Clifford Valued Differential Forms, and Some Issues in Gravitation, Electromagnetism and 'Unified' Theories

In this paper we show how to describe the general theory of a linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the Lie algebra of Clifford bivectors is isomorphic to the Lie algebra of Sl(2,C). In that way the pullback of the linear connection under a local trivialization of the bundle (i.e., a choice of gauge) is represented by a Clifford valued 1-form. That observation makes it possible to realize immediately that Einstein's gravitational theory can be formulated in a way which is similar to a Sl(2,C) gauge theory. Such a theory is compared with other interesting mathematical formulations of Einstein's theory. and particularly with a supposedly "unified" field theory of gravitation and electromagnetism proposed by M. Sachs. We show that his identification of Maxwell equations within his formalism is not a valid one. Also, taking profit of the mathematical methods introduced in the paper we investigate a very polemical issue in Einstein gravitational theory, namely the problem of the 'energy-momentum' conservation. We show that many statements appearing in the literature are confusing or even wrong.Comment: Misprints and errors in some equations of the printed version have been correcte

Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics

We revisit the Mittag-Leffler functions of a real variable $t$, with one, two and three order-parameters $\{\alpha, \beta, \gamma\}$, as far as their Laplace transform pairs and complete monotonicty properties are concerned. These functions, subjected to the requirement to be completely monotone for $t>0$, 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

A Review of Definitions for Fractional Derivatives and Integral

This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering

A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator

In this paper, we propose a generalized Gronwall inequality through the fractional integral with respect to another function. The Cauchy-type problem for a nonlinear differential equation involving the psi-Hilfer fractional derivative and the existence and uniqueness of solutions are discussed. Finally, through generalized Gronwall inequality, we prove the continuous dependence of data on the Cauchy-type problem.1118710

Diferentes abordagens para calcular integrais reais

We present three distinct ways to approach a real integral, a class of integrals, since the integral depends on two parameters. The first way uses a general result, a theorem; the second way, complex variables, through of the residue theorem and Jordan's lemma, while the third way, an artifice through real functions, without using the complex plane. The goal is to make the student choose the best way to approach this class of integrals, or possibly, propose another different way.Apresentamos três distintas maneiras de abordar uma integral real, uma classe de integrais, pois a integral depende de dois parâmetros. A primeira maneira utiliza um resultado geral, um teorema; a segunda as variáveis complexas, através do teorema dos resíduos e o lema de Jordan, enquanto a terceira maneira consiste em um artifício por meio de funções reais, sem utilizar o plano complexo. O objetivo é fazer com que o estudante escolha a melhor forma de abordar essa classe de integrais, ou eventualmente, propondo uma outra maneira diferente

Sobre o notável teorema de Ptolomeu

A geometria elementar do plano foi proposta por Euclides em sua monumental obra Os Elementos, mediante um método axiomático-dedutivo. Assim, partindo de entes fundamentais, axiomas e postulados, resultados são demonstrados por meio de uma estrutura lógica, dentre eles os teoremas, formalizados como o binômio hipótese-tese, ou ainda afirmações que podem ser provadas como verdadeiras. Existem muitos teoremas e talvez o mais famoso, por uma ou outra razão, seja o teorema de Pitágoras associado ao triângulo retângulo. Aqui, vamos abordar o teorema de Ptolomeu, relacionado a um quadrilátero inscrito numa circunferência, também conhecido pelo nome de quadrilátero cíclico. A notabilidade do teorema de Ptolomeu é evidenciada por suas aplicações, dentre outras citamos, os teoremas de Stewart, de Hiparco e de Chadu; relações com polígonos regulares; com a trigonometria, recuperando as expressões para o seno e o cosseno da soma de arcos e, por fim, interessantes relações envolvendo cordas

A note on Pascal's triangle and division by eleven

Divisibility is an old topic that to this day intrigues and fascinates researchers and scholars. Several rules are well-known in particular the divisibility by eleven, since, for example, a palindrome with an even number of digits is divisible by eleven. In current times, divisibility has its applications, for example, in cryptography. Here, in this paper, we will show that applying two somewhat intuitive procedures to the lines of Pascal's triangle shall always yield numbers divisible by eleven. Illustrative examples are presented