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Logical thinking, the analysis of complex relationships, the recognition of und- lying simple structures which are common to a multitude of problems — these are the skills which are needed to do mathematics, and their development is the main goal of mathematics education. Of course, these skills cannot be learned ‘in a vacuum’. Only a continuous struggle with concrete problems and a striving for deep understanding leads to success. A good measure of abstraction is needed to allow one to concentrate on the essential, without being distracted by appearances and irrelevancies. The present book strives for clarity and transparency. Right from the beg- ning, it requires from the reader a willingness to deal with abstract concepts, as well as a considerable measure of self-initiative. For these e?orts, the reader will be richly rewarded in his or her mathematical thinking abilities, and will possess the foundation needed for a deeper penetration into mathematics and its applications. Thisbookisthe?rstvolumeofathreevolumeintroductiontoanalysis.It- veloped from courses that the authors have taught over the last twenty six years at theUniversitiesofBochum,Kiel,Zurich,BaselandKassel.Sincewehopethatthis book will be used also for self-study and supplementary reading, we have included far more material than can be covered in a three semester sequence. This allows us to provide a wide overview of the subject and to present the many beautiful and important applications of the theory. We also demonstrate that mathematics possesses, not only elegance and inner beauty, but also provides e?cient methods for the solution of concrete problems.
analyse (wiskunde) --- Functional analysis --- Mathematics --- wiskunde --- Mathematical analysis --- functies (wiskunde) --- Mathematical analysis. --- Global analysis (Mathematics). --- Analysis. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Global analysis (Mathematics) --- Analysis (Mathematics). --- 517.1 Mathematical analysis
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In this small text the basic theory of the continuum, including the elements of metric space theory and continuity is developed within the system of intuitionistic mathematics in the sense of L.E.J. Brouwer and H. Weyl. The main features are proofs of the famous theorems of Brouwer concerning the continuity of all functions that are defined on "whole" intervals, the uniform continuity of all functions that are defined on compact intervals, and the uniform convergence of all pointwise converging sequences of functions defined on compact intervals. The constructive approach is interesting both in itself and as a contrast to, for example, the formal axiomatic one.
Mathematics. --- Analysis. --- Global analysis (Mathematics). --- Mathématiques --- Analyse globale (Mathématiques) --- Continuity. --- Mathematical analysis. --- Engineering & Applied Sciences --- Applied Mathematics --- Analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Global analysis (Mathematics) --- 517.1 Mathematical analysis --- Mathematical analysis
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International Mathematical Series Volume 12 Around the Research of Vladimir Maz'ya II Partial Differential Equations Edited by Ari Laptev Numerous influential contributions of Vladimir Maz'ya to PDEs are related to diverse areas. In particular, the following topics, close to the scientific interests of V. Maz'ya are discussed: semilinear elliptic equation with an exponential nonlinearity resolvents, eigenvalues, and eigenfunctions of elliptic operators in perturbed domains, homogenization, asymptotics for the Laplace-Dirichlet equation in a perturbed polygonal domain, the Navier-Stokes equation on Lipschitz domains in Riemannian manifolds, nondegenerate quasilinear subelliptic equations of p-Laplacian type, singular perturbations of elliptic systems, elliptic inequalities on Riemannian manifolds, polynomial solutions to the Dirichlet problem, the first Neumann eigenvalues for a conformal class of Riemannian metrics, the boundary regularity for quasilinear equations, the problem on a steady flow over a two-dimensional obstacle, the well posedness and asymptotics for the Stokes equation, integral equations for harmonic single layer potential in domains with cusps, the Stokes equations in a convex polyhedron, periodic scattering problems, the Neumann problem for 4th order differential operators. Contributors include: Catherine Bandle (Switzerland), Vitaly Moroz (UK), and Wolfgang Reichel (Germany); Gerassimos Barbatis (Greece), Victor I. Burenkov (Italy), and Pier Domenico Lamberti (Italy); Grigori Chechkin (Russia); Monique Dauge (France), Sebastien Tordeux (France), and Gregory Vial (France); Martin Dindos (UK); Andras Domokos (USA) and Juan J. Manfredi (USA); Yuri V. Egorov (France), Nicolas Meunier (France), and Evariste Sanchez-Palencia (France); Alexander Grigor'yan (Germany) and Vladimir A. Kondratiev (Russia); Dmitry Khavinson (USA) and Nikos Stylianopoulos (Cyprus); Gerasim Kokarev (UK) and Nikolai Nadirashvili (France); Vitali Liskevich (UK) and Igor I. Skrypnik (Ukraine); Oleg Motygin (Russia) and Nikolay Kuznetsov (Russia); Grigory P. Panasenko (France) and Ruxandra Stavre (Romania); Sergei V. Poborchi (Russia); Jurgen Rossmann (Germany); Gunther Schmidt (Germany); Gregory C. Verchota (USA). Ari Laptev Imperial College London (UK) and Royal Institute of Technology (Sweden) Ari Laptev is a world-recognized specialist in Spectral Theory of Differential Operators. He is the President of the European Mathematical Society for the period 2007- 2010. Tamara Rozhkovskaya Sobolev Institute of Mathematics SB RAS (Russia) and an independent publisher Editors and Authors are exclusively invited to contribute to volumes highlighting recent advances in various fields of mathematics by the Series Editor and a founder of the IMS Tamara Rozhkovskaya. Cover image: Vladimir Maz'ya.
Mathematics. --- Analysis. --- Partial Differential Equations. --- Functional Analysis. --- Global analysis (Mathematics). --- Functional analysis. --- Differential equations, partial. --- Mathématiques --- Analyse globale (Mathématiques) --- Analyse fonctionnelle --- Mathematics --- Global analysis (Mathematics) --- Functional analysis --- Differential equations, Partial --- Mazia, V. G. --- Mathematical analysis. --- Analysis (Mathematics). --- Partial differential equations. --- 517.1 Mathematical analysis --- Mathematical analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations
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The idea of this book is to give an extensive description of the classical complex analysis, here ''classical'' means roughly that sheaf theoretical and cohomological methods are omitted. The first four chapters cover the essential core of complex analysis presenting their fundamental results. After this standard material, the authors step forward to elliptic functions and to elliptic modular functions including a taste of all most beautiful results of this field. The book is rounded by applications to analytic number theory including distinguished pearls of this fascinating subject as for instance the Prime Number Theorem. Great importance is attached to completeness, all needed notions are developed, only minimal prerequisites (elementary facts of calculus and algebra) are required. More than 400 exercises including hints for solutions and many figures make this an attractive, indispensable book for students who would like to have a sound introduction to classical complex analysis. For the second edition the authors have revised the text carefully.
Mathematics. --- Functions of a Complex Variable. --- Functions of complex variables. --- Mathématiques --- Fonctions d'une variable complexe --- Functions of complex variables --- Complex variables --- Elliptic functions --- Functions of real variables --- Problems, exercises, etc. --- Electronic books. -- local. --- Functions of complex variables -- Problems, exercises, etc. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Mathematical analysis. --- Analysis (Mathematics). --- Analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Science --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Geometry, Algebraic
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This book is intended for students who wish to get an introduction to the theory of partial differential equations. The author focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. These are maximum principle methods (particularly important for numerical analysis schemes), parabolic equations, variational methods, and continuity methods. This book also develops the main methods for obtaining estimates for solutions of elliptic equations: Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. Connections between elliptic, parabolic, and hyperbolic equations are explored, as well as the connection with Brownian motion and semigroups. This book can be utilized for a one-year course on partial differential equations. For the new edition the author has added a new chapter on reaction-diffusion equations and systems. There is also new material on Neumann boundary value problems, Poincaré inequalities, expansions, as well as a new proof of the Hölder regularity of solutions of the Poisson equation. Jürgen Jost is Co-Director of the Max Planck Institute for Mathematics in the Sciences and Professor of Mathematics at the University of Leipzig. He is the author of a number of Springer books, including Dynamical Systems (2005), Postmodern Analysis (3rd ed. 2005, also translated into Japanese), Compact Riemann Surfaces (3rd ed. 2006) and Riemannian Geometry and Geometric Analysis (4th ed., 2005). The present book is an expanded translation of the original German version, Partielle Differentialgleichungen (1998). About the first edition: Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations. Teachers will also find in this textbook the basis of an introductory course on second-order partial differential equations. - Alain Brillard, Mathematical Reviews Beautifully written and superbly well-organised, I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of mathematics. - Nick Lord, The Mathematical Gazette.
Mathematics. --- Partial Differential Equations. --- Theoretical, Mathematical and Computational Physics. --- Numerical and Computational Physics. --- Differential equations, partial. --- Mathématiques --- Differential equations, Partial --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Differential equations, Partial. --- Partial differential equations --- Mathematical analysis. --- Analysis (Mathematics). --- Partial differential equations. --- Physics. --- Analysis. --- Global analysis (Mathematics). --- Numerical and Computational Physics, Simulation. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Physical mathematics --- Physics --- 517.1 Mathematical analysis --- Mathematical analysis
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Complex Analysis with Applications in Science and Engineering weaves together theory and extensive applications in mathematics, physics and engineering. In this edition there are many new problems, revised sections, and an entirely new chapter on analytic continuation. This work will serve as a textbook for undergraduate and graduate students in the areas noted above. Key Features of this Second Edition: Excellent coverage of topics such as series, residues and the evaluation of integrals, multivalued functions, conformal mapping, dispersion relations and analytic continuation Systematic and clear presentation with many diagrams to clarify discussion of the material Numerous worked examples and a large number of assigned problems Excerpts from reviews of the 1st edition: "The textbook Fundamentals and Applications of Complex Analysis by Harold Cohen is an idiosyncratic treatment of the subject, written by a physicist, with lots of interesting insights and alternative ways of viewing the ideas and methods of complex analysis. The book includes several excursions into applications of interest to physicists and electrical engineers, as circuit analysis and a chapter on dispersion relations. The book would be of particular interest to physics and electrical engineering students. Mathematicians might find it a good hunting ground for offbeat approaches to familiar themes and for various other serendipities." —Theodore W. Gamelin, professor of mathematics, UCLA "This book might be useful for readers who are still familiar with complex analysis and who are searching for several examples, exercises or applications in physics and engineering…" — Zentralblatt Math.
Mathematics. --- Functions of a Complex Variable. --- Theoretical, Mathematical and Computational Physics. --- Appl.Mathematics/Computational Methods of Engineering. --- Functions of complex variables. --- Engineering mathematics. --- Mathématiques --- Fonctions d'une variable complexe --- Mathématiques de l'ingénieur --- Electronic books. -- local. --- Mathematical analysis. --- Functions of complex variables --- Mathematical analysis --- Mathematics --- Calculus --- Physical Sciences & Mathematics --- 517.1 Mathematical analysis --- Complex variables --- Physics. --- Applied mathematics. --- Elliptic functions --- Functions of real variables --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Mathematical physics. --- Physical mathematics --- Physics
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This volume contains 23 articles on algebraic analysis of differential equations and related topics, most of which were presented as papers at the conference ""Algebraic Analysis of Differential Equations - from Microlocal Analysis to Exponential Asymptotics"" at Kyoto University in 2005. This volume is dedicated to Professor Takahiro Kawai, who is one of the creators of microlocal analysis and who introduced the technique of microlocal analysis into exponential asymptotics.
Mathematics. --- Integral Transforms, Operational Calculus. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Special Functions. --- Integral Transforms. --- Differential Equations. --- Differential equations, partial. --- Functions, special. --- Mathématiques --- Differential equations. --- Mathematical analysis. --- Differential equations --- Mathematical analysis --- Mathematics --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Operations Research --- Calculus --- 517.1 Mathematical analysis --- 517.91 Differential equations --- Integral transforms. --- Operational calculus. --- Partial differential equations. --- Special functions. --- Special functions --- Partial differential equations --- Operational calculus --- Electric circuits --- Integral equations --- Transform calculus --- Transformations (Mathematics) --- Math --- Science
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There has been a flurry of activity in recent years in the loosely defined area of holomorphic spaces. This book discusses the most well-known and widely used spaces of holomorphic functions in the unit ball of C^n. Spaces discussed include the Bergman spaces, the Hardy spaces, the Bloch space, BMOA, the Dirichlet space, the Besov spaces, and the Lipschitz spaces. Most proofs in the book are new and simpler than the existing ones in the literature. The central idea in almost all these proofs is based on integral representations of holomorphic functions and elementary properties of the Bergman kernel, the Bergman metric, and the automorphism group. The unit ball was chosen as the setting since most results can be achieved there using straightforward formulas without much fuss. The book can be read comfortably by anyone familiar with single variable complex analysis; no prerequisite on several complex variables is required. The author has included exercises at the end of each chapter that vary greatly in the level of difficulty. Kehe Zhu is Professor of Mathematics at State University of New York at Albany. His previous books include Operator Theory in Function Spaces (Marcel Dekker 1990), Theory of Bergman Spaces, with H. Hedenmalm and B. Korenblum (Springer 2000), and An Introduction to Operator Algebras (CRC Press 1993).
Mathematical analysis --- Analytical spaces --- analyse (wiskunde) --- Holomorphic functions --- Fonctions holomorphes --- EPUB-LIV-FT SPRINGER-B LIVMATHE --- Holomorphic functions. --- Unit ball. --- Differential equations, partial. --- Global analysis (Mathematics). --- Several Complex Variables and Analytic Spaces. --- Analysis. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Partial differential equations --- Functions of complex variables. --- Mathematical analysis. --- Analysis (Mathematics). --- 517.1 Mathematical analysis --- Complex variables --- Elliptic functions --- Functions of real variables
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Whatdoasupernovaexplosioninouterspace,?owaroundanairfoil and knocking in combustion engines have in common? The physical and chemical mechanisms as well as the sizes of these processes are quite di?erent. So are the motivations for studying them scienti?cally. The super- 8 nova is a thermo-nuclear explosion on a scale of 10 cm. Astrophysicists try to understand them in order to get insight into fundamental properties of the universe. In ?ows around airfoils of commercial airliners at the scale of 3 10 cm shock waves occur that in?uence the stability of the wings as well as fuel consumption in ?ight. This requires appropriate design of the shape and structure of airfoils by engineers. Knocking occurs in combustion, a chemical 1 process, and must be avoided since it damages motors. The scale is 10 cm and these processes must be optimized for e?ciency and environmental conside- tions. The common thread is that the underlying ?uid ?ows may at a certain scale of observation be described by basically the same type of hyperbolic s- tems of partial di?erential equations in divergence form, called conservation laws. Astrophysicists, engineers and mathematicians share a common interest in scienti?c progress on theory for these equations and the development of computational methods for solutions of the equations. Due to their wide applicability in modeling of continua, partial di?erential equationsareamajor?eldofresearchinmathematics. Asubstantialportionof mathematical research is related to the analysis and numerical approximation of solutions to such equations. Hyperbolic conservation laws in two or more spacedimensionsstillposeoneofthemainchallengestomodernmathematics.
Conservation laws (Mathematics) --- Fluid mechanics --- Mathematics. --- Hydromechanics --- Continuum mechanics --- Differential equations, Hyperbolic --- Computer science --- Global analysis (Mathematics). --- Numerical analysis. --- Hydraulic engineering. --- Computational Mathematics and Numerical Analysis. --- Analysis. --- Numerical Analysis. --- Engineering Fluid Dynamics. --- Classical and Continuum Physics. --- Astrophysics and Astroparticles. --- Engineering, Hydraulic --- Engineering --- Hydraulics --- Shore protection --- Mathematical analysis --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Mathematics --- Computer mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Fluid mechanics. --- Continuum physics. --- Astrophysics. --- Astronomical physics --- Astronomy --- Cosmic physics --- Physics --- Classical field theory --- Continuum physics --- 517.1 Mathematical analysis
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The study of CR manifolds lies at the intersection of three main mathematical disciplines: partial differential equations, complex analysis in several complex variables, and differential geometry. While the PDE and complex analytic aspects have been intensely studied in the last fifty years, much effort has recently been made to understand the differential geometric side of the subject. This monograph provides a unified presentation of several differential geometric aspects in the theory of CR manifolds and tangential Cauchy–Riemann equations. It presents the major differential geometric achievements in the theory of CR manifolds, such as the Tanaka–Webster connection, Fefferman's metric, pseudo-Einstein structures and the Lee conjecture, CR immersions, subelliptic harmonic maps as a local manifestation of pseudoharmonic maps from a CR manifold, Yang–Mills fields on CR manifolds, to name a few. It also aims at explaining how certain results from analysis are employed in CR geometry. Motivated by clear exposition, many examples, explicitly worked-out geometric results, and stimulating unproved statements and comments referring to the most recent aspects of the theory, this monograph is suitable for researchers and graduate students in differential geometry, complex analysis, and PDEs.
Geometry, Differential. --- CR submanifolds. --- Differentiable manifolds. --- Differential manifolds --- Manifolds (Mathematics) --- Cauchy-Riemann submanifolds --- Submanifolds, CR --- Differential geometry --- CR submanifolds --- Differentiable manifolds --- Geometry, Differential --- Global differential geometry. --- Global analysis. --- Differential equations, partial. --- Global analysis (Mathematics). --- Differential Geometry. --- Global Analysis and Analysis on Manifolds. --- Partial Differential Equations. --- Several Complex Variables and Analytic Spaces. --- Analysis. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Partial differential equations --- Differential geometry. --- Manifolds (Mathematics). --- Partial differential equations. --- Functions of complex variables. --- Mathematical analysis. --- Analysis (Mathematics). --- 517.1 Mathematical analysis --- Mathematical analysis --- Complex variables --- Elliptic functions --- Functions of real variables --- Topology
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