Calculus and mathematical analysis Books

Book SynopsisThe authors consider applications of singularity theory and computer algebra to bifurcations of Hamiltonian dynamical systems. They restrict themselves to the case were the following simplification is possible. Near the equilibrium or (quasi-) periodic solution under consideration the linear part allows approximation by a normalized Hamiltonian system with a torus symmetry. It is assumed that reduction by this symmetry leads to a system with one degree of freedom. The volume focuses on two such reduction methods, the planar reduction (or polar coordinates) method and the reduction by the energy momentum mapping. The one-degree-of-freedom system then is tackled by singularity theory, where computer algebra, in particular, Gröbner basis techniques, are applied. The readership addressed consists of advanced graduate students and researchers in dynamical systems.Table of ContentsIntroduction.- I. Applications: Methods I: Planar reduction; Method II: The energy-momentum map.- II. Theory: Birkhoff Normalization; Singularity Theory; Gröbner bases and Standard bases; Computing normalizing transformations.- Appendix A.1. Classification of term orders; Appendix A.2. Proof of Proposition 5.8.- References.- Index.

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Book SynopsisThis edition contains detailed solutions of selected exercises. Many readers have requested this, because it makes the book more suitable for self-study. At the same time new exercises (without solutions) have beed added. They have all been placed in the end of each chapter, in order to facilitate the use of this edition together with previous ones. Several errors have been corrected and formulations have been improved. This has been made possible by the valuable comments from (in alphabetical order) Jon Bohlin, Mark Davis, Helge Holden, Patrick Jaillet, Chen Jing, Natalia Koroleva,MarioLefebvre,Alexander Matasov,Thilo Meyer-Brandis, Keigo Osawa, Bjorn Thunestvedt, Jan Uboe and Yngve Williassen. I thank them all for helping to improve the book. My thanks also go to Dina Haraldsson, who once again has performed the typing and drawn the ?gures with great skill. Blindern, September 2002 Bernt Oksendal xv Preface to Corrected Printing, Fifth Edition The main corrections and improvements in this corrected printing are from Chapter 12. I have bene?tted from useful comments from a number of p- ple, including (in alphabetical order) Fredrik Dahl, Simone Deparis, Ulrich Haussmann, Yaozhong Hu, Marianne Huebner, Carl Peter Kirkebo, Ni- lay Kolev, Takashi Kumagai, Shlomo Levental, Geir Magnussen, Anders Oksendal, Jur . . gen Pottho?, Colin Rowat, Stig Sandnes, Lones Smith, S- suo Taniguchi and Bjorn Thunestvedt. I want to thank them all for helping me making the book better. I also want to thank Dina Haraldsson for pro?cient typing.Trade ReviewFrom the reviews of the fifth edition: "This is a highly readable and refreshingly rigorous introduction to stochastic calculus. … This is not a watered-down treatment. It is a serious introduction that starts with fundamental measure-theoretic concepts and ends, coincidentally, with the Black-Scholes formula as one of several examples of applications. This is the best single resource for learning the stochastic calculus … ." (riskbook.com, 2002) From the reviews of the sixth edition: "The book … has evolved from a 200-page typewritten booklet to a modern classic. Part of its charm and success is the fact that the author does not bother too much with the (for the novice) cumbersome rigorous theory … . This does not mean that the book is not rigorous, it is just the timing and dosage of mathematical rigour … that is palatable for undergraduates … . a highly readable account, suitable for self-study and for use in the classroom." (René L. Schilling, The Mathematical Gazette, March, 2005) "This is the sixth edition of the classical and excellent book on stochastic differential equations. The main difference with the next to last edition is the addition of detailed solutions of selected exercises … . This is certainly an excellent idea in view to test its ability of applications of the concepts … . certainly one of the best books on the subject, it will be very helpful to any graduate students and also very valuable for any analysts of financial market." (Stéphane Métens, Physicalia, Vol. 26 (1), 2004) "This is now the sixth edition of the excellent book on stochastic differential equations and related topics. … the presentation is successfully balanced between being easily accessible for a broad audience and being mathematically rigorous. The book is a first choice for courses at graduate level in applied stochastic differential equations. The inclusion of detailed solutions to many of the exercises in this edition also makes it very useful for self-study." (Evelyn Buckwar, Zentralblatt MATH, Vol. 1025, 2003)Table of ContentsSome Mathematical Preliminaries.- Itô Integrals.- The Itô Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary Value Problems.- Application to Optimal Stopping.- Application to Stochastic Control.- Application to Mathematical Finance.

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Table of ContentsSome analytic function algebras.- A local characterization of analytic structure in a commutative Banach algebra.- A differential version of a theorem of mergelyan.- Polynomial approximation on thin sets.- On an example of Stolzenberg.- Flat differential operators.- Fiber integration and some applications.- Parametrizing the compact submanifolds of a period matrix domain by a Stein manifold.- Generalizations of Grauert's direct image theorem.- Cohomology of analytic families of differential complexes.- Families of strongly pseudoconvex manifolds.- Extending analytic subvarieties.- On algebraic divisors in ?K.- Problems.

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Book SynopsisTable of ContentsTopics in C*- and von neumann algebras.- Infinitely divisible probaility measures on compact groups.- Darstellung Verallgemeinerter L1-Algebren II.- Lectures on the trace in a finite von Neumann algebra.- Cohomology of operator algebras.- Generations of von Neumann algebras.- Hyponormal operators and related topics.- Duality and von Neumann algebras.

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Book SynopsisFunctions in R and C, including the theory of Fourier series, Fourier integrals and part of that of holomorphic functions, form the focal topic of these two volumes. Based on a course given by the author to large audiences at Paris VII University for many years, the exposition proceeds somewhat nonlinearly, blending rigorous mathematics skilfully with didactical and historical considerations. It sets out to illustrate the variety of possible approaches to the main results, in order to initiate the reader to methods, the underlying reasoning, and fundamental ideas. It is suitable for both teaching and self-study. In his familiar, personal style, the author emphasizes ideas over calculations and, avoiding the condensed style frequently found in textbooks, explains these ideas without parsimony of words. The French edition in four volumes, published from 1998, has met with resounding success: the first two volumes are now available in English.Trade ReviewFrom the reviews of the original French edition: "... The content is quite classical ... [...] The treatment is less classical: precise although unpedantic (rather far from the definition-theorem-corollary-style), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. [...] The author gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction also contains comments that are very unusual in a book on mathematical analysis, going from pedagogy to critique of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such a way that readers are less surprised than they might be.J. Mawhin in Zentralblatt Mathematik (1999) From the reviews: "Analysis I is the translation of the first volume of Godement’s four-volume work Analyse Mathématique, which offers a development of analysis more or less from the beginning up to some rather advanced topics. … the organization of the material is radically different … . It would … make excellent supplementary reading for honors calculus courses." (Gerald B. Folland, SIAM Review, Vol. 47 (3), 2005) "A book on analysis that is quite different from all other books on this subject. … for those who essentially know the material (the level of an average graduate student, say), and who are interested in mathematics will certainly love reading it. Those who lecture this material may find a lot of inspiration to make their lessons entertaining." (Adhemar Bultheel, Bulletin of the Belgian Mathematical Society, Vol. 12 (2), 2005) "Analysis I is an English translation of the first volume of a four-volume work. Analysis I consists of a spirally organized, organic, non-linear treatment of the introductory areas of ‘mathematical analysis as it was and as it has become’. It is infused with some excellent, sensitive appreciations of the work of pioneers … and reads as a heady blend of both classical concerns and modern refinements, often illuminated by a variety of approaches." (Nick Lord, The Mathematical Gazette, March, 2005)Table of ContentsI Sets and Functions.- §1. Set Theory.- 1 - Membership, equality, empty set.- 2 - The set defined by a relation. Intersections and unions.- 3 - Whole numbers. Infinite sets.- 4 - Ordered pairs, Cartesian products, sets of subsets.- 5 - Functions, maps, correspondences.- 6 - Injections, surjections, bijections.- 7 - Equipotent sets. Countable sets.- 8 - The different types of infinity.- 9 - Ordinals and cardinals.- §2. The logic of logicians.- II - Convergence: Discrete variables.- §1. Convergent sequences and series.- 0 - Introduction: what is a real number?.- 1 - Algebraic operations and the order relation: axioms of ?.- 2 - Inequalities and intervals.- 3 - Local or asymptotic properties.- 4 - The concept of limit. Continuity and differentiability.- 5 - Convergent sequences: definition and examples.- 6 - The language of series.- 7 - The marvels of the harmonic series.- 8 - Algebraic operations on limits.- §2. Absolutely convergent series.- 9 - Increasing sequences. Upper bound of a set of real numbers.- 10 - The function log x. Roots of a positive number.- 11 - What is an integral?.- 12 - Series with positive terms.- 13 - Alternating series.- 14 - Classical absolutely convergent series.- 15 - Unconditional convergence: general case.- 16 - Comparison relations. Criteria of Cauchy and d’Alembert.- 17 - Infinite limits.- 18 - Unconditional convergence: associativity.- §3. First concepts of analytic functions.- 19 - The Taylor series.- 20 - The principle of analytic continuation.- 21 - The function cot x and the series ?1/n2k.- 22 - Multiplication of series. Composition of analytic functions Formal series.- 23 - The elliptic functions of Weierstrass.- III - Convergence: Continuous variables.- §1. The intermediate value theorem.- 1 - Limit values of a function. Open and closed sets.- 2 - Continuous functions.- 3 - Right and left limits of a monotone function.- 4 - The intermediate value theorem.- §2. Uniform convergence.- 5 - Limits of continuous functions.- 6 - A slip up of Cauchy’s.- 7 - The uniform metric.- 8 - Series of continuous functions. Normal convergence.- §3. Bolzano-Weierstrass and Cauchy’s criterion.- 9 - Nested intervals, Bolzano-Weierstrass, compact sets.- 10 - Cauchy’s general convergence criterion.- 11 - Cauchy’s criterion for series: examples.- 12 - Limits of limits.- 13 - Passing to the limit in a series of functions.- §4. Differentiable functions.- 14 - Derivatives of a function.- 15 - Rules for calculating derivatives.- 16 - The mean value theorem.- 17 - Sequences and series of differentiable functions.- 18 - Extensions to unconditional convergence.- §5. Differentiable functions of several variables.- 19 - Partial derivatives and differentials.- 20 - Differentiability of functions of class C1.- 21 - Differentiation of composite functions.- 22 - Limits of differentiable functions.- 23 - Interchanging the order of differentiation.- 24 - Implicit functions.- Appendix to Chapter III.- 1 - Cartesian spaces and general metric spaces.- 2 - Open and closed sets.- 3 - Limits and Cauchy’s criterion in a metric space; complete spaces.- 4 - Continuous functions.- 5 - Absolutely convergent series in a Banach space.- 6 - Continuous linear maps.- 7 - Compact spaces.- 8 - Topological spaces.- IV Powers, Exponentials, Logarithms, Trigonometric Functions.- §1. Direct construction.- 1 - Rational exponents.- 2 - Definition of real powers.- 3 - The calculus of real exponents.- 4 - Logarithms to base a. Power functions.- 5 - Asymptotic behaviour.- 6 - Characterisations of the exponential, power and logarithmic functions.- 7 - Derivatives of the exponential functions: direct method.- 8 - Derivatives of exponential functions, powers and logarithms.- §2. Series expansions.- 9 - The number e. Napierian logarithms.- 10 - Exponential and logarithmic series: direct method.- 11 - Newton’s binomial series.- 12 - The power series for the logarithm.- 13 - The exponential function as a limit.- 14 - Imaginary exponentials and trigonometric functions.- 15 - Euler’s relation chez Euler.- 16 - Hyperbolic functions.- §3. Infinite products.- 17 - Absolutely convergent infinite products.- 18 - The infinite product for the sine function.- 19 - Expansion of an infinite product in series.- 20 - Strange identities.- §4. The topology of the functions Arg(z) and Log z.

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Book SynopsisProceedingsTable of ContentsMy mathematical expectations.- Admissibility and the integral equations of asymptotic theory.- Differential inequalities and boundary problems for functional-differential equations.- Singularly perturbed boundary value problems revisited.- Bounded solutions of nonlinear equations at an irregular type singularity.- On meromorphic solutions of the difference equation y(x+1)=y(x)+1+? / y(x).- Branching of periodic solutions.- Effective solutions for meromorphic second order differential equations.- Optimal control of limit cycles or what control theory can do to cure a heart attack or to cause one.- The stable manifold theorem via an isolating block.- Stability of a lurie type equation.- A nonlinear integral equation relating distillation processes.- Totally implicity methods for numerical solution of singular initial value problems.- Dichotomies for differential and integral equations.- An entire solution of the functional equation f(?)+f(? ?)f(??1?)=1, (?5=1).

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Book SynopsisTable of ContentsI. Überlagerungen.- § 1. Definition der Riemannschen Flächen.- § 2. Einfache Eigenschaften holomorpher Abbildungen.- § 3. Homotopie von Kurven. Fundamentalgruppe.- § 4. Verzweigte und unverzweigte Überlagerungen.- § 5. Universelle Überlagerung, Decktransformationen.- § 6. Garben.- § 7. Analytische Fortsetzung.- § 8. Algebraische Funktionen.- § 9. Differentialformen.- § 10. Integration von Differentialformen.- § 11. Lineare Differentialgleichungen.- II. Kompakte Riemannsche Flächen.- § 12. Cohomologiegruppen.- § 13. Das Dolbeaultsche Lemma.- § 14. Ein Endlichkeitssatz.- § 15. Die exakte Cohomologiesequenz.- § 16. Der Satz von Riemann-Roch.- § 17. Der Serresche Dualitätssatz.- § 18. Funktionen und Differentialformen zu vorgegebenen Hauptteilen.- § 19. Harmonische Differentialformen.- §.20. Das Abelsche Theorem.- § 21. Das Jacobische Umkehrproblem.- III. Nicht-kompakte Riemannsche Flächen.- § 22. Das Dirichletsche Randwertproblem.- § 23. Abzählbarkeit der Topologie.- § 24. Das Weylsche Lemma.- § 25. Der Rungesche Approximationssatz.- § 26. Die Sätze von Mittag-Leffler und Weierstraß..- § 27. Der Riemannsche Abbildungssatz.- § 28. Funktionen zu vorgegebenen Automorphiesummanden.- § 29. Geraden- und Vektorraumbündel.- § 30. Trivialität von Vektorraumbündeln.- § 31. Das Riemann-Hilbertsche Problem.- A. Teilungen der Eins.- B. Topologische Vektorräume.- Literaturhinweise.- Symbolverzeichnis.- Namen- und Sachverzeichnis.

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Table of ContentsElectrostatics.- Poisson's equation.- Fundamental solutions.- Capacity.- Energy.- Existence of the equilibrium distribution.- Maximum principle for potentials.- Uniqueness of the equilibrium distribution.- The cone condition.- Singularities of bounded harmonic functions.- Green's function.- The kelvin transform.- Perron's method.- Barriers.- Kellogg's theorem.- The riesz decomposition theorem.- Applications of the riesz decomposition.- Wiener's criterion.

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Table of ContentsAnalysis in functional spaces.- Banach algebras.- Probabilistic problems.- Operator theory.- Hankel and toeplitz operators.- Singular integrals, BMO, Hp.- Spectral analysis and synthesis.- Approximation and Capacities.- Uniqueness, moments, normality.- Interpolation, bases, multiplers.- Entire, meromorphic and subharmonic functions.- ?n.- Miscellaneous problems.- Solutions.

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Table of ContentsRearrangements.- Maximum principles.

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Table of ContentsMonodromy and poles of ?X |f|2??.- Le groupe de monodromie des familles universelles d'hypersurfaces et d'intersections completes.- Complete families of stable vector bundles over ?2.- Appendix to the paper “complete families of stable vector bundles over ?”.- On the minimal model problem.- Modulräume holomorpher Abbildungen auf komplexen Mannigfaltigkeiten mit 1-konkavem Rand.- Stable rationality of some moduli spaces of vector bundles on P2.- Compact kähler manifolds of nonnegative holomorphic bisectional curvature.- Concavity, convexity and complements in complex spaces.- Subvarieties in homogeneous manifolds.- Rational curves in mois?zon 3-folds.- On the structure of 4 folds with a hyperplane section which is a ?1 bundle over a ruled surface.- Complex surfaces with negative tangent bundle.- Nonequidimensional value distribution theory and subvariety extension.- On the adjunction theoretic structure of projective varieties.- Value distribution theory for moving targets.

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Table of ContentsMultiplicity of spectral measures.- The spectral theorem.- Bochner’s theorem.- Distribution of cocycles.- Cocycles on the line.

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Book SynopsisThese notes will be useful and of interest to mathematicians and physicists active in research as well as for students with some knowledge of the abstract theory of operators in Hilbert spaces. They give a complete spectral theory for ordinary differential expressions of arbitrary order n operating on -valued functions existence and construction of self-adjoint realizations via boundary conditions, determination and study of general properties of the resolvent, spectral representation and spectral resolution. Special attention is paid to the question of separated boundary conditions, spectral multiplicity and absolutely continuous spectrum. For the case nm=2 (Sturm-Liouville operators and Dirac systems) the classical theory of Weyl-Titchmarch is included. Oscillation theory for Sturm-Liouville operators and Dirac systems is developed and applied to the study of the essential and absolutely continuous spectrum. The results are illustrated by the explicit solution of a number of particular problems including the spectral theory one partical Schrödinger and Dirac operators with spherically symmetric potentials. The methods of proof are functionally analytic wherever possible.Table of ContentsFormally self-adjoint differential expressions.- Appendix to section 1: The separation of the Dirac operator.- Fundamental properties and general assumptions.- Appendix to section 2: Proof of the Lagrange identity for n>2.- The minimal operator and the maximal operator.- Deficiency indices and self-adjoint extensions of T0.- The solutions of the inhomogeneous differential equation (?-?)u=f; Weyl's alternative.- Limit point-limit circle criteria.- Appendix to section 6: Semi-boundedness of Sturm-Liouville type operators.- The resolvents of self-adjoint extensions of T0.- The spectral representation of self-adjoint extensions of T0.- Computation of the spectral matrix ?.- Special properties of the spectral representation, spectral multiplicities.- L2-solutions and essential spectrum.- Differential operators with periodic coefficients.- Appendix to section 12: Operators with periodic coefficients on the half-line.- Oscillation theory for regular Sturm-Liouville operators.- Oscillation theory for singular Sturm-Liouville operators.- Essential spectrum and absolutely continuous spectrum of Sturm-Liouville operators.- Oscillation theory for Dirac systems, essential spectrum and absolutely continuous spectrum.- Some explicitly solvable problems.

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Book SynopsisThis research monograph concerns the Nevanlinna factorization of analytic functions smooth, in a sense, up to the boundary. The peculiar properties of such a factorization are investigated for the most common classes of Lipschitz-like analytic functions. The book sets out to create a satisfactory factorization theory as exists for Hardy classes. The reader will find, among other things, the theorem on smoothness for the outer part of a function, the generalization of the theorem of V.P. Havin and F.A. Shamoyan also known in the mathematical lore as the unpublished Carleson-Jacobs theorem, the complete description of the zero-set of analytic functions continuous up to the boundary, generalizing the classical Carleson-Beurling theorem, and the structure of closed ideals in the new wide range of Banach algebras of analytic functions. The first three chapters assume the reader has taken a standard course on one complex variable; the fourth chapter requires supplementary papers cited there. The monograph addresses both final year students and doctoral students beginning to work in this area, and researchers who will find here new results, proofs and methods.Table of ContentsNotations.- The (F)-property.- Moduli of analytic functions smooth up to the boundary.- Zeros and their multiplicities.- Closed ideals in the space X pq ? (?,?).

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Book SynopsisKodaira is a Fields Medal Prize Winner. (In the absence of a Nobel prize in mathematics, they are regarded as the highest professional honour a mathematician can attain.) Kodaira is an honorary member of the London Mathematical Society. Affordable softcover edition of 1986 classicTable of ContentsHolomorphic Functions.- Complex Manifolds.- Differential Forms, Vector Bundles, Sheaves.- Infinitesimal Deformation.- Theorem of Existence.- Theorem of Completeness.- Theorem of Stability.

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Book SynopsisThe goal of the book is to summarize those methods for evaluating Feynman integrals that have been developed over a span of more than fifty years. The book characterizes the most powerful methods and illustrates them with numerous examples starting from very simple ones and progressing to nontrivial examples. The book demonstrates how to choose adequate methods and combine evaluation methods in a non-trivial way. The most powerful methods are characterized and then illustrated through numerous examples. This is an updated textbook version of the previous book (Evaluating Feynman integrals, STMP 211) of the author.Trade ReviewFrom the reviews: "The book is based on the courses of lectures given by the author in the two winter semesters of 2003-2004 and 2005-2006 at the University of Hamburg as a DFG Mercator professor in Hamburg as well as on the course given in 2003-2004 at the University of Karlsruhe. It will be useful for postgraduate students and theoretical physicists specializing in quantum field theory." (Michael B. Mensky, Zentralblatt MATH, Vol. 1111 (8), 2007)Table of ContentsFeynman Integrals: Basic Definitions and Tools.- Evaluating by Alpha and Feynman Parameters.- Evaluating by MB Representation.- IBP and Reduction to Master Integrals.- Reduction to Master Integrals by Baikov’s Method.- Evaluation by Differential Equations.- Tables.- Some Special Functions.- Summation Formulae.- Table of MB Integrals.- Analysis of Convergence and Sector Decompositions.- A Brief Review of Some Other Methods.- Applying Gröbner Bases to Solve IBP Relations.- Solutions.

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Book SynopsisDie ersten vier Kapitel dieser Darstellung der klassischen Funktionentheorie vermitteln mit minimalem Begriffsaufwand und auf geringen Vorkenntnissen aufbauend zentrale Ergebnisse und Methoden der komplexen Analysis einer Veränderlichen. Sie gipfeln in einem Beweis des kleinen Riemannschen Abbildungssatzes und einer Charakterisierung einfach zusammenhängender Gebiete. Weitere Themen sind: elliptische Funktionen (Weierstraßscher, Jacobischer Ansatz), die elementare Theorie der Modulformen einer Variablen, Anwendungen der Funktionen- auf die Zahlentheorie (einschl. eines Beweises des Primzahlsatzes). Plus: über 400 Übungsaufgaben mit Lösungen. Trade Review"... Jeder einzelne Abschnitt enthält sorgfältig ausgewählte Übungsaufgaben." Monatshefte für Mathematik "... Positiv hervorzuheben sind die optisch sehr übersichtliche Aufbereitung und der Versuch der Autoren, alle Begriffsbildungen dem Leser gegenüber soweit wie möglich zu motivieren ..." Internationale Mathematische Nachrichten ÖsterreichTable of ContentsDifferentialrechnung im Komplexen.- Integralrechnung im Komplexen.- Folgen und Reihen analytischer Funktionen, Residuensatz.- Konstruktion analytischer Funktionen.- Elliptische Funktionen.- Elliptische Modulformen.- Analytische Zahlentheorie.

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Book SynopsisAusführlicher Einblick in die Anfänge der Analysis: von der Einführung der reellen Zahlen bis hin zu fortgeschrittenen Themen wie Differentialformen auf Mannigfaltigkeiten, asymptotische Betrachtungen, Fourier-, Laplace- und Legendre-Transformationen, elliptische Funktionen und Distributionen. Ausgerichtet auf naturwissenschaftliche Fragestellungen und in detaillierter Herangehensweise an die Integral- und Differentialrechnung. Mit einer Fülle hilfreicher Beispiele, Aufgaben und Anwendungen. In Band 1: vollständige Übersicht zur Integral- und Differentialrechnung einer Variablen, erweitert um die Differentialrechnung mehrerer Variablen. Trade ReviewAus den Rezensionen der englischen Ausgabe: "Diese profunde Einführung [Math.Analysis I und II] in die Analysis sollte in keiner mathematischen Bibliothek fehlen, selbst bei budgetären Restriktionen, trotz der Überfülle an Einführungsbüchern. Eine genaue, bewußte Lektüre dieses profunden Werks könnte mögliche künftige Autoren mittelmäßiger Analysisbücher vielleicht abschrecken. [...]Meisterhaft wird hier intuitives Verstehen gefördert, vermittelt durch anschauliche geometrische Denkweisen, heuristische Ideen und induktive Vorgangsweisen, ohne Exaktheitsansprüche hintanzustellen oder konkrete Details oder Anwendungen auch nur ansatzweise zu vernachlässigen. Der Aufbau ist in vieler Hinsicht ungewöhnlich, eröffnet frühe Einblicke und Weitblicke und regt zum Denken an [...], ist auch der historischen Entwicklung angemessen und bietet eine wichtige Alternative zu den vielen "eleganten" Zugängen, bei denen die Vermittlung wichtiger nötiger Entwicklungsschritte für ein aktives Verständnis zu kurz kommt. Der umfassende, Nachbardisziplinen laufend berührende Zugang trägt reiche Früchte, ebenso die facettenreiche Fülle an Erklärungen der Wurzeln und Essenz der grundlegenden Konzepte und Resultate, die Beschreibungen von Zusammenhängen und Ausblicke auf weitere Entwicklungen mit vielen in Einführungsbüchern leider eher unüblichen Anwendungen und Querbezügen [...]. Man erwirbt mit diesem Werk zusätzlich ein vollständiges, umfangreiches und wertvolles "Problem-Buch". Bei aller reichhaltiger Fülle stellt sich die Mathematik hier aber immer als eine Einheit dar, in ihrer auf den heutigen Stellenwert Bezug nehmenden historischen und philosophischen Entwicklung, geprägt durch, an passender Stelle kompetent gewürdigte, bedeutende große schöpferische Persönlichkeiten. [...] Dieses vorzügliche Werk atmet den Geist einer bewunderungswürdigen, vielschichtigen Forscher- und Lehrerpersönlichkeit." H.Rindler, Monatshefte für Mathematik 146, Issue 4, 2005 "Die vorliegenden zwei Bände sind die englische Übersetzung eines russischen Werkes, das bereits Anfang der achtziger Jahre erschienen ist und inzwischen bereits zum vierten Mal aufgelegt wurde. Die Bücher beinhalten auf über 1200 Seiten die klassische Analysis in einer zeitgemäßen Darstellung sowie Querverbindungen zu Algebra, Differenzailgleichungen, Differenzialgeometrie, komplexe Analysis und Funktionalanlaysis. Addressaten sind Studenten (und Lehrende), die neben einer strengen mathematischen Theorie auch konkrete Anwendungen suchen... Dieses ausgezeichnete Werk kann Studienanfängern und fortgeschrittenen Studierenden uneingeschränkt empfohlen werden, aber auch Lehrende werden viele Anregungen darin finden." M.Kronfellner (Wien), IMN - Internationale Mathematische Nachrichten 59, Issue 198, 2005, S. 36-37 Aus den Rezensionen: "Der umfangreiche Band enthält den … Stoff einer Analysisvorlesung … Viel Raum wird … der Behandlung der Grundlagen gewidmet. … Im weiteren Verlauf beleben dann immer wieder naturwissenschaftliche und technische Anwendungen die mathematische Theorie. Jeder Abschnitt endet mit Aufgabenstellungen. Bei aller mathematischen Strenge sind die Ausführungen verständlich und vermeiden nicht unbedingt erforderliche abstrakte Ausweitungen … Empfehlenswert als Begleitlektüre zum Studium." (Wolfgang Grölz, in: ekz-Informationsdienst Einkaufszentrale für öffentliche Bibliotheken, 2006, Issue 52)Table of ContentsInhaltsverzeichnis 1 Allgemeine mathematische Begriffe und Schreibweisen . . . . . 1 1.1 Logische Symbole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Bindew¨orter und Klammern . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Hinweise zu Beweisen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Einige besondere Schreibweisen. . . . . . . . . . . . . . . . . . . . . . 3 1.1.4 Abschließende Anmerkungen . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.5 ¨Ubungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Mengen und elementare Mengenoperationen . . . . . . . . . . . . . . . . 5 1.2.1 Der Begriff einer Menge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Teilmengen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Elementare Mengenoperationen . . . . . . . . . . . . . . . . . . . . . 9 1.2.4 ¨Ubungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Funktionen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.1 Der Begriff einer Funktion (Abbildung) . . . . . . . . . . . . . . 12 1.3.2 Elementare Klassifizierung von Abbildungen . . . . . . . . . . 17 1.3.3 Zusammengesetzte Funktionen. Inverse Abbildungen . . . 18 1.3.4 Funktionen als Relationen. Der Graph einer Funktion . . 20 1.3.5 ¨Ubungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4 Erg¨anzungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4.1 Die M¨achtigkeit einer Menge (Kardinalzahlen) . . . . . . . . 27 1.4.2 Axiome der Mengenlehre . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4.3 S¨atze in der Sprache der Mengenlehre . . . . . . . . . . . . . . . . 31 1.4.4 ¨Ubungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 Die reellen Zahlen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1 Axiome und Eigenschaften der reellen Zahlen . . . . . . . . . . . . . . . 38 2.1.1 Definition der Menge der reellen Zahlen . . . . . . . . . . . . . . 38 2.1.2 Algebraische Eigenschaften der reellen Zahlen . . . . . . . . . 42 2.1.3 Das Vollst¨andigkeitsaxiom. Die kleinste obere Schranke 46 2.2 Klassen reeller Zahlen und Berechnungen . . . . . . . . . . . . . . . . . . . 49 2.2.1 Die nat¨urlichen Zahlen. Mathematische Induktion . . . . . 49 XVI Inhaltsverzeichnis 2.2.2 Rationale und irrationale Zahlen . . . . . . . . . . . . . . . . . . . . 52 2.2.3 Das archimedische Prinzip . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.2.4 Geometrische Interpretation. Gesichtspunkte beim Rechnen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.2.5 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.3 Wichtige S¨atze zur Vollst¨andigkeit . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.3.1 Der Satz zur Intervallschachtelung . . . . . . . . . . . . . . . . . . . 74 2.3.2 Der Satz zur endlichen ¨Uberdeckung . . . . . . . . . . . . . . . . . 75 2.3.3 Der Satz vom H¨aufungspunkt . . . . . . . . . . . . . . . . . . . . . . . 76 2.3.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.4 Abz¨ahlbare und ¨uberabz¨ahlbare Mengen . . . . . . . . . . . . . . . . . . . 78 2.4.1 Abz¨ahlbare Mengen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.4.2 Die M¨achtigkeit des Kontinuums . . . . . . . . . . . . . . . . . . . . 80 2.4.3 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3 Grenzwerte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.1 Der Grenzwert einer Folge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.1.1 Definitionen und Beispiele . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.1.2 Eigenschaften des Grenzwertes einer Folge . . . . . . . . . . . . 86 3.1.3 Existenz des Grenzwertes einer Folge . . . . . . . . . . . . . . . . 90 3.1.4 Elementares zu Reihen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.1.5 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.2 Der Grenzwert einer Funktion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.2.1 Definitionen und Beispiele . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.2.2 Eigenschaften des Grenzwertes einer Funktion . . . . . . . . . 116 3.2.3 Grenzwert auf einer Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.2.4 Existenz des Grenzwertes einer Funktion . . . . . . . . . . . . . 137 3.2.5 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4 Stetige Funktionen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.1 Wichtige Definitionen und Beispiele . . . . . . . . . . . . . . . . . . . . . . . . 157 4.1.1 Stetigkeit einer Funktion in einem Punkt . . . . . . . . . . . . . 157 4.1.2 Unstetigkeitsstellen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.2 Eigenschaften stetiger Funktionen . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.2.1 Lokale Eigenschaften . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.2.2 Globale Eigenschaften stetiger Funktionen . . . . . . . . . . . . 167 4.2.3 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5 Differentialrechnung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.1 Differenzierbare Funktionen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.1.1 Problemstellung und einleitende Betrachtungen . . . . . . . 181 5.1.2 In einem Punkt differenzierbare Funktionen . . . . . . . . . . . 186 5.1.3 Tangenten und geometrische Interpretation der Ableitung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5.1.4 Die Rolle des Koordinatensystems . . . . . . . . . . . . . . . . . . . 192 Inhaltsverzeichnis XVII 5.1.5 Einige Beispiele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.1.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.2 Wichtige Ableitungsregeln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.2.1 Differentiation und arithmetische Operationen . . . . . . . . 201 5.2.2 Differentiation einer verketteten Funktion (Kettenregel) 205 5.2.3 Differentiation einer inversen Funktion . . . . . . . . . . . . . . . 208 5.2.4 Ableitungstabelle der Elementarfunktionen . . . . . . . . . . . 213 5.2.5 Differentiation einer sehr einfachen impliziten Funktion 213 5.2.6 Ableitungen h¨oherer Ordnung . . . . . . . . . . . . . . . . . . . . . . . 218 5.2.7 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 5.3 Die zentralen S¨atze der Differentialrechnung . . . . . . . . . . . . . . . . 223 5.3.1 Der Satz von Fermat und der Satz von Rolle . . . . . . . . . . 223 5.3.2 Der Mittelwertsatz und der Satz von Cauchy. . . . . . . . . . 225 5.3.3 Die Taylorschen Formeln . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 5.3.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 5.4 Differentialrechnung zur Untersuchung von Funktionen . . . . . . . 246 5.4.1 Bedingungen f¨ur die Monotonie einer Funktion . . . . . . . . 246 5.4.2 Bedingungen f¨ur ein inneres Extremum einer Funktion . 247 5.4.3 Bedingungen f¨ur die Konvexit¨at einer Funktion . . . . . . . 253 5.4.4 Die Regel von L’Hˆopital . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 5.4.5 Das Konstruieren von Graphen von Funktionen . . . . . . . 263 5.4.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5.5 Komplexe Zahlen und Elementarfunktionen . . . . . . . . . . . . . . . . . 276 5.5.1 Komplexe Zahlen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 5.5.2 Konvergenz in C und Reihen mit komplexen Gliedern . . 280 5.5.3 Eulersche Formel und Elementarfunktionen . . . . . . . . . . . 285 5.5.4 Analytischer Zugang zur Potenzreihendarstellung . . . . . . 288 5.5.5 Algebraische Abgeschlossenheit des K¨orpers C . . . . . . . . 293 5.5.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 5.6 Beispiele zur Differentialrechnung in den Naturwissenschaften . 301 5.6.1 Bewegung eines K¨orpers mit ver¨anderlicher Masse . . . . . 302 5.6.2 Die barometrische H¨ohenformel . . . . . . . . . . . . . . . . . . . . . 304 5.6.3 Radioaktiver Zerfall und Kernreaktoren . . . . . . . . . . . . . . 306 5.6.4 In der Atmosph¨are fallende K¨orper . . . . . . . . . . . . . . . . . . 308 5.6.5 Die Zahl e und ein erneuter Blick auf exp x . . . . . . . . . . . 310 5.6.6 Schwingungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 5.6.7 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 5.7 Stammfunktionen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 5.7.1 Stammfunktionen und das unbestimmte Integral . . . . . . 321 5.7.2 Allgemeine Methoden zur Bestimmung einer Stammfunktion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 5.7.3 Stammfunktionen rationaler Funktionen . . . . . . . . . . . . . . 329 5.7.4 Stammfunktionen der Form R R(cos x, sin x) dx . . . . . . . . 333 5.7.5 Stammfunktionen der Form R R(x, y(x)) dx . . . . . . . . . . . 335 5.7.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 XVIII Inhaltsverzeichnis 6 Integralrechnung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 6.1 Definition des Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 6.1.1 Problemstellung und einf¨uhrende Betrachtungen . . . . . . 345 6.1.2 Definition des Riemannschen Integrals . . . . . . . . . . . . . . . 347 6.1.3 Die Menge der integrierbaren Funktionen . . . . . . . . . . . . . 349 6.1.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 6.2 Linearit¨at, Additivit¨at und Monotonie des Integrals . . . . . . . . . . 365 6.2.1 Das Integral als lineare Funktion auf dem Raum R[a, b] 365 6.2.2 Das Integral als eine additive Intervallfunktion . . . . . . . . 365 6.2.3 Absch¨atzung, Monotonie und Mittelwertsatz . . . . . . . . . . 368 6.2.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 6.3 Das Integral und die Ableitung . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 6.3.1 Das Integral und die Stammfunktion . . . . . . . . . . . . . . . . . 377 6.3.2 Fundamentalsatz der Integral- und Differentialrechnung 380 6.3.3 Partielle Integration und Taylorsche Formel . . . . . . . . . . . 381 6.3.4 ¨Anderung der Variablen in einem Integral . . . . . . . . . . . . 383 6.3.5 Einige Beispiele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 6.3.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 6.4 Einige Anwendungen der Integralrechnung . . . . . . . . . . . . . . . . . . 393 6.4.1 Additive Intervallfunktionen und das Integral . . . . . . . . . 393 6.4.2 Bogenl¨ange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 6.4.3 Die Fl¨ache eines krummlinigen Trapezes . . . . . . . . . . . . . . 402 6.4.4 Volumen eines Drehk¨orpers . . . . . . . . . . . . . . . . . . . . . . . . . 404 6.4.5 Arbeit und Energie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 6.4.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 6.5 Uneigentliche Integrale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 6.5.1 Definition, Beispiele und wichtige Eigenschaften . . . . . . . 413 6.5.2 Konvergenz eines uneigentlichen Integrals . . . . . . . . . . . . 418 6.5.3 Uneigentliche Integrale mit mehr als einer Singularit¨at . 425 6.5.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 7 Funktionen mehrerer Variabler . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 7.1 Der Raum Rm und seine Unterr¨aume . . . . . . . . . . . . . . . . . . . . . . 432 7.1.1 Die Menge Rm und der Abstand in dieser Menge . . . . . . 432 7.1.2 Offene und abgeschlossene Mengen in Rm . . . . . . . . . . . . 433 7.1.3 Kompakte Mengen in Rm . . . . . . . . . . . . . . . . . . . . . . . . . . 436 7.1.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 7.2 Grenzwerte und Stetigkeit von Funktionen mehrerer Variabler . 438 7.2.1 Der Grenzwert einer Funktion . . . . . . . . . . . . . . . . . . . . . . . 438 7.2.2 Stetigkeit einer Funktion mehrerer Variabler . . . . . . . . . . 444 7.2.3 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 Inhaltsverzeichnis XIX 8 Differentialrechnung mit Funktionen mehrerer Variabler . . . 451 8.1 Die lineare Struktur auf Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 8.1.1 Rm als Vektorraum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 8.1.2 Lineare Transformationen L : Rm ! Rn . . . . . . . . . . . . . . 452 8.1.3 Die Norm in Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 8.1.4 Die euklidische Struktur auf Rm . . . . . . . . . . . . . . . . . . . . . 455 8.2 Das Differential einer Funktion mehrerer Variabler . . . . . . . . . . . 456 8.2.1 Differenzierbarkeit und das Differential in einem Punkt . 456 8.2.2 Partielle Ableitung einer Funktion mit reellen Werten . . 457 8.2.3 Die Jacobimatrix in koordinatenweiser Darstellung . . . . 460 8.2.4 Partielle Ableitungen und Differenzierbarkeit in einem Punkt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 8.3 Die wichtigsten Gesetze der Differentiation . . . . . . . . . . . . . . . . . 462 8.3.1 Linearit¨at der Ableitung . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 8.3.2 Ableitung verketteter Abbildungen (Kettenregel) . . . . . . 465 8.3.3 Ableitung einer inversen Abbildung . . . . . . . . . . . . . . . . . . 470 8.3.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 8.4 Reelle Funktionen mehrerer Variabler . . . . . . . . . . . . . . . . . . . . . . 478 8.4.1 Der Mittelwertsatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 8.4.2 Eine hinreichende Bedingung f¨ur die Differenzierbarkeit 480 8.4.3 Partielle Ableitungen h¨oherer Ordnung . . . . . . . . . . . . . . . 481 8.4.4 Die Taylorsche Formel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 8.4.5 Extrema von Funktionen mehrerer Variabler . . . . . . . . . . 486 8.4.6 Einige geometrische Darstellungen . . . . . . . . . . . . . . . . . . . 493 8.4.7 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 8.5 Der Satz zur impliziten Funktion . . . . . . . . . . . . . . . . . . . . . . . . . . 504 8.5.1 Einleitung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 8.5.2 Ein einfacher Satz zur impliziten Funktion . . . . . . . . . . . . 506 8.5.3 ¨Ubergang zur Gleichung F(x1, . . . , xm, y) = 0 . . . . . . . . . 510 8.5.4 Der Satz zur impliziten Funktion . . . . . . . . . . . . . . . . . . . . 513 8.5.5 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 8.6 Einige Korollare zum Satz zur impliziten Funktion . . . . . . . . . . . 522 8.6.1 Der Satz zur inversen Funktion. . . . . . . . . . . . . . . . . . . . . . 522 8.6.2 Lokale Reduktion in kanonische Form . . . . . . . . . . . . . . . . 527 8.6.3 Funktionale Abh¨angigkeit . . . . . . . . . . . . . . . . . . . . . . . . . . 532 8.6.4 Lokale Zerlegung eines Diffeomorphismus . . . . . . . . . . . . . 534 8.6.5 Das Morse-Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 8.6.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 8.7 Fl¨achen in Rn und bedingte Extrema . . . . . . . . . . . . . . . . . . . . . . 542 8.7.1 k-dimensionale Fl¨achen in Rn . . . . . . . . . . . . . . . . . . . . . . . 542 8.7.2 Der Tangentialraum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 8.7.3 Extrema mit Nebenbedingungen . . . . . . . . . . . . . . . . . . . . 552 8.7.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 XX Inhaltsverzeichnis Einige Aufgaben aus den Halbjahrespr¨ufungen . . . . . . . . . . . . . . . . 571 1. Einf¨uhrung der Analysis (Zahlen, Funktionen, Grenzwerte) . . . . . . 571 2. Differentialrechnung in einer Variablen . . . . . . . . . . . . . . . . . . . . . . . 572 3. Integration und Einf¨uhrung mehrerer Variabler . . . . . . . . . . . . . . . . 574 4. Differentialrechnung mehrerer Variabler . . . . . . . . . . . . . . . . . . . . . . 575 Pr¨ufungsgebiete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 1. Erstes Semester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 1.1. Einleitung und Differentialrechnung in einer Variablen . . . . 579 2. Zweites Semester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 2.1. Integration. Differentialrechnung mit mehreren Variablen . 581 Literaturverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 1. Klassische Werke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 1.1 Orginalquellen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 1.2 Wichtige umfassende grundlegende Werke . . . . . . . . . . . . . . . 585 1.3 Klassische Vorlesungen in Analysis aus der ersten H¨alfte des 20. Jahrhunderts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 2. Lehrb¨ucher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 3. Studienunterlagen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 4. Weiterf¨uhrende Literatur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Namensverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 Sachverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

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Book SynopsisFew books on Ordinary Differential Equations (ODEs) have the elegant geometric insight of this one, which puts emphasis on the qualitative and geometric properties of ODEs and their solutions, rather than on routine presentation of algorithms. From the reviews: "Professor Arnold has expanded his classic book to include new material on exponential growth, predator-prey, the pendulum, impulse response, symmetry groups and group actions, perturbation and bifurcation." --SIAM REVIEWTrade ReviewFrom the reviews: "Professor Arnold has expanded his classic book to include new material on exponential growth, predator-prey, the pendulum, impulse response, symmetry groups and group actions, perturbation and bifurcation … . The new edition is highly recommended as a general reference for the essential theory of ordinary differential equations and as a textbook for an introductory course for serious undergraduate or graduate students. … In the US system, it is an excellent text for an introductory graduate course." (Carmen Chicone, SIAM Review, Vol. 49 (2), 2007) "Vladimir Arnold’s is a master, not just of the technical realm of differential equations but of pedagogy and exposition as well. … The writing throughout is crisp and clear. … Arnold’s says that the book is based on a year-long sequence of lectures for second-year mathematics majors in Moscow. In the U.S., this material is probably most appropriate for advanced undergraduates or first-year graduate students." (William J. Satzer, MathDL, August, 2007)Table of ContentsBasic Concepts.- Basic Theorems.- Linear Systems.- Proofs of the Main Theorems.- Differential Equations on Manifolds.

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Book SynopsisSince its inception by von Neumann 70 years ago, the theory of operator algebras has become a rapidly developing area of importance for the understanding of many areas of mathematics and theoretical physics. Accessible to the non-specialist, this first part of a three-volume treatise provides a clear, carefully written survey that emphasizes the theory's analytical and topological aspects.Trade Review"... These three bulky volumes [EMS 124, 125, 127], written by one of the most prominent researchers of the area, provide an introduction to this repidly developing theory. ... These books can be warmly recommended to every graduate student who wants to become acquainted with this exciting branch of matematics. Furthermore, they should be on the bookshelf of every researcher of the area." (László Kérchy, Acta Scientiarum Mathematicarum, Vol. 69, 2003)Table of ContentsFundaments of Banach Algebras and C*-Algebras.- Topologies and Density Theorems in Operator Algebras.- Conjugate Spaces.- Tensor Products of Operator Algebras and Direct Integrals.- Types of von Neumann Algebras and Traces.- Appendix: Polish Spaces and Standard Borel Spaces.

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Book SynopsisSheaf Theory is modern, active field of mathematics at the intersection of algebraic topology, algebraic geometry and partial differential equations. This volume offers a comprehensive and self-contained treatment of Sheaf Theory from the basis up, with emphasis on the microlocal point of view. From the reviews: "Clearly and precisely written, and contains many interesting ideas: it describes a whole, largely new branch of mathematics." –Bulletin of the L.M.S.Table of ContentsA Short History: Les débuts de la théorie des faisceaux.- I. Homological algebra.- II. Sheaves.- III. Poincaré-Verdier duality and Fourier-Sato transformation.- IV. Specialization and microlocalization.- V. Micro-support of sheaves.- VI. Micro-support and microlocalization.- VII. Contact transformations and pure sheaves.- VIII. Constructible sheaves.- IX. Characteristic cycles.- X. Perverse sheaves.- XI. Applications to O-modules and D-modules.- Appendix: Symplectic geometry.- Summary.- A.1. Symplectic vector spaces.- A.2. Homogeneous symplectic manifolds.- A.3. Inertia index.- Exercises to the Appendix.- Notes.- List of notations and conventions.

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Book SynopsisThe volume contains the texts of the main talks delivered at the International Symposium on Complex Geometry and Analysis held in Pisa, May 23-27, 1988. The Symposium was organized on the occasion of the sixtieth birthday of Edoardo Vesentini. The aim of the lectures was to describe the present situation, the recent developments and research trends for several relevant topics in the field. The contributions are by distinguished mathematicians who have actively collaborated with the mathematical school in Pisa over the past thirty years.Table of ContentsHyperkähler manifolds.- Affine differential geometry and holomorphic curves.- The meromorphic continuation of Kloosterman-Selberg zeta functions.- Deformation of compact Riemann surfaces Y of genus p with distinguished points P 1 …, P m ? Y.- On moduli of vector bundles.- Quasiconformal mappings on CR manifolds.- On the stability of positive semigroups generated by operator matrices.- The levi problem on algebraic manifolds.- A Banach-Steinhaus theorem for weak and order continuous operators.- Fixed points of holomorphic mappings.

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Book SynopsisThis monograph provides a systematic treatment of the abstract theory of adjoint semigroups. After presenting the basic elementary results, the following topics are treated in detail: The sigma (X, X )-topology, -reflexivity, the Favard class, Hille-Yosida operators, interpolation and extrapolation, weak -continuous semigroups, the codimension of X in X , adjoint semigroups and the Radon-Nikodym property, tensor products of semigroups and duality, positive semigroups and multiplication semigroups. The major part of the material is reasonably self-contained and is accessible to anyone with basic knowledge of semi- group theory and Banach space theory. Most of the results are proved in detail. The book is addressed primarily to researchers working in semigroup theory, but in view of the "Banach space theory" flavour of many of the results, it will also be of interest to Banach space geometers and operator theorists.Table of ContentsThe adjoint semigroup.- The ?(X,X?)-topology.- Interpolation, extrapolation and duality.- Perturbation theory.- Dichotomy theorems.- Adjoint semigroups and the RNP.- Tensor products.- The adjoint of a positive semigroup.

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Book SynopsisThe monograph is a study of the local bifurcations of multiparameter symplectic maps of arbitrary dimension in the neighborhood of a fixed point.The problem is reduced to a study of critical points of an equivariant gradient bifurcation problem, using the correspondence between orbits ofa symplectic map and critical points of an action functional. New results onsingularity theory for equivariant gradient bifurcation problems are obtained and then used to classify singularities of bifurcating period-q points. Of particular interest is that a general framework for analyzing group-theoretic aspects and singularities of symplectic maps (particularly period-q points) is presented. Topics include: bifurcations when the symplectic map has spatial symmetry and a theory for the collision of multipliers near rational points with and without spatial symmetry. The monograph also includes 11 self-contained appendices each with a basic result on symplectic maps. The monograph will appeal to researchers and graduate students in the areas of symplectic maps, Hamiltonian systems, singularity theory and equivariant bifurcation theory.Table of ContentsGeneric bifurcation of periodic points.- Singularity theory for equivariant gradient bifurcation problems.- Classification of Zq-equivariant gradient bifurcation problems.- Period-3 points of the generalized standard map.- Classification of Dq-equivariant gradient bifurcation problems.- Reversibility and degenerate bifurcation of period-q points of multiparameter maps.- Periodic points of equivariant symplectic maps.- Collision of multipliers at rational points for symplectic maps.- Equivariant maps and the collision of multipliers.

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Book SynopsisThis book deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing-up technique, geometric and topological methods. It explores important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved.Table of Contents1 Riemannian Geometry.- 2 Sobolev Spaces.- 3 Background Material.- 4 Complementary Material.- 5 The Yamabe Problem.- 6 Prescribed Scalar Curvature.- 7 Einstein—Kähler Metrics.- 8 Monge—Ampère Equations.- 9 The Ricci Curvature.- 10 Harmonic Maps.- Bibliography*.- Notation.

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Book SynopsisL'ouvrage traite des fondements de la mécanique des milieux continus en grandes transformations. Les éléments et théories usuels sont présentés. Une modélisation originale de l'état de déformation est proposée. Elle ne s'identifie ni à une mesure de déformation contingente ni au classique tenseur metrique. La théorie matérielle fondée sur cette variable d'état non tensorielle est intrinsèque et cohérente avec les éléments incontestés de l'approche eulérienne. Les théories matérielles usuelles (lagrangiennes, en rotation,...) en sont des approximations dont la pertinence est étudiée. Ce livre est susceptible d'intéresser aussi bien les étudiants de troisième cycle en mécanique ou en mathématiques appliquées que les chercheurs et ingénieurs.Trade ReviewL'ouvrage traite des fondements de la mécanique des milieux continus en grandes transformations. Ce livre est susceptible d'intéresser aussi bien les étudiants de troisième cycle en mécanique ou en mathématiques appliquées que les chercheurs et ingénieurs.Table of ContentsPO - Le cadre classique I. Le cadre classique II. La physique dans l'espace-temps cinématiqueP1 - Le Mouvement III. Milieu continu en mouvement IV. Etude locale des vitesses V. Etude locale de la transformation VI. P2 - Approche matérielle VII. Le modèle matière VIII. Les dérivées matérielles IX. Le référentiel matière X. Dérivée matérielle des tenseurs euclidiensP3 - La Forme XI. Approche spatiale du comportement XII. Elasticité et élastoplasticité XIII. Intégration de D XIV. La variété des métriques XV. Taille, forme et triaxialité XVI. Cinématique de la déformation XVII. Approche matérielle du comportementP4- La Déformation XVIII. Approche spatiale de la déformation XIX. Approche matérielle de la déformation XX. Les approches matérielles classiquesP5 - Compléments

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Book SynopsisFrom the reviews: "The work is one of the real classics of this century; it has had much influence on teaching, on research in several branches of hard analysis, particularly complex function theory, and it has been an essential indispensable source book for those seriously interested in mathematical problems." Bulletin of the American Mathematical SocietyTrade ReviewFrom the reviews: "The present English edition is not a mere translation of the German original. Many new problems have been added. (Jahresb. DMV) "There are some excellent books which are indispensable to the instruction of indeed good mathematicians and this volume is, without any doubt, one of them. The broad horizon of the book, its clear style and logical construction are some of the qualities which assure From the reviews: "The work is one of the real classics of this century; it has had much influence on teaching, on research in several branches of hard analysis, particularly complex function theory, and it has been an essential indispensable source book for those seriously interested in mathematical problems. These volumes contain many extraordinary problems and sequences of problems, mostly from some time past, well worth attention today and tomorrow. Written in the early twenties by two young mathematicians of outstanding talent, taste, breadth, perception, perseverence, and pedagogical skill, this work broke new ground in the teaching of mathematics and how to do mathematical research." -Bulletin of the American Mathematical SocietyTable of ContentsOne Infinite Series and Infinite Sequences.- 1 Operations with Power Series.- Additive Number Theory, Combinatorial Problems, and Applications.- Binomial Coefficients and Related Problems.- Differentiation of Power Series.- Functional Equations and Power Series.- Gaussian Binomial Coefficients.- Majorant Series.- 2 Linear Transformations of Series. A Theorem of Cesàro.- Triangular Transformations of Sequences into Sequences.- More General Transformations of Sequences into Sequences.- Transformations of Sequences into Functions. Theorem of Cesàro.- 3 The Structure of Real Sequences and Series.- The Structure of Infinite Sequences.- Convergence Exponent.- The Maximum Term of a Power Series.- Subseries.- Rearrangement of the Terms.- Distribution of the Signs of the Terms.- 4 Miscellaneous Problems.- Enveloping Series.- Various Propositions on Real Series and Sequences.- Partitions of Sets, Cycles in Permutations.- Two Integration.- 1 The Integral as the Limit of a Sum of Rectangles.- The Lower and the Upper Sum.- The Degree of Approximation.- Improper Integrals Between Finite Limits.- Improper Integrals Between Infinite Limits.- Applications to Number Theory.- Mean Values and Limits of Products.- Multiple Integrals.- 2 Inequalities.- Inequalities.- Some Applications of Inequalities.- 3 Some Properties of Real Functions.- Proper Integrals.- Improper Integrals.- Continuous, Differentiate, Convex Functions.- Singular Integrals. Weierstrass’ Approximation Theorem.- 4 Various Types of Equidistribution.- Counting Function. Regular Sequences.- Criteria of Equidistribution.- Multiples of an Irrational Number.- Distribution of the Digits in a Table of Logarithms and Related Questions.- Other Types of Equidistribution.- 5 Functions of Large Numbers.- Laplace’s Method.- Modifications of the Method.- Asymptotic Evaluation of Some Maxima.- Minimax and Maximin.- Three Functions of One Complex Variable. General Part.- 1 Complex Numbers and Number Sequences.- Regions and Curves. Working with Complex Variables.- Location of the Roots of Algebraic Equations.- Zeros of Polynomials, Continued. A Theorem of Gauss.- Sequences of Complex Numbers.- Sequences of Complex Numbers, Continued: Transformation of Sequences.- Rearrangement of Infinite Series.- 2 Mappings and Vector Fields.- The Cauchy-Riemann Differential Equations.- Some Particular Elementary Mappings.- Vector Fields.- 3 Some Geometrical Aspects of Complex Variables.- Mappings of the Circle. Curvature and Support Function.- Mean Values Along a Circle.- Mappings of the Disk. Area.- The Modular Graph. The Maximum Principle.- 4 Cauchy’s Theorem • The Argument Principle.- Cauchy’s Formula.- Poisson’s and Jensen’s Formulas.- The Argument Principle.- Rouche’s Theorem.- 5 Sequences of Analytic Functions.- Lagrange’s Series. Applications.- The Real Part of a Power Series.- Poles on the Circle of Convergence.- Identically Vanishing Power Series.- Propagation of Convergence.- Convergence in Separated Regions.- The Order of Growth of Certain Sequences of Polynomials.- 6 The Maximum Principle.- The Maximum Principle of Analytic Functions.- Schwarz’s Lemma.- Hadamard’s Three Circle Theorem.- Harmonic Functions.- The Phragmén-Lindelöf Method.- Author Index.

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Book SynopsisFrom the reviews: "...I think the volume is a great success ... a welcome addition to the literature ..." The Mathematical Intelligencer, 1993 "... It is comparable in scope with the great Courant-Hilbert Methods of Mathematical Physics, but it is much shorter, more up to date of course, and contains more elaborate analytical machinery...." The Mathematical Gazette, 1993Trade ReviewFrom the reviews: "...I think the volume is a great success ... a welcome addition to the literature ..." The Mathematical Intelligencer, 1993 "... According to the authors ... the work was written for the nonspecialists and physicists but in my opinion almost every specialist will find something new for herself/himself in the text. ..." Acta Scientiarum Mathematicarum, 1993 "... On the whole, a thorough overview on the classical aspects of the topic may be gained from that volume." Monatshefte für Mathematik, 1993 "... It is comparable in scope with the great Courant-Hilbert Methods of Mathematical Physics, but it is much shorter, more up to date of course, and contains more elaborate analytical machinery...." The Mathematical Gazette, 1993Table of Contents1. Basic Concepts.- 1. Basic Definitions and Examples.- 1.1. The Definition of a Linear Partial Differential Equation.- 1.2. The Role of Partial Differential Equations in the Mathematical Modeling of Physical Processes.- 1.3. Derivation of the Equation for the Longitudinal Elastic Vibrations of a Rod.- 1.4. Derivation of the Equation of Heat Conduction.- 1.5. The Limits of Applicability of Mathematical Models.- 1.6. Initial and Boundary Conditions.- 1.7. Examples of Linear Partial Differential Equations.- 1.8. The Concept of Well-Posedness of a Boundary-value Problem. The Cauchy Problem.- 2. The Cauchy-Kovalevskaya Theorem and Its Generalizations.- 2.1. The Cauchy-Kovalevskaya Theorem.- 2.2. An Example of Nonexistence of an Analytic Solution.- 2.3. Some Generalizations of the Cauchy-Kovalevskaya Theorem. Characteristics.- 2.4. Ovsyannikov’s Theorem.- 2.5. Holmgren’s Theorem.- 3. Classification of Linear Differential Equations. Reduction to Canonical Form and Characteristics.- 3.1. Classification of Second-Order Equations and Their Reduction to Canonical Form at a Point.- 3.2. Characteristics of Second-Order Equations and Reduction to Canonical Form of Second-Order Equations with Two Independent Variables.- 3.3. Ellipticity, Hyperbolicity, and Parabolicity for General Linear Differential Equations and Systems.- 3.4. Characteristics as Solutions of the Hamilton-Jacobi Equation.- 2. The Classical Theory.- 1. Distributions and Equations with Constant Coefficients.- 1.1. The Concept of a Distribution.- 1.2. The Spaces of Test Functions and Distributions.- 1.3. The Topology in the Space of Distributions.- 1.4. The Support of a Distribution. The General Form of Distributions.- 1.5. Differentiation of Distributions.- 1.6. Multiplication of a Distribution by a Smooth Function. Linear Differential Operators in Spaces of Distributions.- 1.7. Change of Variables and Homogeneous Distributions.- 1.8. The Direct or Tensor Product of Distributions.- 1.9. The Convolution of Distributions.- 1.10. The Fourier Transform of Tempered Distributions.- 1.11. The Schwartz Kernel of a Linear Operator.- 1.12. Fundamental Solutions for Operators with Constant Coefficients.- 1.13. A Fundamental Solution for the Cauchy Problem.- 1.14. Fundamental Solutions and Solutions of Inhomogeneous Equations.- 1.15. Duhamel’s Principle for Equations with Constant Coefficients.- 1.16. The Fundamental Solution and the Behavior of Solutions at Infinity.- 1.17. Local Properties of Solutions of Homogeneous Equations with Constant Coefficients. Hypoellipticity and Ellipticity.- 1.18. Liouville’s Theorem for Equations with Constant Coefficients.- 1.19. Isolated Singularities of Solutions of Hypoelliptic Equations.- 2. Elliptic Equations and Boundary-Value Problems.- 2.1. The Definition of Ellipticity. The Laplace and Poisson Equations.- 2.2. A Fundamental Solution for the Laplacian Operator. Green’s Formula.- 2.3. Mean-Value Theorems for Harmonic Functions.- 2.4. The Maximum Principle for Harmonic Functions and the Normal Derivative Lemma.- 2.5. Uniqueness of the Classical Solutions of the Dirichlet and Neumann Problems for Laplace’s Equation.- 2.6. Internal A Priori Estimates for Harmonic Functions. Harnack’s Theorem.- 2.7. The Green’s Function of the Dirichlet Problem for Laplace’s Equation.- 2.8. The Green’s Function and the Solution of the Dirichlet Problem for a Ball and a Half-Space. The Reflection Principle.- 2.9. Harnack’s Inequality and Liouville’s Theorem.- 2.10. The Removable Singularities Theorem.- 2.11. The Kelvin Transform and the Statement of Exterior Boundary-Value Problems for Laplace’s Equation.- 2.12. Potentials.- 2.13. Application of Potentials to the Solution of Boundary-Value Problems.- 2.14. Boundary-Value Problems for Poisson’s Equation in Hölder Spaces. Schauder Estimates.- 2.15. Capacity.- 2.16. The Dirichlet Problem in the Case of Arbitrary Regions (The Method of Balayage). Regularity of a Boundary Point. The Wiener Regularity Criterion.- 2.17. General Second-Order Elliptic Equations. Eigenvalues and Eigenfunctions of Elliptic Operators.- 2.18. Higher-Order Elliptic Equations and General Elliptic Boundary-Value Problems. The Shapiro-Lopatinskij Condition.- 2.19. The Index of an Elliptic Boundary-Value Problem.- 2.20. Ellipticity with a Parameter and Unique Solvability of Elliptic Boundary-Value Problems.- 3. Sobolev Spaces and Generalized Solutions of Boundary-Value Problems.- 3.1. The Fundamental Spaces.- 3.2. Imbedding and Trace Theorems.- 3.3. Generalized Solutions of Elliptic Boundary-Value Problems and Eigenvalue Problems.- 3.4. Generalized Solutions of Parabolic Boundary-Value Problems.- 3.5. Generalized Solutions of Hyperbolic Boundary-Value Problems.- 4. Hyperbolic Equations.- 4.1. Definitions and Examples.- 4.2. Hyperbolicity and Well-Posedness of the Cauchy Problem.- 4.3. Energy Estimates.- 4.4. The Speed of Propagation of Disturbances.- 4.5. Solution of the Cauchy Problem for the Wave Equation.- 4.6. Huyghens’ Principle.- 4.7. The Plane Wave Method.- 4.8. The Solution of the Cauchy Problem in the Plane.- 4.9. Lacunae.- 4.10. The Cauchy Problem for a Strictly Hyperbolic System with Rapidly Oscillating Initial Data.- 4.11. Discontinuous Solutions of Hyperbolic Equations.- 4.12. Symmetric Hyperbolic Operators.- 4.13. The Mixed Boundary-Value Problem.- 4.14. The Method of Separation of Variables.- 5. Parabolic Equations.- 5.1. Definitions and Examples.- 5.2. The Maximum Principle and Its Consequences.- 5.3. Integral Estimates.- 5.4. Estimates in Hölder Spaces.- 5.5. The Regularity of Solutions of a Second-Order Parabolic Equation.- 5.6. Poisson’s Formula.- 5.7. A Fundamental Solution of the Cauchy Problem for a Second-Order Equation with Variable Coefficients.- 5.8. Shilov-Parabolic Systems.- 5.9. Systems with Variable Coefficients.- 5.10. The Mixed Boundary-Value Problem.- 5.11. Stabilization of the Solutions of the Mixed Boundary-Value Problem and the Cauchy Problem.- 6. General Evolution Equations.- 6.1. The Cauchy Problem. The Hadamard and Petrovskij Conditions.- 6.2. Application of the Laplace Transform.- 6.3. Application of the Theory of Semigroups.- 6.4. Some Examples.- 7. Exterior Boundary-Value Problems and Scattering Theory.- 7.1. Radiation Conditions.- 7.2. The Principle of Limiting Absorption and Limiting Amplitude.- 7.3. Radiation Conditions and the Principle of Limiting Absorption for Higher-Order Equations and Systems.- 7.4. Decay of the Local Energy.- 7.5. Scattering of Plane Waves.- 7.6. Spectral Analysis.- 7.7. The Scattering Operator and the Scattering Matrix.- 8. Spectral Theory of One-Dimensional Differential Operators.- 8.1. Outline of the Method of Separation of Variables.- 8.2. Regular Self-Adjoint Problems.- 8.3. Periodic and Antiperiodic Boundary Conditions.- 8.4. Asymptotics of the Eigenvalues and Eigenfunctions in the Regular Case.- 8.5. The Schrödinger Operator on a Half-Line.- 8.6. Essential Self-Adjointness and Self-Adjoint Extensions. The Weyl Circle and the Weyl Point.- 8.7. The Case of an Increasing Potential.- 8.8. The Case of a Rapidly Decaying Potential.- 8.9. The Schrödinger Operator on the Entire Line.- 8.10. The Hill Operator.- 9. Special Functions.- 9.1. Spherical Functions.- 9.2. The Legendre Polynomials.- 9.3. Cylindrical Functions.- 9.4. Properties of the Cylindrical Functions.- 9.5. Airy’s Equation.- 9.6. Some Other Classes of Functions.- References.- Author Index.

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Book SynopsisFrom the reviews: "...one of the best textbooks introducing several generations of mathematicians to higher mathematics. ... This excellent book is highly recommended both to instructors and students." --Acta Scientiarum Mathematicarum, 1991Trade ReviewFrom the reviews: "These books (Introduction to Calculus and Analysis Vol. I/II) are very well written. The mathematics are rigorous but the many examples that are given and the applications that are treated make the books extremely readable and the arguments easy to understand. These books are ideally suited for an undergraduate calculus course. Each chapter is followed by a number of interesting exercises. More difficult parts are marked with an asterisk. There are many illuminating figures...Of interest to students, mathematicians, scientists and engineers. Even more than that." Newsletter on Computational and Applied Mathematics, 1991 "...one of the best textbooks introducing several generations of mathematicians to higher mathematics. ... This excellent book is highly recommended both to instructors and students." Acta Scientiarum Mathematicarum, 1991Table of ContentsFunctions of Several Variables and Their Derivatives: Points and Points Sets in the Plane and in Space; Functions of Several Independent Variables; Continuity; The Partial Derivatives of a Function; The Differential of a Function and Its Geometrical Meaning; Functions of Functions (Compound Functions) and the Introduction of New Independent Variables; The mean Value Theorem and Taylor's Theorem for Functions of Several Variables; Integrals of a Function Depending on a Parameter; Differentials and Line Integrals; The Fundamental Theorem on Integrability of Linear Differential Forms; Appendix.- Vectors, Matrices, Linear Transformations: Operatios with Vectors; Matrices and Linear Transformations; Determinants; Geometrical Interpretation of Determinants; Vector Notions in Analysis.- Developments and Applications of the Differential Calculus: Implicit Functions; Curves and Surfaces in Implicit Form; Systems of Functions, Transformations, and Mappings; Applications; Families of Curves, Families of Surfaces, and Their Envelopes; Alternating Differential Forms; Maxima and Minima; Appendix.- Multiple Integrals: Areas in the Plane; Double Integrals; Integrals over Regions in three and more Dimensions; Space Differentiation. Mass and Density; Reduction of the Multiple Integral to Repeated Single Integrals; Transformation of Multiple Integrals; Improper Multiple Integrals; Geometrical Applications; Physical Applications; Multiple Integrals in Curvilinear Coordinates; Volumes and Surface Areas in Any Number of Dimensions; Improper Single Integrals as Functions of a Parameter; The Fourier Integral; The Eulerian Integrals (Gamma Function); Appendix

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Book SynopsisThis two-volume work focuses on partial differential equations (PDEs) with important applications in mechanical and civil engineering, emphasizing mathematical correctness, analysis, and verification of solutions. The presentation involves a discussion of relevant PDE applications, its derivation, and the formulation of consistent boundary conditions.Table of Contents8. The biharmonic equation.- 9. Poisson’s equation.

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Book SynopsisThis, the fourth edition of Stuwe’s book on the calculus of variations, surveys new developments in this exciting field. It also gives a concise introduction to variational methods. In particular it includes the proof for the convergence of the Yamabe flow and a detailed treatment of the phenomenon of blow-up. Recently discovered results for backward bubbling in the heat flow for harmonic maps or surfaces are discussed. A number of changes have been made throughout the text.Trade ReviewFrom the reviews of the fourth edition:"The fourth edition of Michael Struwe’s book Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems was published in 2008, 18 years after the first edition. … The bibliography alone would make it a valuable reference as it contains nearly 500 references. … Struwe’s book is addressed to researchers in differential geometry and partial differential equations." (John D. Cook, MAA Online, January, 2009)“This is the fourth edition of a standard reference work on direct methods in the calculus of variations. … The book contains a wealth of important results that would otherwise be hard to find in one single place.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 160 (4), July, 2010)Table of ContentsThe Direct Methods in the Calculus of Variations.- Minimax Methods.- Limit Cases of the Palais-Smale Condition.

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Book SynopsisThis text provides a lively introduction to pure mathematics. It begins with sets, functions and relations, proof by induction and contradiction, complex numbers, vectors and matrices, and provides a brief introduction to group theory. It moves onto analysis, providing a gentle introduction to epsilon-delta technology and finishes with continuity and functions. The book features numerous exercises of varying difficulty throughout the text.Table of Contents1. Sets, Functions and Relations.- 1.1 Sets.- 1.2 Subsets.- 1.3 Well-known Sets.- 1.4 Rationals, Reals and Pictures.- 1.5 Set Operations.- 1.6 Sets of Sets.- 1.7 Paradox.- 1.8 Set-theoretic Constructions.- 1.9 Notation.- 1.10 Venn Diagrams.- 1.11 Quantifiers and Negation.- 1.12 Informal Description of Maps.- 1.13 Injective, Surjective and Bijective Maps.- 1.14 Composition of Maps.- 1.15 Graphs and Respectability Reclaimed.- 1.16 Characterizing Bijections.- 1.17 Sets of Maps.- 1.18 Relations.- 1.19 Intervals.- 2. Proof.- 2.1 Induction.- 2.2 Complete Induction.- 2.3 Counter-examples and Contradictions.- 2.4 Method of Descent.- 2.5 Style.- 2.6 Implication.- 2.7 Double Implication.- 2.8 The Master Plan.- 3. Complex Numbers and Related Functions.- 3.1 Motivation.- 3.2 Creating the Complex Numbers.- 3.3 A Geometric Interpretation.- 3.4 Sine, Cosine and Polar Form.- 3.5 e.- 3.6 Hyperbolic Sine and Hyperbolic Cosine.- 3.7 Integration Tricks.- 3.8 Extracting Roots and Raising to Powers.- 3.9 Logarithm.- 3.10 Power Series.- 4. Vectors and Matrices.- 4.1 Row Vectors.- 4.2 Higher Dimensions.- 4.3 Vector Laws.- 4.4 Lengths and Angles.- 4.5 Position Vectors.- 4.6 Matrix Operations.- 4.7 Laws of Matrix Algebra.- 4.8 Identity Matrices and Inverses.- 4.9 Determinants.- 4.10 Geometry of Determinants.- 4.11 Linear Independence.- 4.12 Vector Spaces.- 4.13 Transposition.- 5. Group Theory.- 5.1 Permutations.- 5.2 Inverse Permutations.- 5.3 The Algebra of Permutations.- 5.4 The Order of a Permutation.- 5.5 Permutation Groups.- 5.6 Abstract Groups.- 5.7 Subgroups.- 5.8 Cosets.- 5.9 Cyclic Groups.- 5.10 Isomorphism.- 5.11 Homomorphism.- 6. Sequences and Series.- 6.1 Denary and Decimal Sequences.- 6.2 The Real Numbers.- 6.3 Notation for Sequences.- 6.4 Limits of Sequences.- 6.5 The Completeness Axiom.- 6.6 Limits of Sequences Revisited.- 6.7 Series.- 7. Mathematical Analysis.- 7.1 Continuity.- 7.2 Limits.- 8. Creating the Real Numbers.- 8.1 Dedekind’s Construction.- 8.2 Construction via Cauchy Sequences.- 8.3 A Sting in the Tail: p-adic numbers.- Further Reading.- Solutions.

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Book SynopsisThe main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.Trade ReviewFrom the reviews:"The book under review is an introduction to the field of linear partial differential equations and to standard methods for their numerical solution. … The balanced combination of mathematical theory with numerical analysis is an essential feature of the book. … The book is easily accessible and concentrates on the main ideas while avoiding unnecessary technicalities. It is therefore well suited as a textbook for a beginning graduate course in applied mathematics." (A. Ostermann, IMN - Internationale Mathematische Nachrichten, Vol. 59 (198), 2005)"This book, which is aimed at beginning graduate students of applied mathematics and engineering, provides an up to date synthesis of mathematical analysis, and the corresponding numerical analysis, for elliptic, parabolic and hyperbolic partial differential equations. … This widely applicable material is attractively presented in this impeccably well-organised text. … Partial differential equations with numerical methods covers a lot of ground authoritatively and without ostentation and with a constant focus on the needs of practitioners." (Nick Lord, The Mathematical Gazette, March, 2005)"Larsson and Thomée … discuss numerical solution methods of linear partial differential equations. They explain finite difference and finite element methods and apply these concepts to elliptic, parabolic, and hyperbolic partial differential equations. … The text is enhanced by 13 figures and 150 problems. Also included are appendixes on mathematical analysis preliminaries and a connection to numerical linear algebra. Summing Up: Recommended. Upper-division undergraduates through faculty." (D. P. Turner, CHOICE, March, 2004)"This book presents a very well written and systematic introduction to the finite difference and finite element methods for the numerical solution of the basic types of linear partial differential equations (PDE). … the book is very well written, the exposition is clear, readable and very systematic." (Emil Minchev, Zentralblatt MATH, Vol. 1025, 2003)"The author’s purpose is to give an elementary, relatively short, and readable account of the basic types of linear partial differential equations, their properties, and the most commonly used methods for their numerical solution. … We warmly recommend it to advanced undergraduate and beginning graduate students of applied mathematics and/or engineering at every university of the world." (Ferenc Móricz, Acta Scientiarum Mathematicarum, Vol. 71, 2005)"The presentation of the book is smart and very classical; it is more a reference book for applied mathematicians … . The convergence results, error estimates, variation formulations, all the theorems proofs, are very clear and well presented, the annexes A and B summary the necessary background for the understanding, without redundant generalisation or forgotten matter. The bibliography is presented by theme, well targeted on the topic of the book." (Anne Lemaitre, Physicalia Magazine, Vol. 28 (1), 2006)“Offers basic theory of linear partial differential equations and discusses the most commonly used numerical methods to solve these equations. … There are two appendices providing some extra basic material, useful to help understanding some of the theoretical principles that might be unfamiliar to unexperienced readers and students. The text is elementary and meant for students in mathematics, physics, engineering. … The bibliography is well arranged according to the important issues, which makes it easy to get informed about possible references for further study.” (Paula Bruggen, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)Table of ContentsA Two-Point Boundary Value Problem.- Elliptic Equations.- Finite Difference Methods for Elliptic Equations.- Finite Element Methods for Elliptic Equations.- The Elliptic Eigenvalue Problem.- Initial-Value Problems for Ordinary Differential Equations.- Parabolic Equations.- Finite Difference Methods for Parabolic Problems.- The Finite Element Method for a Parabolic Problem.- Hyperbolic Equations.- Finite Difference Methods for Hyperbolic Equations.- The Finite Element Method for Hyperbolic Equations.- Some Other Classes of Numerical Methods.

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Book SynopsisAny method of fitting equations to data may be called regression. Such equations are valuable for at least two purposes: making predictions and judging the strength of relationships. Because they provide a way of em­ pirically identifying how a variable is affected by other variables, regression methods have become essential in a wide range of fields, including the soeial seiences, engineering, medical research and business. Of the various methods of performing regression, least squares is the most widely used. In fact, linear least squares regression is by far the most widely used of any statistical technique. Although nonlinear least squares is covered in an appendix, this book is mainly ab out linear least squares applied to fit a single equation (as opposed to a system of equations). The writing of this book started in 1982. Since then, various drafts have been used at the University of Toronto for teaching a semester-Iong course to juniors, seniors and graduate students in a number of fields, including statistics, pharmacology, pharmacology, engineering, economics, forestry and the behav­ ioral seiences. Parts of the book have also been used in a quarter-Iong course given to Master's and Ph.D. students in public administration, urban plan­ ning and engineering at the University of Illinois at Chicago (UIC). This experience and the comments and critieisms from students helped forge the final version.Table of Contents1 Introduction.- 2 Multiple Regression.- 3 Tests and Confidence Regions.- 4 Indicator Variables.- 5 The Normality Assumption.- 6 Unequal Variances.- 7 *Correlated Errors.- 8 Outliers and Influential Observations.- 9 Transformations.- 10 Multicollinearity.- 11 Variable Selection.- 12 *Biased Estimation.- A Matrices.- A.1 Addition and Multiplication.- A.2 The Transpose of a Matrix.- A.3 Null and Identity Matrices.- A.4 Vectors.- A.5 Rank of a Matrix.- A.6 Trace of a Matrix.- A.7 Partitioned Matrices.- A.8 Determinants.- A.9 Inverses.- A.10 Characteristic Roots and Vectors.- A.11 Idempotent Matrices.- A.12 The Generalized Inverse.- A.13 Quadratic Forms.- A.14 Vector Spaces.- Problems.- B Random Variables and Random Vectors.- B.1 Random Variables.- B.1.1 Independent Random Variables.- B.1.2 Correlated Random Variables.- B.1.3 Sample Statistics.- B.1.4 Linear Combinations of Random Variables.- B.2 Random Vectors.- B.3 The Multivariate Normal Distribution.- B.4 The Chi-Square Distributions.- B.5 The F and t Distributions.- B.6 Jacobian of Transformations.- B.7 Multiple Correlation.- Problems.- C Nonlinear Least Squares.- C.1 Gauss-Newton Type Algorithms.- C.1.1 The Gauss-Newton Procedure.- C.1.2 Step Halving.- C.1.3 Starting Values and Derivatives.- C.1.4 Marquardt Procedure.- C.2 Some Other Algorithms.- C.2.1 Steepest Descent Method.- C.2.2 Quasi-Newton Algorithms.- C.2.3 The Simplex Method.- C.2.4 Weighting.- C.3 Pitfalls.- C.4 Bias, Confidence Regions and Measures of Fit.- C.5 Examples.- Problems.- Tables.- References.- Author Index.

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Book SynopsisThe present book is about the Askey scheme and the q-Askey scheme, which are graphically displayed right before chapter 9 and chapter 14, respectively. The fa- lies of orthogonal polynomials in these two schemes generalize the classical orth- onal polynomials (Jacobi, Laguerre and Hermite polynomials) and they have pr- erties similar to them. In fact, they have properties so similar that I am inclined (f- lowing Andrews & Askey [34]) to call all families in the (q-)Askey scheme classical orthogonal polynomials, and to call the Jacobi, Laguerre and Hermite polynomials very classical orthogonal polynomials. These very classical orthogonal polynomials are good friends of mine since - most the beginning of my mathematical career. When I was a fresh PhD student at the Mathematical Centre (now CWI) in Amsterdam, Dick Askey spent a sabbatical there during the academic year 1969–1970. He lectured to us in a very stimulating wayabouthypergeometricfunctionsandclassicalorthogonalpolynomials. Evenb- ter, he gave us problems to solve which might be worth a PhD. He also pointed out to us that there was more than just Jacobi, Laguerre and Hermite polynomials, for instance Hahn polynomials, and that it was one of the merits of the Higher Transc- dental Functions (Bateman project) that it included some newer stuff like the Hahn polynomials (see [198, §10. 23]).Trade ReviewFrom the reviews:“The book starts with a brief but valuable foreword by Tom Koornwinder on the history of the classification problem for orthogonal polynomials. … the ideal text for a graduate course devoted to the classification, and it is a valuable reference, which everyone who works in orthogonal polynomials will want to own.” (Warren Johnson, The Mathematical Association of America, August, 2010)Table of ContentsDefinitions and Miscellaneous Formulas.- Classical orthogonal polynomials.- Orthogonal Polynomial Solutions of Differential Equations.- Orthogonal Polynomial Solutions of Real Difference Equations.- Orthogonal Polynomial Solutions of Complex Difference Equations.- Orthogonal Polynomial Solutions in x(x+u) of Real Difference Equations.- Orthogonal Polynomial Solutions in z(z+u) of Complex Difference Equations.- Hypergeometric Orthogonal Polynomials.- Polynomial Solutions of Eigenvalue Problems.- Classical q-orthogonal polynomials.- Orthogonal Polynomial Solutions of q-Difference Equations.- Orthogonal Polynomial Solutions in q?x of q-Difference Equations.- Orthogonal Polynomial Solutions in q?x+uqx of Real

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Book SynopsisA collection of five surveys on dynamical systems, indispensable for graduate students and researchers in mathematics and theoretical physics. Written in the modern language of differential geometry, the book covers all the new differential geometric and Lie-algebraic methods currently used in the theory of integrable systems.Table of ContentsContents: Nonholonomic Dynamical Systems, Geometry of Distributions and Variational Problems by A.M. Vershik, V.Ya. Gershkovich.- Integrable Systems and Infinite Dimensional Lie Algebras by M.A. Olshanetsky, M.A. Perelomov.- Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems by A.G. Reyman, M.A. Semenov-Tian-Shansky.- Quantization of Open Toda Lattices by M.A. Semenov-Tian-Shansky.- Geometric and Algebraic Mechanisms of the Integrability of Hamiltonian Systems on Homogeneous Spaces and Lie Algebras by V.V. Trofimov, A.T. Fomenko.

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Book SynopsisM. Brelot: Historical introduction.- H. Bauer: Harmonic spaces and associated Markov processes.- J.M. Bony: Opérateurs elliptiques dégénérés associés aux axiomatiques de la theorie du potentiel.- J. Deny: Méthodes hilbertiennes en theory du potentiel.- J.L. Doob: Martingale theory – Potential theory.- G. Mokobodzki: Cônes de potentiels et noyaux subordonnés.Table of ContentsM. Brelot: Historical introduction.- H. Bauer: Harmonic spaces and associated Markov processes.- J.M. Bony: Opérateurs elliptiques dégénérés associés aux axiomatiques de la theorie du potentiel.- J. Deny: Méthodes hilbertiennes en theory du potentiel.- J.L. Doob: Martingale theory – Potential theory.- G. Mokobodzki: Cônes de potentiels et noyaux subordonnés.

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Book SynopsisFractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations. Trade ReviewFrom the reviews:“This book treats a fast growing field of fractional differential equations, i.e., differential equations with derivatives of non-integer order. … The book consists of two parts, eight chapters, an appendix, references and an index. … The book is well written and easy to read. It could be used for, a course in the application of fractional calculus for students of applied mathematics and engineering.” (Teodor M. Atanacković, Mathematical Reviews, Issue 2011 j)“This monograph is intended for use by graduate students, mathematicians and applied scientists who have an interest in fractional differential equations. The Caputo derivative is the main focus of the book, because of its relevance to applications. … The monograph may be regarded as a fairly self-contained reference work and a comprehensive overview of the current state of the art. It contains many results and insights brought together for the first time, including some new material that has not, to my knowledge, appeared elsewhere.” (Neville Ford, Zentralblatt MATH, Vol. 1215, 2011)Table of ContentsFundamentals of Fractional Calculus.- Riemann-Liouville Differential and Integral Operators.- Caputo’s Approach.- Mittag-Leffler Functions.- Theory of Fractional Differential Equations.- Existence and Uniqueness Results for Riemann-Liouville Fractional Differential Equations.- Single-Term Caputo Fractional Differential Equations: Basic Theory and Fundamental Results.- Single-Term Caputo Fractional Differential Equations: Advanced Results for Special Cases.- Multi-Term Caputo Fractional Differential Equations.

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Book SynopsisIn recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decomposition have proved to be very efficient for the study of evolution equations. The present book aims at presenting self-contained, state- of- the- art models of those techniques with applications to different classes of partial differential equations: transport, heat, wave and Schrödinger equations. It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity. It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.Trade ReviewFrom the reviews:“The authors did make impressive contributions to a broad area of fluid dynamics. It is the first time that a coherent presentation of those research results is available, which will give easier access to the whole area to a broader audience. … It is a valuable contribution in the important area of the interest of the authors and will without question find its place in the mathematical libraries, and on the shelves of people working in those areas.” (Herbert Koch, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 115, 2014)“The aim of the present monograph is to introduce methods from Fourier analysis, and in particular techniques based on the Littlewood–Paley decomposition, for the solution of nonlinear partial differential equations. … The presentation is fairly self-contained and only requires a solid background in measure theory and functional analysis. It will be of value to both graduate students and researchers interested in application of Fourier analysis to partial differential equations.” (G. Teschl, Monatshefte für Mathematik, Vol. 165 (3-4), March, 2012)“This book is a well-written introduction to Fourier analysis, Littlewood-Paley theory and some of their applications to the theory of evolution equations. It is suitable for readers with a solid undergraduate background in analysis. A feature that distinguishes it from other books of this sort is its emphasis on using Littlewood-Paley decomposition to study nonlinear differential equations. … the references, historical background, and discussion of possible future developments at the end of each chapter are very convenient for its readers.” (Lijing Sun, Zentralblatt MATH, Vol. 1227, 2012)“This book intends to prepare the reader how to apply tools from Fourier analysis to directly solve problems arising in the theory of non linear partial differential equations. … The presentation is well structured and easy to follow. … This is a textbook for advanced undergraduate or beginning graduate students with a good background in real and functional analysis. … even active researchers or mathematicians interested in the application of Fourier-analytic tools will find this book very useful.” (Peter R. Massopust, Mathematical Reviews, Issue 2011 m)Table of ContentsPreface.- 1. Basic analysis.- 2. Littlewood-Paley theory.- 3. Transport and transport-diffusion equations.- 4. Quasilinear symmetric systems.- 5. Incompressible Navier-Stokes system.- 6. Anisotropic viscosity.- 7. Euler system for perfect incompressible fluids.- 8. Strichartz estimates and applications to semilinear dispersive equations.- 9. Smoothing effect in quasilinear wave equations.- 10.- The compressible Navier-Stokes system.- References. - List of notations.- Index.

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Book SynopsisDas Lehrbuch bietet umfangreiche Anleitungen und Übungsaufgaben zum Band „Mathematik für Physiker“ desselben Autors. Die Studienanleitungen mit Fragen und Kontrollaufgaben erleichtern Lesern das eigenständige Erarbeiten des Stoffs. In zusätzlichen Erläuterungen vertieft der Autor einzelne Themenfelder und geht auf individuelle Lernschwierigkeiten ein. Band 2 des Übungswerks enthält über 700 Aufgaben mit ausführlichen Lösungen und ist der ideale Begleiter für Bachelor-Studierende der Physik während des zweiten Semesters.Table of ContentsFunktionen mehrerer Variablen.- Partielle Ableitung, totales Differential.- Mehrfachintegrale, Parameterdarstellung.- Oberflächenintegrale.- Divergenz und Rotation.- Koordinatentransformation, Matrizen.- Lineare Gleichungssysteme.- Eigenwerte, Eigenvektoren.- Fourierreihen.- Fourier-Integrale, Fourier-Transformation.- Laplace-Transformation.- Wellengleichungen.

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Book SynopsisIn recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decomposition have proved to be very efficient for the study of evolution equations. The present book aims at presenting self-contained, state- of- the- art models of those techniques with applications to different classes of partial differential equations: transport, heat, wave and Schrödinger equations. It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity. It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.Trade ReviewFrom the reviews:“The authors did make impressive contributions to a broad area of fluid dynamics. It is the first time that a coherent presentation of those research results is available, which will give easier access to the whole area to a broader audience. … It is a valuable contribution in the important area of the interest of the authors and will without question find its place in the mathematical libraries, and on the shelves of people working in those areas.” (Herbert Koch, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 115, 2014)“The aim of the present monograph is to introduce methods from Fourier analysis, and in particular techniques based on the Littlewood–Paley decomposition, for the solution of nonlinear partial differential equations. … The presentation is fairly self-contained and only requires a solid background in measure theory and functional analysis. It will be of value to both graduate students and researchers interested in application of Fourier analysis to partial differential equations.” (G. Teschl, Monatshefte für Mathematik, Vol. 165 (3-4), March, 2012)“This book is a well-written introduction to Fourier analysis, Littlewood-Paley theory and some of their applications to the theory of evolution equations. It is suitable for readers with a solid undergraduate background in analysis. A feature that distinguishes it from other books of this sort is its emphasis on using Littlewood-Paley decomposition to study nonlinear differential equations. … the references, historical background, and discussion of possible future developments at the end of each chapter are very convenient for its readers.” (Lijing Sun, Zentralblatt MATH, Vol. 1227, 2012)“This book intends to prepare the reader how to apply tools from Fourier analysis to directly solve problems arising in the theory of non linear partial differential equations. … The presentation is well structured and easy to follow. … This is a textbook for advanced undergraduate or beginning graduate students with a good background in real and functional analysis. … even active researchers or mathematicians interested in the application of Fourier-analytic tools will find this book very useful.” (Peter R. Massopust, Mathematical Reviews, Issue 2011 m)Table of ContentsPreface.- 1. Basic analysis.- 2. Littlewood-Paley theory.- 3. Transport and transport-diffusion equations.- 4. Quasilinear symmetric systems.- 5. Incompressible Navier-Stokes system.- 6. Anisotropic viscosity.- 7. Euler system for perfect incompressible fluids.- 8. Strichartz estimates and applications to semilinear dispersive equations.- 9. Smoothing effect in quasilinear wave equations.- 10.- The compressible Navier-Stokes system.- References. - List of notations.- Index.

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Book SynopsisDie Mathematik gilt als schwierig, und ganz besonders die Analysis 1 wird von Studienanfängern als Stolperstein empfunden. Dabei bräuchten die meisten nur etwas mehr Anleitung und vor allem viel Übung, kurz, ein intensives Training. Dieses Buch bietet ein solches Training an.Der Aufbau orientiert sich am Grundkurs Analysis 1 des Autors, aber dank ausführlicher Literaturhinweise mit inhaltlichen Zuordnungen kann das Training Analysis 1 als Begleitung zu jedem gängigen Lehrbuch und jeder Analysisvorlesung erfolgreich eingesetzt werden.Auf eine Zusammenfassung der Theorie folgen in jedem Abschnitt Tutorien mit ausführlichen Erklärungen zu ausgewählten, wichtigen Themen. Danach werden zahlreiche durchgerechnete Beispiele und schließlich eine Reihe von Aufgaben mit mehr oder weniger ausführlichen Lösungshinweisen angeboten. Unterstützt wird das Ganze durch viele Illustrationen, und ein Anhang enthält ausführlich durchgerechnete Musterlösungen zu allen Aufgaben.Table of ContentsDie Sprache der Analysis. Mengen von Zahlen. Induktion. Vollständigkeit. Funktionen. Vektoren und komplexe Zahlen. Polynome und rationale Funktionen.- Der Grenzwertbegriff. Konvergenz. Unendliche Reihen. Grenzwerte von Funktionen. Potenzreihen. Flächen als Grenzwerte.- Der Calculus. Differenzierbare Funktionen. Der Mittelwertsatz. Stammfunktionen und Integrale. Integrationsmethoden. Bogenlänge und Krümmung. Lineare Differentialgleichungen.- Vertauschung von Grenzprozessen. Gleichmäßige Konvergenz. Die Taylorentwicklung. Numerische Anwendungen. Uneigentliche Integrale. Parameterintegrale.- Anhang: Lösungen.- Literaturverzeichnis.- Stichwortverzeichnis.

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Book Synopsis Dieses vierfarbige Lehrbuch wendet sich an Studierende der Mathematik in Bachelor-Studiengängen. Es bietet in einem Band ein lebendiges Bild der mathematischen Inhalte, die üblicherweise im zweiten und dritten Studienjahr behandelt werden (mit Ausnahme der Algebra).Mathematik-Studierende finden wichtige Begriffe, Sätze und Beweise ausführlich und mit vielen Beispielen erklärt und werden an grundlegende Konzepte und Methoden herangeführt.Im Mittelpunkt stehen das Verständnis der mathematischen Zusammenhänge und des Aufbaus der Theorie sowie die Strukturen und Ideen wichtiger Sätze und Beweise. Es wird nicht nur ein in sich geschlossenes Theoriengebäude dargestellt, sondern auch verdeutlicht, wie es entsteht und wozu die Inhalte später benötigt werden.Herausragende Merkmale sind: durchgängig vierfarbiges Layout mit mehr als 350 Abbildungen prägnant formulierte Kerngedanken bilden die Abschnittsüberschriften Selbsttests in kurzen Abständen ermöglichen Lernkontrollen während des Lesens farbige Merkkästen heben das Wichtigste hervor „Unter-der-Lupe“-Boxen zoomen in Beweise hinein, motivieren und erklären Details „Hintergrund-und-Ausblick“-Boxen stellen Zusammenhänge zu anderen Gebieten und weiterführenden Themen her Zusammenfassungen zu jedem Kapitel sowie Übersichtsboxen mehr als 500 Verständnisfragen, Rechenaufgaben und Aufgaben zu Beweisen Der inhaltliche Schwerpunkt liegt auf dem weiteren Ausbau der Analysis sowie auf den Themen der Vorlesungen Numerik sowie Wahrscheinlichkeitstheorie und Statistik. Behandelt werden darüber hinaus Inhalte und Methodenkompetenzen, die vielerorts im zweiten und dritten Studienjahr der Mathematikausbildung vermittelt werden.Auf der Website zum Buch Matheweb finden Sie Hinweise, Lösungswege und Ergebnisse zu allen Aufgaben die Möglichkeit, zu den Kapiteln Fragen zu stellen Das Buch wird allen Studierenden der Mathematik ein verlässlicher Begleiter sein.Trade Review“ ... Ausführliche Erklärungen und über 400 Abbildungen verdeutlichen abstrakte Sachverhalte, kompakte Übersichten liefern zentrale Ergebnisse, Kontrollfragen ermöglichen eine fortlaufende Verständniskontrolle und Übungsaufgaben dienen der eingehenden Beschäftigung mit dem Stoff ... Der Zielgruppe als Lehrbuch und Nachschlagewerk auch neben der Studienliteratur zu den einzelnen Teilgebieten sehr dienlich.” (Philipp Kastendieck, in: ekz-Informationsdienst, Jg. 11, 2016)“... für den Autodidakt kann dieses Lehrbuch empfohlen werden, da eine Vielfalt an Beispielen , Übungsaufgaben und entsprechenden Abfragen den Einstieg im nicht immer leichten Lehrstoff erleichtert. ... kann dieses Lehrbuch für das Mathematikstudium empfohlen werden. Es beinhaltet die Grundlagen des Mathematikstudiums und hat den Vorteil ...” (La, in: Amazon.de, 10. November 2015)Table of Contents1 Mathematik – eine Wissenschaft für sich.- 2 Lineare Differenzialgleichungen – Systeme und Gleichungen höherer Ordnung.- 3 Randwertprobleme und nichtlineare Differenzialgleichungen – Funktionen sind gesucht.- 4 Qualitative Theorie – jenseits von analytischen und mehr als numerische Lösungen.- 5 Funktionentheorie – Analysis im Komplexen.- 6 Differenzialformen und der allgemeine Satz von Stokes.- 7 Grundzüge der Maß- und Integrationstheorie vom Messen und Mitteln.- 8 Lineare Funktionalanalysis – Operatoren statt Matrizen.- 9 Fredholm-Gleichungen – kompakte Störungen der Identität.- 10 Hilberträume – fast wie im Anschauungsraum.- 11 Warum Numerische Mathematik? – Modellierung, Simulation und Optimierung.- 12 Interpolation – Splines und mehr.- 13 Quadratur – numerische Integrationsmethoden.- 14 Numerik linearer Gleichungssysteme – Millionen von Variablen im Griff.- 15 Eigenwertprobleme – Einschließen und Approximieren.- 16 Lineare Ausgleichsprobleme – im Mittel das Beste.- 17 Nichtlineare Gleichungen und Systeme – numerisch gelöst.- 18 Numerik gewöhnlicher Differenzialgleichungen – Schritt für Schritt zur Trajektorie.- 19 Wahrscheinlichkeitsräume – Modelle für stochastische Vorgänge.- 20 Bedingte Wahrscheinlichkeit und Unabhängigkeit – Meister Zufall hängt (oft) ab.- 21 Diskrete Verteilungsmodelle – wenn der Zufall zählt.- 22 Stetige Verteilungen und allgemeine Betrachtungen – jetzt wird es analytisch.- 23 Konvergenzbegriffe und Grenzwertsätze – Stochastik für große Stichproben.- 24 Grundlagen der Mathematischen Statistik – vom Schätzen und Testen.

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Book Synopsis1. Our essential objective is the study of the linear, non-homogeneous problems: (1) Pu = I in CD, an open set in RN, (2) fQjtl = gj on am (boundary of m), lor on a subset of the boundm"J am 1 v, where Pis a linear differential operator in m and where the Q/s are linear differential operators on am. In Volumes 1 and 2, we studied, for particular c1asses of systems {P, Qj}, problem (1), (2) in c1asses of Sobolev spaces (in general constructed starting from P) of positive integer or (by interpolation) non-integer order; then, by transposition, in c1asses of Sobolev spaces of negative order, until, by passage to the limit on the order, we reached the spaces of distributions of finite order. In this volume, we study the analogous problems in spaces of inlinitely dilferentiable or analytic Itlnctions or of Gevrey-type I~mctions and by duality, in spaces 01 distribtltions, of analytic Itlnctionals or of Gevrey- type ultra-distributions. In this manner, we obtain a c1ear vision (at least we hope so) of the various possible formulations of the boundary value problems (1), (2) for the systems {P, Qj} considered here.Table of Contents7 Scalar and Vector Ultra-Distributions.- 1. Scalar-Valued Functions of Class Mk.- 1.1 The Sequences {Mk}.- 1.2 The Space $${D_{{M_k}}}\left( H \right)$$.- 1.3 The Spaces $${D_{{M_k}}}\left( H \right)$$ and $${\varepsilon _{{M_k}}}\left( H \right)$$.- 2. Scalar-Valued Ultra-Distributions of Class Mk; Generalizations.- 2.1 The Space $$D{'_{{M_k}}}\left( \Omega \right)$$.- 2.2 Non-Symmetric Spaces of Class Mk.- 2.3 Scalar Ultra-Distributions of Beurling-Type.- 3. Spaces of Analytic Functions and of Analytic Functionals.- 3.1 The Spaces H(H) and H’(H).- 3.2 The Spaces H(?) and H(?).- 4. Vector-Valued Functions of Class Mk.- 4.1 The Space $${D_{{M_k}}}\left( {\phi ;F} \right)$$.- 4.2 The Spaces $${D_{{M_k}}}\left( {H,F} \right)$$ and $${E_{{M_k}}}\left( {\phi ;F} \right)$$.- 4.3 The Spaces $${D_{ \pm ,{M_k}}}\left( {\phi ;F} \right)$$.- 4.4 Remarks on the Topological Properties of the Spaces $${D_{{M_k}}}\left( {\phi ;F} \right),{E_{{M_k}}}\left( {\phi ;F} \right),{D_{ \pm ,{M_k}}}\left( {\phi ;F} \right)$$.- 5. Vector-Valued Ultra-Distributions of Class Mk; Generalizations.- 5.1 Recapitulation on Vector-Valued Distributions.- 5.2 The Space $$D{'_{{M_k}}}\left( {\phi ;F} \right)$$.- 5.3 The Space $$D{'_{ \pm ,{M_k}}}\left( {\phi ;F} \right)$$.- 5.4 Vector-Valued Ultra-Distributions of Beurling-Type.- 5.5 The Particular Case: F = Banach Space.- 6. Comments.- 8 Elliptic Boundary Value Problems in Spaces of Distributions and Ultra-Distributions.- 1. Regularity of Solutions of Elliptic Boundary Value Problems in Spaces of Analytic Functions and of Class Mk; Statement of the Problems and Results.- 1.1 Recapitulation on Elliptic Boundary Value Problems.- 1.2 Statement of the Mk-Regularity Results.- 1.3 Reduction of the Problem to the Case of the Half-Ball.- 2. The Theorem on “Elliptic Iterates”: Proof.- 2.1 Some Lemmas.- 2.2 The Preliminary Estimate.- 2.3 Bounds for the Tangential Derivatives.- 2.4 Bounds for the Normal Derivatives.- 2.5 Proof of Theorem 1.3.- 2.6 Complements and Remarks.- 3. Application of Transposition; Existence of Solutions in the Space D’(?) of Distributions.- 3.1 Generalities.- 3.2 Choice of the Form L; the Space ?(?) and its Dual.- 3.3 Final Choice of the Form L; the Space Y.- 3.4 Density Theorem.- 3.5 Trace Theorem and Green’s Formula in Y.- 3.6 The Existence of Solutions in the Space Y.- 3.7 Continuity of Traces on Surfaces Neighbouring ?.- 4. Existence of Solutions in the Space $$D{'_{{M_k}}}\left( \Omega \right)$$ of Ultra-Distributions.- 4.1 Generalities.- 4.2 The Space $${\Xi _{{M_k}}}\left( \Omega \right)$$ and its Dual.- 4.3 The Space $${Y_{{M_k}}}$$ and the Existence of Solutions in $${Y_{{M_k}}}$$.- 4.4 Application to the Regularity in the Interior of Ultra-Distribution Solutions of the Equation Au = f.- 5. Comments.- 6. Problems.- 9 Evolution Equations in Spaces of Distributions and Ultra-Distributions.- 1. Regularity Results. Equations of the First Order in t.- 1.1 Orientation and Notation.- 1.2 Regularity in the Spaces D+.- 1.3 Regularity in the Spaces $${D_{ + ,{M_k}}}$$.- 1.4 Regularity in Beurling Spaces.- 1.5 First Applications.- 2. Equations of the Second Order in t.- 2.1 Statement of the Main Results.- 2.2 Proof of Theorem 2.1.- 2.3 Proof of Theorem 2.2.- 3. Singular Equations of the Second Order in t.- 3.1 Statement of the Main Results.- 3.2 Proof of Theorem 3.1.- 4. Schroedinger-Type Equations.- 4.1 Statement of the Main Results.- 4.2 Proof of Theorem 4.1.- 4.3 Proof of Theorem 4.2.- 5. Stability Results in Mk-Classes.- 5.1 Parabolic Regularization.- 5.2 Approximation by Systems of Cauchy-Kowaleska Type (I).- 5.3 Approximation by Systems of Cauchy-Kowaleska Type (II).- 6. Transposition.- 6.1 Orientation.- 6.2 The Parabolic Case.- 6.3 The Second Order in t Case and the Schroedinger Case.- 7. Semi-Groups.- 7.1 Orientation.- 7.2 The Space of Vectors of Class Mk.- 7.3 The Semi-Group G in the Spaces D(A?; Mk). Applications.- 7.4 The Transposed Settings. Applications.- 7.5 Another Mk-Regularity Result.- 8. Mk -Classes and Laplace Transformation.- 8.1 Orientation-Hypotheses.- 8.2 Mk -Regularity Result.- 8.3 Transposition.- 9. General Operator Equations.- 9.1 General Results.- 9.2 Application. Periodic Problems.- 9.3 Transposition.- 10. The Case of a Finite Interval ]0, T[.- 10.1 Orientation. General Problems.- 10.2 Space Described by v(0) as v Describes X.- 10.3 The Space $${\Xi _{{M_k}}}$$.- 10.4 Choice of L.- 10.5 The Space Y and Trace Theorems.- 10.6 Non-Homogeneous Problems.- 11. Distribution and Ultra-Distribution Semi-Groups.- 11.1 Distribution Semi-Groups.- 11.2 Ultra-Distribution Semi-Groups.- 12. A General Local Existence Result.- 12.1 Statement of the Result.- 12.2 Examples.- 13. Comments.- 14. Problems.- 10 Parabolic Boundary Value Problems in Spaces of Ultra-Distributions.- 1. Regularity in the Interior of Solutions of Parabolic Equations.- 1.1 The Hypoellipticity of Parabolic Equations.- 1.2 The Regularity in the Interior in Gevrey Spaces.- 2. The Regularity at the Boundary of Solutions of Parabolic Boundary Value Problems.- 2.1 The Regularity in the Space $$D\left( {\bar Q} \right)$$.- 2.2 The Regularity in Gevrey Spaces.- 3. Application of Transposition: The Finite Cylinder Case.- 3.1 The Existence of Solutions in the Space D’(Q): Generalities, the Spaces X and Y.- 3.2 Space Described by ?v as v Describes X.- 3.3 Trace and Existence Theorems in the Space Y.- 3.4 The Existence of Solutions in the Spaces D’s,r(Q) of Gevrey Ultra-Distributions, with r > 1, s ? 2m.- 4. Application of Transposition: The Infinite Cylinder Case.- 4.1 The Existence of Solutions in the Space D’ (R; D’(?)): The Space X_.- 4.2 The Existence of Solutions in the Space D’+ (R; D’(?)): The Space Y+ and the Trace and Existence Theorems.- 4.3 The Existence of Solutions in the Spaces D’+,s(R;D’r(?)), with r > 1, s ? 2m.- 4.4 Remarks on the Existence of Solutions and the Trace Theorems in other Spaces of Ultra-Distributions.- 5. Comments.- 6. Problems.- 11 Evolution Equations of the Second Order in t and of Schroedinger Type.- 1. Equations of the Second Order in t; Regularity of the Solutions of Boundary Value Problems.- 1.1 The Regularity in the Space $$D\left( {\bar Q} \right)$$.- 1.2 The Regularity in Gevrey Spaces.- 2. Equations of the Second Order in t; Application of Transposition and Existence of Solutions in Spaces of Distributions.- 2.1 Generalities.- 2.2 The Space $${D_{ - ,\gamma }}\left( {\left[ {0,T} \right];{D_\gamma }\left( {\bar \Omega } \right)} \right)$$ and its Dual.- 2.3 The Spaces X and Y.- 2.4 Study of the Operator ?.- 2.5 Trace and Existence Theorems in the Space Y.- 2.6 Complements on the Trace Theorems.- 2.7 The Infinite Cylinder Case.- 3. Equations of the Second Order in t; Application of Transposition and Existence of Solutions in Spaces of Ultra-Distributions.- 3.1 The Difficulties in the Finite Cylinder Case.- 3.2 The Infinite Cylinder Case for m > 1.- 4. Schroedinger Equations; Complements for Parabolic Equations.- 4.1 Regularity Results for the Schroedinger Equation.- 4.2 The Non-Homogeneous Boundary Value Problems for the Schroedinger Equation.- 4.3 Remarks on Parabolic Equations.- 5. Comments.- 6. Problems.- Appendix. Calculus of Variations in Gevrey-Type Spaces.

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Book Synopsis...Je mehr ich tiber die Principien der Functionentheorie nachdenke - und ich thue dies unablassig -, urn so fester wird meine Uberzeugung, dass diese auf dem Fundamente algebraischer Wahrheiten aufgebaut werden muss (WEIERSTRASS, Glaubensbekenntnis 1875, Math. Werke II, p. 235). 1. Sheaf Theory is a general tool for handling questions which involve local solutions and global patching. "La notion de faisceau s'introduit parce qu'il s'agit de passer de donnees 'locales' a l'etude de proprietes 'globales'" [CAR], p. 622. The methods of sheaf theory are algebraic. The notion of a sheaf was first introduced in 1946 by J. LERAY in a short note Eanneau d'homologie d'une representation, C. R. Acad. Sci. 222, 1366-68. Of course sheaves had occurred implicitly much earlier in mathematics. The "Monogene analytische Functionen", which K. WEIERSTRASS glued together from "Func- tionselemente durch analytische Fortsetzung", are simply the connected components of the sheaf of germs of holomorphic functions on a RIEMANN surface*'; and the "ideaux de domaines indetermines", basic in the work of K. OKA since 1948 (cf. [OKA], p. 84, 107), are just sheaves of ideals of germs of holomorphic functions. Highly original contributions to mathematics are usually not appreciated at first. Fortunately H. CARTAN immediately realized the great importance of LERAY'S new abstract concept of a sheaf. In the polycopied notes of his Semina ire at the E. N. S.Table of Contents1. Complex Spaces.- § 1. The Notion of a Complex Space.- 0. Ringed Spaces — 1. The Space (?n, (O) — 2. Zero Sets and Complex Model Spaces — 3. Sheaves of Local ?-Algebras. ?-ringed Spaces — 4. Morphisms of ?-ringed Spaces — 5. Complex Spaces — 6. Sections and Functions — 7. Construction of Complex Spaces by Gluing — 8. The Complex Projective Space ?n — 9. Historical Notes.- § 2. General Properties of Complex Spaces.- 1. Zero Sets of Ideal Sheaves — 2. Closed Complex Subspaces — 3. Factorization of Holomorphic Maps — 4. Complex Spaces and Coherent Analytic Sheaves. Extension Principle — 5. Analytic Image Sheaves — 6. Analytic Inverse Image Sheaves — 7. Holomorphic Embeddings.- § 3. Direct Products and Graphs.- 1. The Bijection ?ol(X, ?n)?O(X)n. Extension of Holomorphic Maps — 2. Complex Direct Products — 3. Existence of Canonical Products. Local Case — 4. Existence of Canonical Products. Global Case — 5. Graph Space of a Holomorphic Map.- § 4. Complex Spaces and Cohomology.- 1. Divisors — 2. Holomorphic Vector Bundles — 3. Line Bundles and Divisors — 4. Holomorphically Convex Spaces and Stein Spaces — 5. ?ech Cohomology of Analytic Sheaves — 6. Cohomology of Coherent Sheaves with Respect to Stein Coverings — 7. Higher Dimensional Direct Images.- 2. Local Weierstrass Theory.- § 1. The Weierstrass Theorems.- 0. Generalities — 1. The WeierstraB Division Theorem — 2. The Weierstraß Preparation Theorem — 3. A Simple Observation.- § 2. Algebraic Structure of $${O_{{C^n},0}}$$.- 1. Noether Property and Factoriality — 2. Hensel’s Lemma — 3. Closedness of Sub-modules.- § 3. Finite Maps.- 1. Closed Maps — 2. Finite Maps. Local Description — 3. Local Representation of Image Sheaves — 4. Exactness of the Functor f* for Finite Maps — 5. Weierstraß Maps.- §4. The Weierstrass Isomorphism.- 1. The Generalized Weierstraß Division Theorem — 2. The Weierstraß Isomorphism — 3. A Coherence Lemma — 4. A Further Generalization of the Generalized Weierstraß Division Theorem.- § 5. Coherence of Structure Sheaves.- 1. Formal Coherence Criterion — 2. The Coherence of $${O_{{C^n}}}$$ — 3. Coherence of all Structure Sheaves OX.- 3. Finite Holomorphic Maps.- § 1. Finite Mapping Theorem.- 1. Projection Lemma — 2. Finite Holomorphic Maps and Isolated Points — 3. Finite Mapping Theorem.- § 2. Rückert Nullstellensatz for Coherent Sheaves.- 1. Preliminary Version — 2. Rückert Nullstellensatz.- § 3. Finite Open Holomorphic Maps.- 1. A Necessary Condition for Openness — 2. Torsion Sheaves and Criterion of Openness — 3. Coherence of Torsion Sheaves and Open Mapping Lemma — 4. Existence of Finite Open Projections.- § 4. Local Description of Complex Subspaces in ?n.- 1. The Local Description Lemma — 2. Proof of the Local Description Lemma.- 4. Analytic Sets. Coherence of Ideal Sheaves.- § 1. Analytic Sets and their Ideal Sheaves.- 1. Analytic Sets — 2. Ideal Sheaf of an Analytic Set — 3. Local Decomposition Lemma — 4. Prime Components. Criterion of Reducibility — 5. Rückert Nullstellensatz for Ideal Sheaves — 6. Analytic Sets and Finite Holomorphic Maps.- § 2. Coherence of the Sheaves i (A).- 1. Proof of Coherence in a Special Case — 2. Reduction to Analytic Sets in Domains of ?n — 3. Further Reduction to a Lemma — 4. Verification of the Assumptions of Lemma 3–5. Coherence of Radical Sheaves.- § 3. Applications of the Fundamental Theorem and of the Nullstellensatz.- 1. Analytic Sets and Reduced Closed Complex Subspaces — 2. Reduction of Complex Spaces — 3. Reduced Complex Spaces.- § 4. Coherent and Locally Free Sheaves.- 1. Corank of a Coherent Sheaf — 2. Characterization of Locally Free Sheaves.- 5. Dimension Theory.- § 1. Analytic and Algebraic Dimension.- 1. Analytic Dimension of Complex Spaces. Upper Semi-Continuity — 2. Analytic and Algebraic Dimension — 3. Dimension of the Reduction and of Analytic Sets.- § 2. Active Germs and the Active Lemma.- 1. The Sheaf of Active Germs — 2. Criterion of Activity — 3. Existence of Active Functions. Lifting Lemma — 4. Active Lemma.- § 3. Applications of the Active Lemma.- 1. Basic Properties of Dimension. Ritt’s Lemma — 2. Analytic Sets of Maximal Dimension — 3. Computation of the Dimension of Analytic Sets in ?n.- § 4. Dimension and Finite Maps. Pure Dimensional Spaces.- 1. Invariance of Dimension under Finite Maps — 2. Pure Dimensional Complex Spaces — 3. Open Finite Maps and Dimension. Open Mapping Theorem — 4. Local Prime Components (revisited).- § 5. Maximum Principle.- 1. Open Mapping Theorem for Holomorphic Functions — 2. Local and Absolute Maximum Principle — 3. Maximum Principle for Complex Spaces with Boundary.- § 6. Noether Lemma for Coherent Analytic Sheaves.- 1. Statement of the Lemma and Applications — 2. Proof of the Lemma.- 6. Analyticity of the Singular Locus. Normalization of the Structure Sheaf.- § 1. Embedding Dimension.- 1. Embedding Dimension. Jacobi Criterion — 2. Analyticity of the Sets X(k). Algebraic Description of embxX.- § 2. Smooth Points and the Singular Locus.- 1. Smooth Points and Singular Locus — 2. Analyticity of the Singular Locus — 3. A Property of the Ideals i(S(X))x, x?S(X).- § 3. The Sheaf M of Germs of Meromorphic Functions.- 1. The Sheaf M — 2. The Zero Set and the Polar Set of a Meromorphic Function — 3. The Lifting Monomorphism MY?f*(MX).- § 4. The Normalization Sheaf $${\hat O_X}$$.- 1. The Normalization Sheaf Normal Points $${\hat O_X}$$ — 2. Normality and Irreducibility at a Point.- § 5. Criterion of Normality. Theorem of Oka.- 1. The Canonical OX homomorphism $$\sigma :Hom\left( {f,f} \right) \to M$$ — 2. Criterion of Normality. Theorem of Oka — 3. Singular Locus and Normal Points.- 7. Riemann Extension Theorem and Analytic Coverings.- § 1. Riemann Extension Theorem on Complex Manifolds.- 1. First Riemann Theorem — 2. Second Riemann Theorem — 3. Riemann Extension Theorem on Complex Manifolds. Criterion of Connectedness.- § 2. Analytic Coverings.- 1. Definition and Elementary Properties — 2. Covering Lemma and Existence of Open Coverings — 3. Open Analytic Coverings.- § 3. Theorem of Primitive Element.- 1. Theorem of Integral Dependence — 2. A Lemma about Holomorphic Determinants. Discriminants — 3. Theorem of Primitive Element. Universal Denominators — 4. The Sheaf Monomorphism $${\pi _*}\left( {{{\hat O}_X}} \right) \to O_Y^b$$.- § 4. Applications of the Theorem of Primitive Element.- 1. Riemann Extension Theorem on Locally Pure Dimensional Complex Spaces — 2. Characterization of Normality by the Riemann Extension Theorem — 3. Weierstraß Convergence Theorem on Locally Pure Dimensional Complex Spaces.- § 5. Analytically Normal Vector Bundles.- 1. General Remarks — 2. Decent Vector Bundles — 3. Analytically Normal Vector Bundles and Normal Cones — 4. Whitney Sums of Analytically Normal Bundles — 5. Discussion of the Cones Akm.- 8. Normalization of Complex Spaces.- § 1. One-Sheeted Analytic Coverings.- 1. Examples — 2. General Structure of One-Sheeted Coverings — 3. The Isomorphisms $$\tilde v:{M_Y}\tilde \to {\tilde v_*}\left( {{M_X}} \right) $$ and $$\tilde v:{\hat O_Y}\tilde \to {v_*}\left( {{{\hat O}_X}} \right)$$.- § 2. The Local Existence Theorem. Coherence of the Normalization Sheaf.- 1. Admissible Sheaves and the Local Existence Theorem — 2. Proof of the Local Existence Theorem — 3. Coherence of the Normalization Sheaf.- § 3. The Global Existence Theorem. Existence of Normalization Spaces.- 1. Linking Isomorphisms — 2. The Global Existence Theorem — 3. Existence of a Normalization.- § 4. Properties of the Normalization.- 1. The Space of Prime Germs. Topological Structure of Normalization Spaces — 2. Uniqueness of the Normalization — 3. Lifting of Holomorphic Maps — 4. Injective Holomorphic Maps.- 9. Irreducibility and Connectivity. Extension of Analytic Sets.- § 1. Irreducible Complex Spaces.- 1. Identity Lemma — 2. Irreducible Complex Spaces — 3. Properties of Irreducible Complex Spaces.- § 2. Global Decomposition of Complex Spaces.- 1. Connected Components — 2. Global Decomposition Theorem — 3. Global and Local Decomposition. Global Maximum Principle — 4. Proper Maps — 5. Holomorphically Spreadable Spaces.- § 3. Local and Arcwise Connectedness of Complex Spaces.- 1. Local Connectedness — 2. Arcwise Connectedness — 3. Finite Holomorphic Surjections and Covering Maps.- § 4. Removable Singularities of Analytic Sets.- 1. Analyticity of Closures of Coverings — 2. Extension Theorem for Analytic Sets — 3. Proof of Proposition 2–4. Historical Note.- § 5. Theorems of Chow, Levi and Hurwitz-Weierstrass.- 1. Theorem of Chow — 2. Levi Extension Theorem — 3. Theorem of Hurwitz-Weierstraß — 4. Historical Notes.- 10. Direct Image Theorem.- § 1. Polydisc Modules.- 1. The Protonorm System on O(E) — 2. Polydisc Modules — 3. Morphisms and Morphism Systems — 4. Complexes of Polydisc Modules — 5. Cohomology of Poly-disc Modules. Quasi-Isomorphisms — 6. Finiteness Lemma F(q) and Projection Lemma Z(q) for Cocycles.- § 2. Proof of Lemmata F(q) and Z(q).- 1. Homotopy — 2. Z(q) ? Z(q-1) — 3. F(q), Z(q)?F(q-1) begin — 4. Smoothing — 5. Construction of Lq-1, ? - 6. Basic Property of ? - 7. Vanishing of Hq-1(t, ?, K).- § 3. Sheaves of Polydisc Modules.- 1. Definitions for $$U \subset \dot E$$ — 2. The Natural Functor — 3. The Paragraphs 1.4–1.6 for Polydisc Sheaves — 4. Coherence of Cohomology Sheaves. Main Theorem.- § 4. Coherence of Direct Image Sheaves.- 1. Mounting Complex Spaces — 2. Resolutions — 3. Complexes of Polydisc Modules — 4. Complexes of Sheaves — 5. Application of the Main Theorem — 6. The Direct Image Theorem.- § 5. Regular Families of Compact Complex Manifolds.- 1. Regular Families — 2. Complex Subspaces Y’ ? Y of Codimension 1 — 3. The Maps fy,i — 4. Upper Semi-Continuity — 5. The Case $${\dim _C}{H^i}\left( {{X_y},{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{V} }_y}} \right) = $$ constant — 6. Rigid Complex Manifolds.- § 6. Stein Factorization and Applications.- 1. Stein Factorization of Proper Holomorphic Maps — 2. Proper Modifications of Normal Complex Spaces — 3. Graph of a Finite System of Meromorphic Functions — 4. Analytic and Algebraic Dependence — 5. Base Space of a Finite System of Meromorphic Functions — 6. Properties of Base Spaces — 7. Analytic Closures and Structure of the Field M(X) — 8. Reduction Theorem for Holomorphically Convex Spaces.- Annex. Theory of Sheaves. Notion of Coherence.- §0. Sheaves.- 1. Sheaves and Morphisms — 2. Restrictions, Subsheaves and Sums of Sheaves — 3. Sections. Hausdorff Sheaves.- § 1. Construction of Sheaves from Presheaves.- 1. Presheaves — 2. The Sheaf Associated to a Preshaf — 3. Canonical Presheaves — 4. Image Sheaves.- § 2. Sheaves and Presheaves with Algebraic Structure.- 1. Sheaves of Groups, Rings and A-Modules — 2. The Category of A-Modules. Quotient Sheaves — 3. Presheaves with Algebraic Structure — 4. The Functor Hom — 5. The Functor ?.- § 3. Coherent Sheaves.- 1. Sheaves of Finite Type — 2. Sheaves of Relation Finite Type — 3. Coherent Sheaves.- § 4. Yoga of Coherent Sheaves.- 1. Three Lemma — 2. Consequences of the Three Lemma — 3. Coherence of Trivial Extensions — 4. Coherence of the Functors Hom and ? — 5. Annihilator Sheaves.- Index of Names.

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Book SynopsisDie Bewältigung des Grundstudiums Mathematik entscheidet sich größtenteils am erfolgreichen Lösen der gestellten Übungsaufgaben. Dies erfordert jedoch eine Professionalität, in die Studierende erst langsam hineinwachsen müssen. Das vorliegende Buch möchte sie bei diesem Prozess unterstützen. Es schafft Vorbilder in Gestalt ausführlicher Musterlösungen zu typischen Aufgaben aus der Analysis und der Linearen Algebra. Zusätzlich liefert es Anleitungen, wesentliche Strategien und Techniken zu verstehen, einzuüben und zu reflektieren. Das Buch hat den Anspruch, die kompletten Lösungswege inklusive der Ideengewinnung und etwaiger Alternativen darzustellen. Im Übungsteil wird das Hin- und Herschalten zwischen komprimierten und ausführlichen Musterlösungen geschult. In der vorliegenden Neuauflage wurde ein Kapitel mit Musterlösungen eingefügt, die sich mit Grundlagen mathematischen Arbeitens beschäftigen.Table of ContentsLerntheoretische Grundlagen.- Teilprozesse beim Aufgabenlösen.- Musterlösungen zu mathematischen Grundlagen.- Musterlösungen aus der Analysis 1.- Musterlösungen aus der Analysis 2.- Musterlösungen aus der Linearen Algebra 1.- Musterlösungen aus der Linearen Algebra 2.- Verfassen ausführlicher Musterlösungen.- Lösungsvorschläge.

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Book SynopsisAnimationen im Internet veranschaulichen z. B. die Wellengleichung durch eine schwingende Membran, die Wärmeleitung durch eine abnehmende Temperaturverteilung und die Potentialgleichung durch ein von der Randbelegung aufgeprägtes Potenzial. Welche Methoden verbergen sich dahinter, wie erzeugt man diese Animationen? Darauf soll der Leser eine erschöpfende Antwort geben können. Auf ausführliche, formale Beweise wird verzichtet. Die Begriffe werden mittels Beispielen und Graphiken in ihren Grundideen veranschaulicht und motiviert. Der Leser soll Hintergrundwissen und Lösungskompetenz bekommen, damit er sich nicht mit der Formelmanipulation zufrieden geben muss. Studierende sollen in die Lage versetzt werden, Probleme, die sich aus ihrer Bachelor/Masterarbeit oder aus den Anwendungen ergeben, zu bearbeiten. Table of ContentsTensoren.- Tensorfelder.- Kurven- und Flächenintegrale.- Orthogonale Systeme von Polynomen.- Lineare Differentialgleichungen im Komplexen.- Stabilität dynamischer Systeme.- Partielle Differentialgleichungen.- Gleichungen erster Ordnung.- Gleichungen zweiter Ordnung.- Wellengleichung.- Wärmeleitungsgleichung.- Potentialgleichung.

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Book SynopsisDieser Band für Studierende der Mathematik ab dem zweiten Semester setzt den ersten Band »Etwas Analysis« fort und führt in die klassische mehrdimensionale Analysis ein. Wieder wird Wert auf eine klare Darstellung mit einer möglichst einfachen Notation gelegt, die Denkweisen der Analysis werden herausgearbeitet. Ausgehend von Kurven werden die mehrdimensionale Differenziation und Analysis entwickelt. Dies führt bis zum Begriff der eingebetteten Mannigfaltigkeit, dem natürlichen Ort der Theorie der Extrema mit Nebenbedingungen. Die gewöhnlichen Differenzialgleichungen nehmen einen breiten Raum ein und bieten gleichzeitig einen Einstieg in die Theorie der dynamischen Systeme. Zu jedem Kapitel gibt es zahlreiche Aufgaben, deren vollständige Lösungen auf der Website des Verlages unter „Zusätzliche Informationen“ bereit gestellt werden. Dieser Band findet seine Fortsetzung im dritten Band "Noch mehr Analysis".Table of ContentsKurven und Wege.- Mehrdimensionale Differenziation.- Mehrdimensionale Analysis.– Wegintegrale.- Lineare Differenzialgleichungen.- Gewöhnliche Differenzialgleichungen.- Integration von Vektorfeldern.- Gleichgewichtspunkte.- Periodische Lösungen.

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