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Hyperbolic Differential Operators And Related Problems

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Hyperbolic Differential Operators And Related Problems

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Hyperbolic Differential Operators And Related Problems by Vincenzo Ancona,Jean Vaillant Book Summary:

Presenting research from more than 30 international authorities, this reference provides a complete arsenal of tools and theorems to analyze systems of hyperbolic partial differential equations. The authors investigate a wide variety of problems in areas such as thermodynamics, electromagnetics, fluid dynamics, differential geometry, and topology. Renewing thought in the field of mathematical physics, Hyperbolic Differential Operators defines the notion of pseudosymmetry for matrix symbols of order zero as well as the notion of time function. Surpassing previously published material on the topic, this text is key for researchers and mathematicians specializing in hyperbolic, Schrödinger, Einstein, and partial differential equations; complex analysis; and mathematical physics.

Pseudo-Differential Operators and Related Topics

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Pseudo-Differential Operators and Related Topics by Paolo Boggiatto,Luigi Rodino,Joachim Toft,M. W. Wong Book Summary:

Contains articles based on lectures given at the International Conference on Pseudo-differential Operators and Related Topics at Vaxjo University in Sweden from June 22 to June 25, 2005. Sixteen refereed articles cover a spectrum of topics such as partial differential equations, Wigner transforms, mathematical physics, and more.

Phase Space Analysis of Partial Differential Equations

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Phase Space Analysis of Partial Differential Equations by Antonio Bove,Ferruccio Colombini,Daniele Del Santo Book Summary:

Covers phase space analysis methods, including microlocal analysis, and their applications to physics Treats the linear and nonnlinear aspects of the theory of PDEs Original articles are self-contained with full proofs; survey articles give a quick and direct introduction to selected topics evolving at a fast pace Excellent reference and resource for grad students and researchers in PDEs and related fields

Differential Operators and Related Topics

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Differential Operators and Related Topics by V. M. Adami͡an,Israel Gohberg,M. Gorbachuk,V. Gorbachuk,M. A. Kaashoek,H. Langer,G. Popov Book Summary:

The present book is the first of the two volume proceedings of the Mark Krein International Conference on Operator Theory and Applications. This conference, which was dedicated to the 90th anniversary of the prominent mathematician Mark Krein, was held in Odessa, Ukraine, from August 18-22, 1997. The conference focused on the main ideas, methods, results, and achievements of M. G. Krein. This first volume is devoted to the theory of differential operators and related topics. It opens with a description of the conference, biographical material and a number of survey papers about the work of M. G. Krein. The main part of the book consists of original research papers presenting the state of the art in the area of differential operators. The second volume of these proceedings, entitled Operator Theory and Related Topics, concerns the other aspects of the conference. The two volumes will be of interest to a wide range of readership in pure and applied mathematics, physics and engineering sciences.

Progress in Partial Differential Equations

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Progress in Partial Differential Equations by Michael Reissig,Michael Ruzhansky Book Summary:

Progress in Partial Differential Equations is devoted to modern topics in the theory of partial differential equations. It consists of both original articles and survey papers covering a wide scope of research topics in partial differential equations and their applications. The contributors were participants of the 8th ISAAC congress in Moscow in 2011 or are members of the PDE interest group of the ISAAC society. This volume is addressed to graduate students at various levels as well as researchers in partial differential equations and related fields. The readers will find this an excellent resource of both introductory and advanced material. The key topics are: • Linear hyperbolic equations and systems (scattering, symmetrisers) • Non-linear wave models (global existence, decay estimates, blow-up) • Evolution equations (control theory, well-posedness, smoothing) • Elliptic equations (uniqueness, non-uniqueness, positive solutions) • Special models from applications (Kirchhoff equation, Zakharov-Kuznetsov equation, thermoelasticity)

Mathematical Reviews

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Mathematical Reviews by N.A Book Summary:

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General Theory of Partial Differential Equations and Microlocal Analysis

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General Theory of Partial Differential Equations and Microlocal Analysis by Min-You Qi,L Rodino Book Summary:

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Hyperbolic Equations and Related Topics

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Hyperbolic Equations and Related Topics by Shigeru Mizohata Book Summary:

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Bibliographic Index

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Bibliographic Index by N.A Book Summary:

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Hyperbolic Problems and Regularity Questions

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Hyperbolic Problems and Regularity Questions by Mariarosaria Padula,Luisa Zanghirati Book Summary:

This book discusses new challenges in the quickly developing field of hyperbolic problems. Particular emphasis lies on the interaction between nonlinear partial differential equations, functional analysis and applied analysis as well as mechanics. The book originates from a recent conference focusing on hyperbolic problems and regularity questions. It is intended for researchers in functional analysis, PDE, fluid dynamics and differential geometry.

Hyperbolic Partial Differential Equations and Geometric Optics

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Hyperbolic Partial Differential Equations and Geometric Optics by Jeffrey Rauch Book Summary:

This book introduces graduate students and researchers in mathematics and the sciences to the multifaceted subject of the equations of hyperbolic type, which are used, in particular, to describe propagation of waves at finite speed. Among the topics carefully presented in the book are nonlinear geometric optics, the asymptotic analysis of short wavelength solutions, and nonlinear interaction of such waves. Studied in detail are the damping of waves, resonance, dispersive decay, and solutions to the compressible Euler equations with dense oscillations created by resonant interactions. Many fundamental results are presented for the first time in a textbook format. In addition to dense oscillations, these include the treatment of precise speed of propagation and the existence and stability questions for the three wave interaction equations. One of the strengths of this book is its careful motivation of ideas and proofs, showing how they evolve from related, simpler cases. This makes the book quite useful to both researchers and graduate students interested in hyperbolic partial differential equations. Numerous exercises encourage active participation of the reader. The author is a professor of mathematics at the University of Michigan. A recognized expert in partial differential equations, he has made important contributions to the transformation of three areas of hyperbolic partial differential equations: nonlinear microlocal analysis, the control of waves, and nonlinear geometric optics.

Method of Averaging for Differential Equations on an Infinite Interval

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Method of Averaging for Differential Equations on an Infinite Interval by Vladimir Burd Book Summary:

Explains the modern theory of the method of averaging and provides a understanding of the results obtained when applying this theory. This book starts with the less complicated theory of averaging linear differential equations (LDEs), focusing on almost periodic functions. It also includes chapters devoted to systems with a rapidly rotating phase.

Asymptotic Methods for Investigating Quasiwave Equations of Hyperbolic Type

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Asymptotic Methods for Investigating Quasiwave Equations of Hyperbolic Type by Yuri A. Mitropolsky,G. Khoma,M. Gromyak Book Summary:

The theory of partial differential equations is a wide and rapidly developing branch of contemporary mathematics. Problems related to partial differential equations of order higher than one are so diverse that a general theory can hardly be built up. There are several essentially different kinds of differential equations called elliptic, hyperbolic, and parabolic. Regarding the construction of solutions of Cauchy, mixed and boundary value problems, each kind of equation exhibits entirely different properties. Cauchy problems for hyperbolic equations and systems with variable coefficients have been studied in classical works of Petrovskii, Leret, Courant, Gording. Mixed problems for hyperbolic equations were considered by Vishik, Ladyzhenskaya, and that for general two dimensional equations were investigated by Bitsadze, Vishik, Gol'dberg, Ladyzhenskaya, Myshkis, and others. In last decade the theory of solvability on the whole of boundary value problems for nonlinear differential equations has received intensive development. Significant results for nonlinear elliptic and parabolic equations of second order were obtained in works of Gvazava, Ladyzhenskaya, Nakhushev, Oleinik, Skripnik, and others. Concerning the solvability in general of nonlinear hyperbolic equations, which are connected to the theory of local and nonlocal boundary value problems for hyperbolic equations, there are only partial results obtained by Bronshtein, Pokhozhev, Nakhushev.

Partial Differential Equations of Hyperbolic Type and Applications

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Partial Differential Equations of Hyperbolic Type and Applications by Giuseppe Geymonat Book Summary:

This book introduces the general aspects of hyperbolic conservation laws and their numerical approximation using some of the most modern tools: spectral methods, unstructured meshes and ?-formulation. The applications of these methods are found in some significant examples such as the Euler equations. This book, a collection of articles by the best authors in the field, exposes the reader to the frontier of the research and many open problems.

Fundamental Solutions for Differential Operators and Applications

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Fundamental Solutions for Differential Operators and Applications by Prem Kythe Book Summary:

A self-contained and systematic development of an aspect of analysis which deals with the theory of fundamental solutions for differential operators, and their applications to boundary value problems of mathematical physics, applied mathematics, and engineering, with the related computational aspects.

Mathematical Research Letters

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Mathematical Research Letters by N.A Book Summary:

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Partial Differential Equations in Clifford Analysis

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Partial Differential Equations in Clifford Analysis by Elena Obolashvili Book Summary:

Clifford analysis represents one of the most remarkable fields of modern mathematics. With the recent finding that almost all classical linear partial differential equations of mathematical physics can be set in the context of Clifford analysis-and that they can be obtained without applying any physical laws-it appears that Clifford analysis itself can suggest new equations or new generalizations of classical equations that may have some physical content. Partial Differential Equations in Clifford Analysis considers-in a multidimensional space-elliptic, hyperbolic, and parabolic operators related to Helmholtz, Klein-Gordon, Maxwell, Dirac, and heat equations. The author addresses two kinds of parabolic operators, both related to the second-order parabolic equations whose principal parts are the Laplacian and d'Alembertian: an elliptic-type parabolic operator and a hyperbolic-type parabolic operator. She obtains explicit integral representations of solutions to various boundary and initial value problems and their properties and solves some two-dimensional and non-local problems. Written for the specialist but accessible to non-specialists as well, Partial Differential Equations in Clifford Analysis presents new results, reformulations, refinements, and extensions of familiar material in a manner that allows the reader to feel and touch every formula and problem. Mathematicians and physicists interested in boundary and initial value problems, partial differential equations, and Clifford analysis will find this monograph a refreshing and insightful study that helps fill a void in the literature and in our knowledge.

Cauchy Problem for Differential Operators with Double Characteristics

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Cauchy Problem for Differential Operators with Double Characteristics by Tatsuo Nishitani Book Summary:

Combining geometrical and microlocal tools, this monograph gives detailed proofs of many well/ill-posed results related to the Cauchy problem for differential operators with non-effectively hyperbolic double characteristics. Previously scattered over numerous different publications, the results are presented from the viewpoint that the Hamilton map and the geometry of bicharacteristics completely characterizes the well/ill-posedness of the Cauchy problem. A doubly characteristic point of a differential operator P of order m (i.e. one where Pm = dPm = 0) is effectively hyperbolic if the Hamilton map FPm has real non-zero eigen values. When the characteristics are at most double and every double characteristic is effectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order terms. If there is a non-effectively hyperbolic characteristic, solvability requires the subprincipal symbol of P to lie between −Pμj and Pμj , where iμj are the positive imaginary eigenvalues of FPm . Moreover, if 0 is an eigenvalue of FPm with corresponding 4 × 4 Jordan block, the spectral structure of FPm is insufficient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role.

New Trends in the Theory of Hyperbolic Equations

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New Trends in the Theory of Hyperbolic Equations by Michael Reissig,Bert-Wolfgang Schulze Book Summary:

Presenting several developments in the theory of hyperbolic equations, this book's contributions deal with questions of low regularity, critical growth, ill-posedness, decay estimates for solutions of different non-linear hyperbolic models, and introduce new approaches based on microlocal methods.

Partial Differential Equations I

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Partial Differential Equations I by Michael Eugene Taylor,Eberhard Zeidler Book Summary:

This book is intended to be a comprehensive introduction to the subject of partial differential equations. It should be useful to graduate students at all levels beyond that of a basic course in measure theory. It should also be of interest to professional mathematicians in analysis, mathematical physics, and differential geometry. This work will be divided into three volumes, the first of which focuses on the theory of ordinary differential equations and a survey of basic linear PDEs.

Pseudo-Differential Operators: Analysis, Applications and Computations

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Pseudo-Differential Operators: Analysis, Applications and Computations by Luigi Rodino,M. W. Wong,Hongmei Zhu Book Summary:

This volume consists of eighteen peer-reviewed papers related to lectures on pseudo-differential operators presented at the meeting of the ISAAC Group in Pseudo-Differential Operators (IGPDO) held at Imperial College London on July 13-18, 2009. Featured in this volume are the analysis, applications and computations of pseudo-differential operators in mathematics, physics and signal analysis. This volume is a useful complement to the volumes “Advances in Pseudo-Differential Operators”, “Pseudo-Differential Operators and Related Topics”, “Modern Trends in Pseudo-Differential Operators”, “New Developments in Pseudo-Differential Operators” and “Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations” published in the same series in, respectively, 2004, 2006, 2007, 2009 and 2010.

Variable Lebesgue Spaces and Hyperbolic Systems

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Variable Lebesgue Spaces and Hyperbolic Systems by David Cruz-Uribe,Alberto Fiorenza,Michael Ruzhansky,Jens Wirth Book Summary:

This book targets graduate students and researchers who want to learn about Lebesgue spaces and solutions to hyperbolic equations. It is divided into two parts. Part 1 provides an introduction to the theory of variable Lebesgue spaces: Banach function spaces like the classical Lebesgue spaces but with the constant exponent replaced by an exponent function. These spaces arise naturally from the study of partial differential equations and variational integrals with non-standard growth conditions. They have applications to electrorheological fluids in physics and to image reconstruction. After an introduction that sketches history and motivation, the authors develop the function space properties of variable Lebesgue spaces; proofs are modeled on the classical theory. Subsequently, the Hardy-Littlewood maximal operator is discussed. In the last chapter, other operators from harmonic analysis are considered, such as convolution operators and singular integrals. The text is mostly self-contained, with only some more technical proofs and background material omitted. Part 2 gives an overview of the asymptotic properties of solutions to hyperbolic equations and systems with time-dependent coefficients. First, an overview of known results is given for general scalar hyperbolic equations of higher order with constant coefficients. Then strongly hyperbolic systems with time-dependent coefficients are considered. A feature of the described approach is that oscillations in coefficients are allowed. Propagators for the Cauchy problems are constructed as oscillatory integrals by working in appropriate time-frequency symbol classes. A number of examples is considered and the sharpness of results is discussed. An exemplary treatment of dissipative terms shows how effective lower order terms can change asymptotic properties and thus complements the exposition.

Blowup for Nonlinear Hyperbolic Equations

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Blowup for Nonlinear Hyperbolic Equations by Serge Alinhac Book Summary:

'The style is both lively and rigorous, and the proofs are well-organized making the main ingredients transparent. The examples are concrete and well-chosen...it nevertheless is a very good snapshot of the subject. This book is very stimulating and enjoyable. It should encourage students and researchers alike to contribute to the subject.' - Zbl. Math. Solutions to partial differential equations or systems often, over specific time periods, exhibit smooth behavior. Though given sufficient time they almost invariably undergo a brutal change in behavior, and this phenomenon has become known as 'blowup'. In this book, the author provides an overview of what is known about this situation and discusses many of the open problems concerning it. The book deals with classical solutions of global Cauchy problems for hyperbolic equations or systems. The approach is based on the display and study of two local blowup mechanisms, which the author calls the 'Ordinary Differential Equation mechanism' and the 'Geometric Blowup mechanism'. It introduces, via energy methods, the concept of lifespan, related to the nonlinear propagation of regularity (from the past to the future). It specifically addresses the question of whether or not there will be blowup in a solution, and it classifies those methods used to give positive answers to the question. The material corresponds with a one semester course for students or researchers with a basic elementary knowledge of Partial Differential Equations, especially of hyperbolic type including such topics as the Cauchy problem, wave operators, energy inequalities, finite speed of propagation, and symmetric systems. It contains a complete biography reflecting the high degree of activity among mathematicians interested in the problem. Series: Progress in Nonlinear Differential Equations and Their Applications, Volume 17 Condensed Table of Contents Chapter I. The two basic blowup mechanisms Introduction A. The ODE mechanism B. The geometric blowup mechanism C. Combinations of the two mechanisms Notes Chapter II. First concepts on global Cauchy problems Introduction Notes Chapter III. Semilinear wave equations Introduction Notes Chapter IV. Quasilinear equations in one space dimension Introduction Notes Chapter V. Nonlinear geometrical optics and applications Introduction Notes Bibliography Index

Solution Sets of Differential Equations in Abstract Spaces

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Solution Sets of Differential Equations in Abstract Spaces by Robert Dragoni,Paolo Nistri,Pietro Zecca,Jack W Macki Book Summary:

This book presents results on the geometric/topological structure of the solution set S of an initial-value problem x(t) = f(t, x(t)), x(0) =xo, when f is a continuous function with values in an infinite-dimensional space. A comprehensive survey of existence results and the properties of S, e.g. when S is a connected set, a retract, an acyclic set, is presented. The authors also survey results onthe properties of S for initial-value problems involving differential inclusions, and for boundary-value problems. This book will be of particular interest to researchers in ordinary and partial differential equations and some workers in control theory.

Doklady

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Doklady by N.A Book Summary:

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Partial Differential Equations and Related Topics

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Partial Differential Equations and Related Topics by J.A. Goldstein Book Summary:

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Boundary Value Problems and Markov Processes

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Boundary Value Problems and Markov Processes by Kazuaki Taira Book Summary:

This is a thorough and accessible exposition on the functional analytic approach to the problem of construction of Markov processes with Ventcel’ boundary conditions in probability theory. It presents new developments in the theory of singular integrals.

Monotone Operators in Banach Space and Nonlinear Partial Differential Equations

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Monotone Operators in Banach Space and Nonlinear Partial Differential Equations by R. E. Showalter Book Summary:

The objectives of this monograph are to present some topics from the theory of monotone operators and nonlinear semigroup theory which are directly applicable to the existence and uniqueness theory of initial-boundary-value problems for partial differential equations and to construct such operators as realizations of those problems in appropriate function spaces. A highlight of this presentation is the large number and variety of examples introduced to illustrate the connection between the theory of nonlinear operators and partial differential equations. These include primarily semilinear or quasilinear equations of elliptic or of parabolic type, degenerate cases with change of type, related systems and variational inequalities, and spatial boundary conditions of the usual Dirichlet, Neumann, Robin or dynamic type. The discussions of evolution equations include the usual initial-value problems as well as periodic or more general nonlocal constraints, history-value problems, those which may change type due to a possibly vanishing coefficient of the time derivative, and other implicit evolution equations or systems including hysteresis models. The scalar conservation law and semilinear wave equations are briefly mentioned, and hyperbolic systems arising from vibrations of elastic-plastic rods are developed. The origins of a representative sample of such problems are given in the appendix.

Hyperbolic Problems and Related Topics

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Hyperbolic Problems and Related Topics by Ferruccio Colombini,Tatsuo Nishitani Book Summary:

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Basic Research Resumés

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Basic Research Resumés by United States. Air Force. Research Division,Herner and Company Book Summary:

The document also contains a Contractor Index, Principal Investigator Index and Subject Index.

Reproducing Kernel Spaces and Applications

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Reproducing Kernel Spaces and Applications by Daniel Alpay Book Summary:

The notions of positive functions and of reproducing kernel Hilbert spaces play an important role in various fields of mathematics, such as stochastic processes, linear systems theory, operator theory, and the theory of analytic functions. Also they are relevant for many applications, for example to statistical learning theory and pattern recognition. The present volume contains a selection of papers which deal with different aspects of reproducing kernel Hilbert spaces. Topics considered include one complex variable theory, differential operators, the theory of self-similar systems, several complex variables, and the non-commutative case. The book is of interest to a wide audience of pure and applied mathematicians, electrical engineers and theoretical physicists.

Nonlinear Evolution Equations and Applications

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Nonlinear Evolution Equations and Applications by Gheorghe Morosanu Book Summary:

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Advances in Microlocal Analysis

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Advances in Microlocal Analysis by H.G. Garnir Book Summary:

The 1985 Castel vecchio-Pas coli NATO Advanced Study Institute is aimed to complete the trilogy with the two former institutes I organized : "Boundary Value Problem for Evolution Partial Differential Operators", Liege, 1976 and "Singularities in Boundary Value Problems", Maratea, 1980. It was indeed necessary to record the considerable progress realized in the field of the propagation of singularities of Schwartz Distri butions which led recently to the birth of a new branch of Mathema tical Analysis called Microlocal Analysis. Most of this theory was mainly built to be applied to distribution solutions of linear partial differential problems. A large part of this institute still went in this direction. But, on the other hand, it was also time to explore the new trend to use microlocal analysis In non linear differential problems. I hope that the Castelvecchio NATO ASI reached its purposes with the help of the more famous authorities in the field. The meeting was held in Tuscany (Italy) at Castelvecchio-Pascoli, little village in the mountains north of Lucca on September 2-12, 1985. It was hosted by "11 Ciocco" an international vacation Center, In a comfortable hotel located in magnificent mountain surroundings and provided with all conference and sport facilities.

The Analysis of Linear Partial Differential Operators III

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The Analysis of Linear Partial Differential Operators III by Lars Hörmander Book Summary:

From the reviews: "Volumes III and IV complete L. Hörmander's treatise on linear partial differential equations. They constitute the most complete and up-to-date account of this subject, by the author who has dominated it and made the most significant contributions in the last decades.....It is a superb book, which must be present in every mathematical library, and an indispensable tool for all - young and old - interested in the theory of partial differential operators." L. Boutet de Monvel in Bulletin of the American Mathematical Society, 1987. "This treatise is outstanding in every respect and must be counted among the great books in mathematics. It is certainly no easy reading (...) but a careful study is extremely rewarding for its wealth of ideas and techniques and the beauty of presentation." J. Brüning in Zentralblatt MATH, 1987.

Functional analysis and related topics, 1991

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Functional analysis and related topics, 1991 by Hikosaburō Komatsu Book Summary:

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Functional Analysis and Related Topics, 1991

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Functional Analysis and Related Topics, 1991 by Hikosaburo Komatsu Book Summary:

In these proceedings of the international conference held in Kyoto in memoryof the late Professor K saku Yosida, twenty six invited speakers display in their many facets of functional analysis and its applications in the research tradition of Yosida's school. Many of the topics are related tolinear and non-linear partial differential equations, including the Schr|dinger equations, the Navier-Stokes equations and quasilinear hyperbolic equations. Several of the papers are survey articles, the others are original (unpublished) and refereed research articles. Also included is a full listing of the publications of K. Yosida. Recommendedto students and research workers looking for a bird's-eye view of current research activity in functional analysis and its applications. FROM THE CONTENTS: K. Ito: Semigroups in probability theory.- T. Kato: Abstract evolution equations, linear and quasilinear, revisited.- J.L. Lions: Remarkson systems with incompletely given initial data and incompletely given part of the boundary.- H. Brezis: New energies for harmonic maps and liquid crystals.- D. Fujiwara: Some Feynman path integrals as oscillatory integrals over a Sobolev manifold.- M. Giga, Y. Giga, H. Sohr: L estimates for the Stokes system.- Y. Kawahigashi: Exactly solvable orbifold models and subfactors.- H. Kitada: Asymptotic completeness of N-body wave operators II. A new proof for the short-range case and the asymptotic clustering for the long-range systems. Y. Kobayashi, S. Oharu: Semigroups oflocally Lipschitzian operators and applications.- H. Komatsu: Operational calculus and semi-groups of operators.

Linear Theory of Colombeau Generalized Functions

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Linear Theory of Colombeau Generalized Functions by M Nedeljkov,S Pilipovic,D Scarpalezos Book Summary:

Results from the now-classical distribution theory involving convolution and Fourier transformation are extended to cater for Colombeau's generalized functions. Indications are given how these particular generalized functions can be used to investigate linear equations and pseudo differential operators. Furthermore, applications are also given to problems with nonregular data.

Partial Differential Equations and Their Applications

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Partial Differential Equations and Their Applications by Canadian Mathematical Society. Seminar Book Summary:

This volume presents lectures given at the 1995 Annual Seminar of the Canadian Mathematical Society on Partial Differential Equations and Their Applications held at the University of Toronto in June 1995. The conference consisted of a combination of minicourses, invited presentations, and contributed talks. In this volume readers will find contributions on a variety of topics related to PDE, such as spectral asymptotics, harmonic analysis, differential operators in hyperbolic manifolds, applications to geometry, mathematical physics, hydrodynamics, and the interaction between theory and numerical methods in PDE.

Extending Modules

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Extending Modules by Nguyen Viet Dung,Dinh Van Huynh,P F Smith,Robert Wisbauer Book Summary:

Module theory is an important tool for many different branches of mathematics, as well as being an interesting subject in its own right. Within module theory, the concept of injective modules is particularly important. Extending modules form a natural class of modules which is more general than the class of injective modules but retains many of its desirable properties. This book gathers together for the first time in one place recent work on extending modules. It is aimed at anyone with a basic knowledge of ring and module theory.