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26079 results in Abstract analysis

Page 882 of 1304

  • Preface

  • George Boros, Xavier University of Louisiana, Victor Moll, Tulane University, Louisiana
  • Book:
    Irresistible Integrals
    Published online:
    13 February 2010
    Print publication:
    21 June 2004, pp xi-xiv

  • Summary

    The idea of writing a book on all the areas of mathematics that appear in the evaluation of integrals occurred to us when we found many beautiful results scattered throughout the literature.

    The original idea was naive: inspired by the paper “Integrals: An Introduction to Analytic Number Theory” by Lian Vardi (1988) we decided to write a text in which we would prove every formula in Table of Integrals, Series, and Products by I. S. Gradshteyn and I. M. Rhyzik (1994) and its precursor by Bierens de Haan (1867). It took a short time to realize that this task was monumental.

    In order to keep the book to a reasonable page limit, we have decided to keep the material at a level accesible to a junior/senior undergraduate student. We assume that the reader has a good knowledge of one-variable calculus and that he/she has had a class in which there has been some exposure to a rigorous proof. At Tulane University this is done in Discrete Mathematics, where the method of mathematical induction and the ideas behind recurrences are discussed in some detail, and in Real Analysis, where the student is exposed to the basic material of calculus, now with rigorous proofs. It is our experience that most students majoring in mathematics will have a class in linear algebra, but not all (we fear, few) study complex analysis. Therefore we have kept the use of these subjects to a minimum. In particular we have made an effort not to use complex analysis.

    The goal of the book is to present to the reader the many facets involved in the evaluation of definite integrals.

  • On the analysis of a non-isothermal model for superconductivity

  • DAN PHILLIPS, EUNJEE SHIN
  • Journal:
    European Journal of Applied Mathematics / Volume 15 / Issue 2 / April 2004
    Published online by Cambridge University Press:
    07 June 2004, pp. 147-179

  • We analyze a model for non-isothermal superconductivity, derived independently by G. Maugin, and K. Miya and S. A. Zhou. The model is described by a parabolic system based on the Time-Dependent Ginzburg–Landau (TDGL) equation, the Maxwell equations, and an energy equation such that the Clausius–Duhem inequality holds. The principal unknown fields are the complex valued Ginzburg–Landau order parameter $\psi$, the magnetic vector potential $A$, and the temperature $T$. A significant feature for this model is that it accounts for the interchange of thermal and electro-magnetic energies through Joule heating. The sensitivity of superconducting materials to thermal variations is a major obstacle to their applications. In practice one sees that time varying currents and magnetic fields generate thermal energy, producing ‘hot spots’ and increasing the temperature. This can result in suppressing superconductivity, i.e. $\psi$ tending to $0$. Our principal result is that we exhibit this phenomenon. We prove that if the electro-magnetic energy of the superconductor is sufficiently large at time $t\,{=}\,0$ then $T$ will rise beyond the critical temperature, $T_c$, and $\psi\to 0$ as $t\to\infty$. Earlier analytic work investigating electro-dynamic effects in superconductivity centered on an isothermal model consisting of the TDGL equation and Maxwell equations. This setting takes into account just electro-magnetic effects and produces markedly different evolutions.

Page 882 of 1304