By the first year of graduate school, a young mathematician will have encountered at least three separate definitions of the integral. The associated integrals are typically studied in isolation, with little attention paid to the relationships between them or to the historical issues that motivated their definitions. This book redresses this situation by introducing the Riemann, Darboux, Lebesgue, and gauge integrals using a common set of examples. This allows the reader to see how the definitions influence proof techniques and computational strategies. Then the properties of the integrals are compared in three major areas: the class of integrable functions, the convergence properties of the integral, and the best form of the Fundamental Theorems of Calculus. With a thorough set of appendices and exercises, and interesting historical context, this book is equally useful as a reference for mathematicians or as a text for a second undergraduate course in real analysis.

This book studies the large-time asymptotic behavior of solutions of the pure initial value problem for linear dispersive equations with constant coefficients and homogeneous symbols in one space dimension. Complete matched and uniformly-valid asymptotic expansions are obtained and sharp error estimates are proved. Using the method of steepest descent much new information on the regularity and spatial asymptotics of the solutions are also obtained. Applications to nonlinear dispersive equations are discussed. This monograph is intended for researchers and graduate students of partial differential equations. Familiarity with basic asymptotic, complex and Fourier analysis is assumed.

A rigorous introduction to the abstract theory of partial differential equations progresses from the theory of distribution and Sobolev spaces to Fredholm operations, the Schauder fixed point theorem and Bochner integrals.

The final aim of the book is to construct effective discretization methods to solve multidimensional weakly singular integral equations of the second kind on a region of Rn e.g. equations arising in the radiation transfer theory. To this end, the smoothness of the solution is examined proposing sharp estimates of the growth of the derivatives of the solution near the boundary G. The superconvergence effect of collocation methods at the collocation points is established. This is a book for graduate students and researchers in the fields of analysis, integral equations, mathematical physics and numerical methods. No special knowledge beyond standard undergraduate courses is assumed.

In three dimensional boundary element analysis, computation of integrals is an important aspect since it governs the accuracy of the analysis and also because it usually takes the major part of the CPU time. The integrals which determine the influence matrices, the internal field and its gradients contain (nearly) singular kernels of order lIr a (0:= 1,2,3,4,.**) where r is the distance between the source point and the integration point on the boundary element. For planar elements, analytical integration may be possible 1,2,6. However, it is becoming increasingly important in practical boundary element codes to use curved elements, such as the isoparametric elements, to model general curved surfaces. Since analytical integration is not possible for general isoparametric curved elements, one has to rely on numerical integration. When the distance d between the source point and the element over which the integration is performed is sufficiently large compared to the element size (d> 1), the standard Gauss-Legendre quadrature formula 1,3 works efficiently. However, when the source is actually on the element (d=O), the kernel 1I~ becomes singular and the straight forward application of the Gauss-Legendre quadrature formula breaks down. These integrals will be called singular integrals. Singular integrals occur when calculating the diagonals of the influence matrices.

Professor Arnold is a prolific and versatile mathematician who has done striking work in differential equations and geometrical aspects of analysis. In this volume are collected seven of his survey articles from Russian Mathematical Surveys on singularity theory, the area to which he has made most contribution. These surveys contain Arnold's own analysis and synthesis of a decade's work. All those interested in singularity theory will find this an invaluable compilation. Professor C. T. C. Wall has written an introduction outlining the significance and content of the articles.

CONTENTS: M.I. Freidlin: Semi-linear PDE's and limit theorems for large deviations.- J.F. Le Gall: Some properties of planar Brownian motion.

At the date of this writing, there is no question that the boundary element method has emerged as one of the major revolutions on the engineering science of computational mechanics. The emergence of the technique from relative obscurity to a cutting edge engineering analysis tool in the short space of basically a ten to fifteen year time span is unparalleled since the advent of the finite element method. At the recent international conference BEM XI, well over one hundred papers were presented and many were pub lished in three hard-bound volumes. The exponential increase in interest in the subject is comparable to that shown in the early days of finite elements. The diversity of appli cations of BEM, the broad base of interested parties, and the ever-increasing presence of the computer as an engineering tool are probably the reasons for the upsurge in pop ularity of BEM among researchers and industrial practitioners. Only in the past few years has the BEM audience become large enough that we have seen the development of specialty books on specific applications of the boundary element method. The present text is one such book. In this work, we have attempted to present a self-contained treatment of the analysis of physical phenomena governed by equations containing biharmonic operators. The biharmonic operator defines a very important class of fourth-order PDE problems which includes deflections of beams and thin plates, and creeping flow of viscous fluids."

In many fields of application of mathematics, progress is crucially dependent on the good flow of information between (i) theoretical mathematicians looking for applications, (ii) mathematicians working in applications in need of theory, and (iii) scientists and engineers applying mathematical models and methods. The intention of this book is to stimulate this flow of information. In the first three chapters (accessible to third year students of mathematics and physics and to mathematically interested engineers) applications of Abel integral equations are surveyed broadly including determination of potentials, stereology, seismic travel times, spectroscopy, optical fibres. In subsequent chapters (requiring some background in functional analysis) mapping properties of Abel integral operators and their relation to other integral transforms in various function spaces are investi- gated, questions of existence and uniqueness of solutions of linear and nonlinear Abel integral equations are treated, and for equations of the first kind problems of ill-posedness are discussed. Finally, some numerical methods are described. In the theoretical parts, emphasis is put on the aspects relevant to applications.

This book provides a simple introduction to a nonlinear theory of generalized functions introduced by J.F. Colombeau, which gives a meaning to any multiplication of distributions. This theory extends from pure mathematics (it presents a faithful generalization of the classical theory of C? functions and provides a synthesis of most existing multiplications of distributions) to physics (it permits the resolution of ambiguities that appear in products of distributions), passing through the theory of partial differential equations both from the theoretical viewpoint (it furnishes a concept of weak solution of pde's leading to existence-uniqueness results in many cases where no distributional solution exists) and the numerical viewpoint (it introduces new and efficient methods developed recently in elastoplasticity, hydrodynamics and acoustics). This text presents basic concepts and results which until now were only published in article form. It is in- tended for mathematicians but, since the theory and applications are not dissociated it may also be useful for physicists and engineers. The needed prerequisites for its reading are essentially reduced to the classical notions of differential calculus and the theory of integration over n-dimensional euclidean spaces.

Developments in numerical initial value ode methods were the focal topic of the meeting at L'Aquila which explord the connections between the classical background and new research areas such as differental-algebraic equations, delay integral and integro-differential equations, stability properties, continuous extensions (interpolants for Runge-Kutta methods and their applications, effective stepsize control, parallel algorithms for small- and large-scale parallel architectures). The resulting proceedings address many of these topics in both research and survey papers.

The volume contains a selection of papers presented at the 7th Symposium on differential geometry and differential equations (DD7) held at the Nankai Institute of Mathematics, Tianjin, China, in 1986. Most of the contributions are original research papers on topics including elliptic equations, hyperbolic equations, evolution equations, non-linear equations from differential geometry and mechanics, micro-local analysis.

The homotopy index theory was developed by Charles Conley for two sided flows on compact spaces. The homotopy or Conley index, which provides an algebraic-topologi cal measure of an isolated invariant set, is defined to be the ho motopy type of the quotient space N /N, where is a certain 1 2 1 2 compact pair, called an index pair. Roughly speaking, N1 isolates the invariant set and N2 is the "exit ramp" of N . 1 It is shown that the index is independent of the choice of the in dex pair and is invariant under homotopic perturbations of the flow. Moreover, the homotopy index generalizes the Morse index of a nQnde generate critical point p with respect to a gradient flow on a com pact manifold. In fact if the Morse index of p is k, then the homo topy index of the invariant set {p} is Ik - the homotopy type of the pointed k-dimensional unit sphere."

The field of nonlinear hyperbolic problems has been expanding very fast over the past few years, and has applications - actual and potential - in aerodynamics, multifluid flows, combustion, detonics amongst other. The difficulties that arise in application are of theoretical as well as numerical nature. In fact, the papers in this volume of proceedings deal to a greater extent with theoretical problems emerging in the resolution of nonlinear hyperbolic systems than with numerical methods. The volume provides an excellent up-to-date review of the current research trends in this area.

This set features: Foundations of Differential Geometry, Volume 1 by Shoshichi Kobayashi and Katsumi Nomizu (978-0-471-15733-5) Foundations of Differential Geometry, Volume 2 by Shoshichi Kobayashi and Katsumi Nomizu (978-0-471-15732-8) Differential and Integral Calculus, Volume 1 by Richard Courant (978-0-471-60842-4) Differential and Integral Calculus, Volume 2 by Richard Courant (978-0-471-60840-0) Linear Operators, Part 1: General Theory by Neilson Dunford and Jacob T. Schwartz (978-0-471-60848-6) Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert Space Theory by Neilson Dunford and Jacob T. Schwartz (978-0-471-60847-9) Linear Operators, Part 3: Spectral Operators by Neilson Dunford and Jacob T. Schwartz (978-0-471-60846-2) Applied and Computational Complex Analysis, Volume 1, Power Series Integration Conformal Mapping Location of Zero by Peter Henrici (978-0-471-60841-7) Applied and Computational Complex Analysis, Volume 2, Special Functions-Integral Transforms- Asymptotics-Continued Fractions by Peter Henrici (978-0-471-54289-6) Applied and Computational Complex Analysis, Volume 3, Discrete Fourier Analysis, Cauchy Integrals, Construction of Conformal Maps, Univalent Functions by Peter Henrici (978-0-471-58986-0)"

The proposed book is one of a series called "A Course of Higher Mathematics and Mathematical Physics" edited by A. N. Tikhonov, V. A. Ilyin and A. G. Sveshnikov. The book is based on a lecture course which, for a number of years now has been taught at the Physics Department and the Department of Computational Mathematics and Cybernetics of Moscow State University. The exposition reflects the present state of the theory of differential equations, as far as it is required by future specialists in physics and applied mathematics, and is at the same time elementary enough. An important part of the book is devoted to approximation methods for the solution and study of differential equations, e.g. numerical and asymptotic methods, which at the present time play an essential role in the study of mathematical models of physical phenomena. Less attention is paid to the integration of differential equations in elementary functions than to the study of algorithms on which numerical solution methods of differential equations for computers are based.

In many scientific or engineering applications, where ordinary differen tial equation (OOE), partial differential equation (POE), or integral equation (IE) models are involved, numerical simulation is in common use for prediction, monitoring, or control purposes. In many cases, however, successful simulation of a process must be preceded by the solution of the so-called inverse problem, which is usually more complex: given meas ured data and an associated theoretical model, determine unknown para meters in that model (or unknown functions to be parametrized) in such a way that some measure of the "discrepancy" between data and model is minimal. The present volume deals with the numerical treatment of such inverse probelms in fields of application like chemistry (Chap. 2,3,4, 7,9), molecular biology (Chap. 22), physics (Chap. 8,11,20), geophysics (Chap. 10,19), astronomy (Chap. 5), reservoir simulation (Chap. 15,16), elctrocardiology (Chap. 14), computer tomography (Chap. 21), and control system design (Chap. 12,13). In the actual computational solution of inverse problems in these fields, the following typical difficulties arise: (1) The evaluation of the sen sitivity coefficients for the model. may be rather time and storage con suming. Nevertheless these coefficients are needed (a) to ensure (local) uniqueness of the solution, (b) to estimate the accuracy of the obtained approximation of the solution, (c) to speed up the iterative solution of nonlinear problems. (2) Often the inverse problems are ill-posed. To cope with this fact in the presence of noisy or incomplete data or inev itable discretization errors, regularization techniques are necessary."

This is a collection of research-oriented monographs, reports, and notes arising from lectures and seminars on the Weil representation, the Maslov index, and the Theta series. It is good contribution to the international scientific community, particularly for researchers and graduate students in the field.