of 1-forms.Comment: 17 pages, to appear in Adv. Appl. Clifford Algebras 16 (3) (2006
The Clifford algebra of physical space and Dirac theory
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term βψ in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation. © 2016 IOP Publishing Ltd.The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term βψ in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation37512
Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics
We revisit the Mittag-Leffler functions of a real variable , with one, two
and three order-parameters , as far as their Laplace
transform pairs and complete monotonicty properties are concerned. These
functions, subjected to the requirement to be completely monotone for ,
are shown to be suitable models for non--Debye 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.Comment: 22 pages, 6 figures, Second Revised Versio
On Fractional Spherically Restricted Hyperbolic Diffusion Random Field
The paper investigates solutions of the fractional hyperbolic diffusion
equation in its most general form with two fractional derivatives of distinct
orders. The solutions are given as spatial-temporal homogeneous and isotropic
random fields and their spherical restrictions are studied. The spectral
representations of these fields are derived and the associated angular spectrum
is analysed. The obtained mathematical results are illustrated by numerical
examples. In addition, the numerical investigations assess the dependence of
the covariance structure and other properties of these fields on the orders of
fractional derivatives.Comment: 32 pages, 18 figure
Theorem for Series in Three-Parameter Mittag-Leffler Function
Mathematics Subject Classification 2010: 26A33, 33E12.The new result presented here is a theorem involving series in the three-parameter Mittag-Leffler function. As a by-product, we recover some known results and discuss corollaries. As an application, we obtain the solution of a fractional differential equation associated with a RLC electrical circuit in a closed form, in terms of the two-parameter Mittag-Leffler function
A algebra do espaço-tempo, o spinor de Dirac-Hestenes e a teoria do eletron
Orientador: Waldyr A. Rodrigues Jr.Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientíficaResumo: A relação entre a teoria do elétron e o eletromagnetismo é discutida com base no uso da álgebra do espaço-tempo e do spinor de Dirac-Hestenes. Desta relação surge uma equação não-linear como uma alternativa, a princípio mais satisfatória, à equação de Dirac. Este estudo é possível uma vez formulada a teoria do spinor de Dirac-Hestenes como uma classe de equivalência de elementos da sub-álgebra par da álgebra do espaço-tempo.Abstract: The relationship between the theory of electron and electromagnetism is discussed by using the spacetime algebra and the Dirac-Hestenes spinor. From this relationship it emerges a non-linear equation which seems to be more satisfactory than Dirac equation. This study is possible once it is formulated the theory of Dirac- Hestenes spinor as an equivalence class of elements of the even subalgebra of the spacetime algebra.DoutoradoDoutor em Matemática Aplicad