Heaviside generalized functions and shock waves for a Burger kind equation

The purpose of the work is to study the existence and nonexistence of shock wave solutions for the Burger equations. The study is developed in the context of Colombeau's theory of generalized functions (GFs). This study uses the equality in the strict sense and the weak equality of GFs. The shock wave solutions are given in terms of GFs that have the Heaviside function, in x and ( x, t) variables, as macroscopic aspect. This means that solutions are sought in the form of sequences of regularizations to the Heaviside function, in R-n and R-n x R, in the distributional limit sense.

GENERALIZED HEAVISIDE FUNCTIONS IN THE COLOMBEAU THEORY CONTEXT

We defined generalized Heaviside functions for a variable x in R-n, and for variables (x, t) in R-n x R-m. Then study properties such as: composition, invertibility, and association relation (the weak equality). This work is developed in the Colombeau generalized functions context.

Evans functions for integral neural field equations with Heaviside firing rate function

In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction and stability analysis of front solutions to a scalar neural field model and a limiting case is shown to recover recent results of L. Zhang [On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16, (2003), pp.513-536.]. Traveling fronts and pulses are considered in more general models possessing either a linear or piecewise constant recovery variable. We establish the stability of coexisting traveling fronts beyond a front bifurcation and consider parameter regimes that support two stable traveling fronts of different speed. Such fronts may be connected and depending on their relative speed the resulting region of activity can widen or contract. The conditions for the contracting case to lead to a pulse solution are established. The stability of pulses is obtained for a variety of examples, in each case confirming a previously conjectured stability result. Finally we show how this theory may be used to describe the dynamic instability of a standing pulse that arises in a model with slow recovery. Numerical simulations show that such an instability can lead to the shedding of a pair of traveling pulses.

Plane wave and Coulomb asymptotics

A simple plane wave solution of the Schrodinger-Helmholtz equation is a quantum eigenfunction obeying both energy and linear momentum correspondence principles. Inclusion of the outgoing wave with scattering amplitude f asymptotic development of the plane wave, we show that there is a problem with angular momentum when we consider forward scattering at the point of closest approach and at large impact parameter given semiclassically by (l + 1/2)/k where l is the azimuthal quantum number and may be large (J. Leech et al., Phys. Rev. Lett. 88. 257901 (2002)). The problem is resolved via non- uniform, non-standard analysis involving the Heaviside step function, unifying classical, semiclassical and quantum mechanics, and the treatment is extended to the case of pure Coulomb scattering.

Some pioneers of the applications of fractional calculus

In the last decades fractional calculus (FC) became an area of intensive research and development. This paper goes back and recalls important pioneers that started to apply FC to scientific and engineering problems during the nineteenth and twentieth centuries. Those we present are, in alphabetical order: Niels Abel, Kenneth and Robert Cole, Andrew Gemant, Andrey N. Gerasimov, Oliver Heaviside, Paul Lévy, Rashid Sh. Nigmatullin, Yuri N. Rabotnov, George Scott Blair.

Branching random motions, nonlinear hyperbolic systems and traveling waves

A branching random motion on a line, with abrupt changes of direction, is studied. The branching mechanism, being independient of random motion, and intensities of reverses are defined by a particle's current direction. A soluton of a certain hyperbolic system of coupled non-linear equations (Kolmogorov type backward equation) have a so-called McKean representation via such processes. Commonly this system possesses traveling-wave solutions. The convergence of solutions with Heaviside terminal data to the travelling waves is discussed.This Paper realizes the McKean programme for the Kolmogorov-Petrovskii-Piskunov equation in this case. The Feynman-Kac formula plays a key role.

Interface dynamics in planar neural field models

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.

Determinação de parâmetros de criticalidade, fluxo e potência em placas planas pelo método LTSn

O objetivo deste trabalho consiste em aplicar o método LTSn em cálculos de parâmetros críticos como Keff, espessura e concentração atômica e obtenção do fiuxo escalar, da potência específica e do enriquecimento do combustível em placa plana homogenea e heterogênea, considerando modelo multigrupo e em diversas ordens de quadraturas. O método LTSn consiste na aplicação da transformada de Laplace em um conjunto de equações~de ordenadas discretas gerado pela aproximação SN, resultando em um sistema de equações algébricas simbólicas dependentes do parâmetro complexo s e reconstrução dos fluxos angulares pela técnica de expansão de Heaviside. A aplicação do método LTSn reduz a soluçào de um problema de autovalor, a solução de uma equação transcedental, possibilitando a obtenção de parâmetros críticos. Simulações numéricas são apresentadas.

Solução da equação de transporte de fótons para uma placa plana heterogênea, modelo de multigrupo com núcleo de espalhamento de Klein-Nishina

Neste trabalho o método LTSN é utilizado para resolver a equação de transporte de fótons para uma placa plana heterogênea, modelo de multigrupo, com núcleo de espalhamento de Klein-Nishina, obtendo-se o fluxo de fótons em valores discretos de energia. O fluxo de fótons, juntamente com os parâmetros da placa foram usados para o cálculo da taxa de dose absorvida e do fator de buildup. O método LTSN consiste na aplicação da transformada de Laplace num conjunto de equações de ordenadas discretas, fornece uma solução analítica do sistema de equações lineares algébricas e a construção dos fluxos angulares pela técnica de expansão de Heaviside. Essa formulação foi aplicada ao cálculo de dose absorvida e ao fator de Buildup, considerando cinco valores de energia. Resultados numéricos são apresentados.

Solução analítica das equações difusivas da teoria geral de perturbação pelo método da transformada de Laplace

Neste trabalho, apresentamos uma solução analítica para as equações difusivas unidimensionais da Teoria Geral de Perturbação em uma placa heterogênea, isto é, apresentamos as soluções analíticas para os problemas de autovalor para o fluxo de nêutrons e para o fluxo adjunto de nêutrons, para o cálculo do fator de multiplicação efetivo (keff), para o problema de fonte fixa e para o problema de função auxiliar. Resolvemos todos os problemas mencionados aplicando a Transformada de Laplace em uma placa heterogênea considerando um modelo de dois grupos de energia e realizamos a inversão de Laplace do fluxo transformado analiticamente através da técnica da expansão de Heaviside. Conhecendo o fluxo de nêutrons, exceto pelas constantes de integração, aplicamos as condições de contorno e de interface e resolvemos as equações algébricas homogêneas para o fator de multiplicação efetivo pelo método da bissecção. Obtemos o fluxo de nêutrons através da avaliação das constantes de integração para uma potência prescrita. Exemplificamos a metodologia proposta para uma placa com duas regiões e comparamos os resultados obtidos com os existentes na literatura.

An approach to generalized holomorphic functions

A study of the generalized holomorphic functions, HG(Omega), having in mind its strict elements, i.e. those which are in HG(Omega) - H(Omega), as well as the possibility of the existence of hybrid elements, i.e. elements which have, in a part of a domain Omega subset of C-n, the strict behaviour and, in another part of the same domain, the classical behaviour, is carried out in this work. The study of hybrid elements is important in the approach of a concept of generalized domain of holomorphy.

Colombeau's theory and shock wave solutions for systems of PDEs

In this article we study the existence of shock wave solutions for systems of partial differential equations of hydrodynamics with viscosity in one space dimension in the context of Colombeau's theory of generalized functions. This study uses the equality in the strict sense and the association of generalized functions (that is the weak equality). The shock wave solutions are given in terms of generalized functions that have the classical Heaviside step function as macroscopic aspect. This means that solutions are sought in the form of sequences of regularizations to the Heaviside function that have to satisfy part of the equations in the strict sense and part of the equations in the sense of association.

Electromagnetic theory /

Date of first publication of each article is given.

Account of the operations of the Great trigonometrical survey of India ...

XVIII. Astronomical observations for latitude made during the period 1885 to 1895 and deduced values of the deflections of the plumb-line. Prepared under the directions of...S.G. Burrard. 1906.--XIX. Levelling of precision in India (1858-190) by S.G. Burrard..1910.--XIXA. Descriptions and heights of bench-marks on southern lines of levelling. Prepared under the directions of S.G. Burrard. 1910.--XIXB. Descriptions and heights of bench-marks on the northern lines of levelling. Prepared under the directions of S.G. Burrard. 1910.

Electromagnetic theory.

Date of first publication of each article is given.