Calculus and mathematical analysis Books

Book SynopsisThis monograph presents recent developments in comparison geometry and geometric analysis on Finsler manifolds. Generalizing the weighted Ricci curvature into the Finsler setting, the author systematically derives the fundamental geometric and analytic inequalities in the Finsler context. Relying only upon knowledge of differentiable manifolds, this treatment offers an accessible entry point to Finsler geometry for readers new to the area. Divided into three parts, the book begins by establishing the fundamentals of Finsler geometry, including Jacobi fields and curvature tensors, variation formulas for arc length, and some classical comparison theorems. Part II goes on to introduce the weighted Ricci curvature, nonlinear Laplacian, and nonlinear heat flow on Finsler manifolds. These tools allow the derivation of the Bochner–Weitzenböck formula and the corresponding Bochner inequality, gradient estimates, Bakry–Ledoux’s Gaussian isoperimetric inequality, and functional inequalities in the Finsler setting. Part III comprises advanced topics: a generalization of the classical Cheeger–Gromoll splitting theorem, the curvature-dimension condition, and the needle decomposition. Throughout, geometric descriptions illuminate the intuition behind the results, while exercises provide opportunities for active engagement. Comparison Finsler Geometry offers an ideal gateway to the study of Finsler manifolds for graduate students and researchers. Knowledge of differentiable manifold theory is assumed, along with the fundamentals of functional analysis. Familiarity with Riemannian geometry is not required, though readers with a background in the area will find their insights are readily transferrable.Trade Review“Finsler geometry is an active area of research in mathematics and has led to numerous real-world applications. This book is a comprehensive introduction to Finsler geometry and its applications. It covers the basic concepts of this geometry. More intuitively, this book provides an accessible introduction to recent developments in comparison geometry and geometric analysis on Finsler manifolds. … this book offers a valuable perspective for those familiar with comparison geometry and geometric analysis.” (Behroz Bidabad, Mathematical Reviews, May, 2023)Table of ContentsI Foundations of Finsler Geometry.- 1. Warm-up: Norms and inner products.- 2. Finsler manifolds.- 3. Properties of geodesics.- 4. Covariant derivatives.- 5. Curvature.- 6. Examples of Finsler manifolds.- 7. Variation formulas for arclength.- 8. Some comparison theorems.- II Geometry and analysis of weighted Ricci curvature.- 9. Weighted Ricci curvature.- 10. Examples of measured Finsler manifolds.- 11. The nonlinear Laplacian.- 12. The Bochner-Weitzenbock formula.- 13. Nonlinear heat flow.- 14. Gradient estimates.- 15. Bakry-Ledoux isoperimetric inequality.- 16. Functional inequalities.- III Further topics.- 17. Splitting theorems.- 18. Curvature-dimension condition.- 19. Needle decompositions.

Read more less

Book SynopsisThis book is designed to serve as a textbook for courses offered to undergraduate and graduate students enrolled in Mathematics. The first edition of this book was published in 2015. As there is a demand for the next edition, it is quite natural to take note of the several suggestions received from the users of the earlier edition over the past six years. This is the prime motivation for bringing out a revised second edition with a thorough revision of all the chapters. The book provides a clear understanding of the basic concepts of differential and integral calculus starting with the concepts of sequences and series of numbers, and also introduces slightly advanced topics such as sequences and series of functions, power series, and Fourier series which would be of use for other courses in mathematics for science and engineering programs. The salient features of the book are - precise definitions of basic concepts; several examples for understanding the concepts and for illustrating the results; includes proofs of theorems; exercises within the text; a large number of problems at the end of each chapter as home-assignments. The student-friendly approach of the exposition of the book would be of great use not only for students but also for the instructors. The detailed coverage and pedagogical tools make this an ideal textbook for students and researchers enrolled in a mathematics course. Table of ContentsSequence and Series of Real Numbers.- Limit, Continuity and Differentiability of Functions.- Definite Integral.- Improper Integrals.- Sequence and Series of Functions.- Fourier Series.- References.- Index.

Read more less

Book SynopsisThis text provides a detailed presentation of the main results for infinite products, as well as several applications. The target readership is a student familiar with the basics of real analysis of a single variable and a first course in complex analysis up to and including the calculus of residues. The book provides a detailed treatment of the main theoretical results and applications with a goal of providing the reader with a short introduction and motivation for present and future study. While the coverage does not include an exhaustive compilation of results, the reader will be armed with an understanding of infinite products within the course of more advanced studies, and, inspired by the sheer beauty of the mathematics. The book will serve as a reference for students of mathematics, physics and engineering, at the level of senior undergraduate or beginning graduate level, who want to know more about infinite products. It will also be of interest to instructors who teach courses that involve infinite products as well as mathematicians who wish to dive deeper into the subject. One could certainly design a special-topics class based on this book for undergraduates. The exercises give the reader a good opportunity to test their understanding of each section.Trade Review“This is an excellent textbook … . It must be very satisfactory for students to learn the subject from such a nicely written book.” (Marcel G. de Bruin, zbMATH 1491.40001, 2022)Table of ContentsPreface.- 1. Introduction.- 2. Infinite Products.- 3. The Gamma Function.- 4. Prime Numbers, Partitions and Products.- 5. Epilogue.- 6. Tables of Products.- References.

Read more less

Book SynopsisPart 1 Fundamentals. I Weighted Potentials.- II Recovery of Measures, Green Functions and Balayage.- III Weighted Polynomials.- IV Determination of the Extremal Measure.- Part 2 Applications and Generalizations.- V Extremal Point Methods.- VI Weights on the Real Line.- VII Applications Concerning Orthogonal Polynomials.- VIII Signed Measures.- IX Some Problems from Physics.- X Generalizations.- Part 3 Appendices.- A.I Basic Tools.- A.II The Dirichlet Problem and Harmonic Measures.- A.III Weighted approximation in C?.- A.IV Classical Logarithmic Potential Theory.

Read more less

Book Synopsis- 1. Guido Weiss: a few memories of a friend and an influential mathematician.- 2. Optimal non-absolute domains for the Cesàro operator minus the identity.- 3. An applicable variant of spectral synthesis for wavelets.- 4. An update on the compactness of bilinear commutators.- 5. Irregularities of distribution on two-point homogeneous spaces.- 6. Analytic families of operators in extrapolation theory with application to average operators.- 7. On Complex Analytic tools, and the Holomorphic Rotation methods.- 8. Speed of convergence in an ergodic theorem.- 9. Characterizations of product Hardy spaces on stratified groups by singular integrals and maximal functions.- 10. A sufficient condition for Haar multipliers in Triebel-Lizorkin spaces.- 11. On frames of smooth, compactly-supported wave packets adapted to tilings of frequency space.- 12. Singular integral operators and Hölder spaces in Dunkl Setting.- 13. An homage to Guido Weiss and his leadership of the Saint Louis team: Commutators of Singular Integrals and Sobolev inequalities.- 14. Time-Frequency Analysis Meets Adversarial Learning.- 15. On Low-Rank Convex-Convex Quadratic Fractional Programming.- 16. Crystalline measures and wave front sets.- 17. Muckenhoupt matrix weights for general bases.

Read more less

Book Synopsis- Introduction.- The Riemann Mapping Theorem.- The Ahlfors Map.- A Riemann Mapping Theorem for Two-Connected Domains in the Plane.- Riemann Multiply Connected Domains.- Quasiconformal Mappings.- Manifolds.- Riemann Surfaces.- The Uniformization Theorem.- Automorphism Groups.- Ridigity of Holomorphic Mappings and a New Schwarz Lemma at the Boundary.- The Schwarz Lemma and Its Generalizations.- Invariant Distances on Complex Manifolds.- Hyperbolic Manifolds.- The Fatou Theory and Related Matters.- The Theorem of Bun Wong and Rosay.- Smoothness to the Boundary of Biholomorphic Mappings.- Solution ? problem.- Harmonic measure.- Quadrature.- Teichmüller Theory.- Bibliography.- Index.

Read more less

Book Synopsis- Introduction. - Convection-Di usion-Reaction Problems and Maximum Principles.- Discrete Maximum Principles.- Partitions of the Domain.- Finite Element Methods.- Finite Element Methods for Diffusion Problems.- Finite Element Methods for Reaction-Diffusion Problems.- Linear Finite Element Methods for Convection-Diffusion-Reaction Problems.- Nonlinear Finite Element Methods for Convection-Diffusion-Reaction Problems: Discretizations Based on Modi ed Variational Forms.- Nonlinear Finite Element Methods for Convection-Diffusion-Reaction Problems: Algebraically Stabilized Methods.- Finite Difference Methods.- Finite Volume Methods.- A Numerical Study for a Problem with Different Regimes.- Outlook.

Read more less

Book Synopsis- 1. Introduction.- Part I: The Sobolev Spaces and the Boundary Value Problems.- 2. Main notations and basic formulas.- 3. Overview of measure theory and functional analysis.- 4. Notes on the distribution theory and Fourier transform.- 5. The Sobolev spaces.- 6. The boundary value problems for second–order elliptic equations and the Dirichlet to Neumann map.- Part II: Cauchy Problem for PDEs and Stability Estimates.- 7. The Cauchy problem for the first–order PDEs.- 8. Real analytic functions.- 9. The Cauchy problem for PDEs with analytic coefficients.- 10. Uniqueness for an inverse problem.- 11. The Hadamard example. Solvability of the Cauchy problem and continuous dependence by the data.- 12. Ill–posed problems. Conditional stability.- 13. The John stability Theorem for the Cauchy problem for PDEs with analytic coefficients.- Part III: Carleman Estimates and Unique Continuation Properties.- 14. Carleman estimates: a first look with simple examples and basic applications.- 15. Carleman estimates and the Cauchy problem for operators with ??∞ coefficients in the principal part.- 16. Carleman estimates for reduced regularity coefficients.- 17. Carleman estimates for second–order operators with real coefficients in the principal part.- 18. Optimal three sphere and doubling inequality for second–order elliptic equations.- 19. Miscellanea.

Read more less

Book SynopsisThe Białowieża workshops on Geometric Methods in Physics are among the most important meetings in the field. Every year some 80 to 100 participants from both mathematics and physics join to discuss new developments and to interchange ideas. This volume contains contributions by selected speakers at the XXX meeting in 2011 as well as additional review articles and shows that the workshop remains at the cutting edge of ongoing research. The 2011 workshop focussed on the works of the late Felix A. Berezin (1931–1980) on the occasion of his 80th anniversary as well as on Bogdan Mielnik and Stanisław Lech Woronowicz on their 75th and 70th birthday, respectively. The groundbreaking work of Berezin is discussed from today’s perspective by presenting an overview of his ideas and their impact on further developments. He was, among other fields, active in representation theory, general concepts of quantization and coherent states, supersymmetry and supermanifolds. Another focus lies on the accomplishments of Bogdan Mielnik and Stanisław Lech Woronowicz. Mielnik’s geometric approach to the description of quantum mixed states, the method of quantum state manipulation and their important implications for quantum computing and quantum entanglement are discussed as well as the intricacies of the quantum time operator. Woronowicz’ fruitful notion of a compact quantum group and related topics are also addressed.

Read more less

Book SynopsisThis book gives a systematic treatment of the basic theory of analytic semigroups and abstract parabolic equations in general Banach spaces, and of how such a theory may be used in parabolic PDE's. It takes into account the developments of the theory during the last fifteen years, and it is focused on classical solutions, with continuous or Holder continuous derivatives. On one hand, working in spaces of continuous functions rather than in Lebesgue spaces seems to be appropriate in view of the number of parabolic problems arising in applied mathematics, where continuity has physical meaning; on the other hand it allows one to consider any type of nonlinearities (even of nonlocal type), even involving the highest order derivatives of the solution, avoiding the limitations on the growth of the nonlinear terms required by the LP approach. Moreover, the continuous space theory is, at present, sufficiently well established. For the Hilbert space approach we refer to J. L. LIONS - E. MAGENES [128], M. S. AGRANOVICH - M. l. VISHIK [14], and for the LP approach to V. A. SOLONNIKOV [184], P. GRISVARD [94], G. DI BLASIO [72], G. DORE - A. VENNI [76] and the subsequent papers [90], [169], [170]. Many books about abstract evolution equations and semigroups contain some chapters on analytic semigroups. See, e. g. , E. HILLE - R. S. PHILLIPS [100]' S. G. KREIN [114], K. YOSIDA [213], A. PAZY [166], H. TANABE [193], PH.Table of Contents0 Preliminary material: spaces of continuous and Hölder continuous functions.- 0.1 Spaces of bounded and/or continuous functions.- 0.2 Spaces of Hölder continuous functions.- 0.3 Extension operators.- 1 Interpolation theory.- 1.1 Interpolatory inclusions.- 1.2 Interpolation spaces.- 1.2.1 The K-method.- 1.2.2 The trace method.- 1.2.3 The Reiteration Theorem.- 1.2.4 Some examples.- 1.3 Bibliographical remarks.- 2 Analytic semigroups and intermediate spaces.- 2.1 Basic properties of etA.- 2.1.1 Identification of the generator.- 2.1.2 A sufficient condition to be a sectorial operator.- 2.2 Intermediate spaces.- 2.2.1 The spaces DA(?, p) and DA(?).- 2.2.2 The domains of fractional powers of —A.- 2.3 Spectral properties and asymptotic behavior.- 2.3.1 Estimates for large t.- 2.3.2 Spectral properties of etA.- 2.4 Perturbations of generators.- 2.5 Bibliographical remarks.- 3 Generation of analytic semigroups by elliptic operators.- 3.1 Second order operators.- 3.1.1 Generation in Lp(?), 1 < p < ?.- 3.1.2 Generation in L? (Rn) and in spaces of continuous functions in Rn.- 3.1.3 Characterization of interpolation spaces and generation results in Hölder spaces in Rn.- 3.1.4 Generation in C1(Rn).- 3.1.5 Generation in L? (?) and in spaces of continuous functions in $$ \overline \Omega $$.- 3.2 Higher order operators and bibliographical remarks.- 4 Nonhomogeneous equations.- 4.1 Solutions of linear problems.- 4.2 Mild solutions.- 4.3 Strict and classical solutions, and optimal regularity.- 4.3.1 Time regularity.- 4.3.2 Space regularity.- 4.3.3 A further regularity result.- 4.4 The nonhomogeneous problem in unbounded time intervals.- 4.4.1 Bounded solutions in [0, +?[.- 4.4.2 Bounded solutions in ] - ?, 0].- 4.4.3 Bounded solutions in R.- 4.4.4 Exponentially decaying and exponentially growing solutions.- 4.5 Bibliographical remarks.- 5 Linear parabolic problems.- 5.1 Second order equations.- 5.1.1 Initial value problems in [0,T] × Rn.- 5.1.2 Initial boundary value problems in $$ \left[ {0,T} \right] \times \overline \Omega $$.- 5.2 Bibliographical remarks.- 6 Linear nonautonomous equations.- 6.1 Construction and properties of the evolution operator.- 6.2 The variation of constants formula.- 6.3 Asymptotic behavior in the periodic case.- 6.3.1 The period map.- 6.3.2 Estimates on the evolution operator.- 6.3.3 Asymptotic behavior in nonhomogeneous problems.- 6.4 Bibliographical remarks.- 7 Semilinear equations.- 7.1 Local existence and regularity.- 7.1.1 Local existence results.- 7.1.2 The maximally defined solution.- 7.1.3 Further regularity, classical and strict solutions.- 7.2 A priori estimates and existence in the large.- 7.3 Some examples.- 7.3.1 Reaction-diffusion systems.- 7.3.2 A general semilinear equation.- 7.3.3 Second order equations with nonlinearities in divergence form.- 7.3.4 The Cahn-Hilliard equation.- 7.4 Bibliographical remarks for Chapter 7.- 8 Fully nonlinear equations.- 8.1 Local existence, uniqueness and regularity.- 8.2 The maximally defined solution.- 8.3 Further regularity properties and dependence on the data.- 8.3.1 Ck regularity with respect to (x, ?).- 8.3.2 Ck regularity with respect to time.- 8.3.3 Analyticity.- 8.4 The case where X is an interpolation space.- 8.5 Examples and applications.- 8.5.1 An equation from detonation theory.- 8.5.2 An example of existence in the large.- 8.5.3 A general second order problem.- 8.5.4 Motion of hypersurfaces by mean curvature.- 8.5.5 Bellman equations.- 8.6 Bibliographical remarks.- 9 Asymptotic behavior in fully nonlinear equations.- 9.1 Behavior near stationary solutions.- 9.1.1 Stability and instability by linearization.- 9.1.2 The saddle point property.- 9.1.3 The case where X is an interpolation space.- 9.1.4 Bifurcation of stationary solutions.- 9.1.5 Applications to nonlinear parabolic problems, I.- 9.1.6 Stability of travelling waves in two-phase free boundary problems.- 9.2 Critical cases of stability.- 9.2.1 The center-unstable manifold.- 9.2.2 Applications to nonlinear parabolic problems, II.- 9.2.3 The case where the linear part generates a bounded semigroup.- 9.2.4 Applications to nonlinear parabolic problems, III.- 9.3 Periodic solutions.- 9.3.1 Hopf bifurcation.- 9.3.2 Stability of periodic solutions.- 9.3.3 Applications to nonlinear parabolic problems, IV.- 9.4 Bibliographical remarks.- Appendix: Spectrum and resolvent.- A.1 Spectral sets and projections.- A.2 Isolated points of the spectrum.- A.3 Perturbation results.

Read more less

Book SynopsisThe present volume is an extensive monograph on the analytic and geometric aspects of Markov diffusion operators. It focuses on the geometric curvature properties of the underlying structure in order to study convergence to equilibrium, spectral bounds, functional inequalities such as Poincaré, Sobolev or logarithmic Sobolev inequalities, and various bounds on solutions of evolution equations. At the same time, it covers a large class of evolution and partial differential equations. The book is intended to serve as an introduction to the subject and to be accessible for beginning and advanced scientists and non-specialists. Simultaneously, it covers a wide range of results and techniques from the early developments in the mid-eighties to the latest achievements. As such, students and researchers interested in the modern aspects of Markov diffusion operators and semigroups and their connections to analytic functional inequalities, probabilistic convergence to equilibrium and geometric curvature will find it especially useful. Selected chapters can also be used for advanced courses on the topic.Trade Review“The book is friendly written and is of a rich content. With simple examples, main ideas of the study are clearly explained and naturally extended to a general framework, so that main progress in the field made for the past decades is presented in a smooth way. The monograph is undoubtedly a significant reference for further development of diffusion semigroups and related topics.” (Feng-Yu Wang, zbMATH 1376.60002, 2018)“It is extremely rich. It is more original and inspirational than a treatise. One can use it and benefit from it in many ways: as a reference book, as an inspiration source, by focusing on a property or on an example. … From the beginning to the end, this book definitely has a strong personality and a characteristic taste. … anybody who wants to explore analytic, probabilistic or geometric properties of Markov semigroups to have a look at it first.” (Thierry Coulhon, Jahresbericht der Deutschen Math-Vereinigung, Vol. 119, 2017)“This impressive monograph is about an important and highly active area that straddles the fertile land occupied by both probability and analysis. … It is written with great clarity and style, and was clearly a labour of love for the authors. I am convinced that it will be a valuable resource for researchers in analysis and probability for many years to come.” (David Applebaum, The Mathematical Gazette, Vol. 100 (548), July, 2016)Table of ContentsIntroduction.- Part I Markov semigroups, basics and examples: 1.Markov semigroups.- 2.Model examples.- 3.General setting.- Part II Three model functional inequalities: 4.Poincaré inequalities.- 5.Logarithmic Sobolev inequalities.- 6.Sobolev inequalities.- Part III Related functional, isoperimetric and transportation inequalities: 7.Generalized functional inequalities.- 8.Capacity and isoperimetry-type inequalities.- 9.Optimal transportation and functional inequalities.- Part IV Appendices: A.Semigroups of bounded operators on a Banach space.- B.Elements of stochastic calculus.- C.Some basic notions in differential and Riemannian geometry.- Notations and list of symbols.- Bibliography.- Index.

Read more less

Book SynopsisBased on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points — which are rarely treated in books at this level — and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.Trade Review"This book, in its second edition, provides the basic concepts of real analysis. ... I strongly recommend it to everyone who wishes to study real mathematical analysis." (Catalin Barbu, zbMATH 1329.26003, 2016)Table of ContentsReal Numbers.- A Taste of Topology.- Functions of a Real Variable.- Function Spaces.- Multivariable Calculus.- Lebesgue Theory.

Read more less

Book SynopsisThis book presents in a systematic and almost self-contained way the striking analogy between classical function theory, in particular the value distribution theory of holomorphic curves in projective space, on the one hand, and important and beautiful properties of the Gauss map of minimal surfaces on the other hand. Both theories are developed in the text, including many results of recent research. The relations and analogies between them become completely clear. The book is written for interested graduate students and mathematicians, who want to become more familiar with this modern development in the two classical areas of mathematics, but also for those, who intend to do further research on minimal surfaces.Table of ContentsContents: The Gauss map of minimal surfaces in R3 - The derived curves of a holomorphic curve - The classical defect relations for holomorphic curves - Modified defect relation for holomorphic curves - The Gauss Map of complete minimal surfaces in Rm.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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).

Read more less

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.

Read more less

Table of ContentsRearrangements.- Maximum principles.

Read more less

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.

Read more less

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

Read more less

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 ? (?,?).

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

Book SynopsisDieses Lehr- und Arbeitsbuch bietet dem Studienanfänger aus Physik und Ingenieurwissenschaften, der Praxis im Umgang mit der Mathematik erwerben möchte, durch Darstellung und didaktische Gestaltung wertvolle Hilfestellung bei der Erarbeitung mathematischen Grundwissens. Die Gestaltung des Textes, die den Leser immer wieder anregt, Gedankenschritte selbst zu vollziehen, weiterzuführen, Verbindungen herzustellen, Rechnungen nachzuvollziehen und die eigenen Kenntnisse zu überprüfen, bietet hier größtmögliche Unterstützung.Immer wieder werden anwendungsbezogene Beispiele gegeben und ausführlich bearbeitet. Definitionen und Sätze sind vollständig formuliert. Beweise werden nur da weggelassen, wo sie weder dem Verständnis des Satzes noch dem Einüben bestimmter Schlußweisen oder Begriffe dienen. Bei der Bearbeitung der ca. 250 Aufgaben wird dem Studenten eine gestufte Hilfestellung in Form von Lösungshinweisen und der kompletten Lösung gegeben.Table of Contents1. Die reellen Zahlen.- § 1 Mengen.- § 2 Funktionen.- Definitionen und Beispiele.- Die Komposition von Funktionen.- Die Umkehrfunktion.- Bijektive Funktionen.- § 3 Die reellen Zahlen.- Die Zahlengerade.- Die arithmetischen Eigenschaften von JR.- Ungleichungen.- Intervalle.- Definition und Eigenschaften der Wurzel.- Der Betrag.- Zusammenfassung.- 2. Vollständige Induktion.- § 1 Beweis durch vollständige Induktion.- Erklärung des Suinmenzeichens.- § 2 Rekursive Definitionen.- § 3 n-te Potenz und n-te Wurzel.- Eigenschaften der n-ten Potenz.- Die n-te Wurzel.- Die binomische Formel.- Zusammenfassung.- 3. Die komplexen zahlen.- § 1 Definition und Veranschaulichung.- § 2 Der Körper ? der komplexen Zahlen.- Rechengesetze in ?.- IR als Teilmenge von ?.- § 3 Realteil, Imaginärteil, Betrag.- Realteil, Imaginârteil, Konjugierte.- Der Betrag.- § 4 Die Polarform.- § 5 n-te Wurzeln einer komplexen Zahl.- Zusammenfassung.- 4. Reelle und komplexe Funktionen.- § 1 Definition der reellen Funktionen und Beispiele.- § 2 Monotone Funktionen.- § 3 Beispiele aus der Wechselstrom-lehre.- § 4 Rechnen mit reellen Funktionen.- § 5 Polynome.- Das Horner-Schema.- Nullstellen von Polynomen.- § 6 Komplexe Funktionen.- Komplexe Funktionen mit reellen Argumenten.- Zusammenfassung.- 5. Das Supremum.- § 1 Schranken, Maximum, Minimum, Supremum, Infimum.- § 2 Das Supremumsaxiom.- § 3 Eigenschaften von Supremum und Infimum.- § 4 Supremum und Maximum bei Funktionen.- § 5 Dual-, Dezimal-und Hexadezimal-zahlen.- Zusammenfassung.- 6. Folgen.- § 1 Definition.- § 2 Monotonie und Beschrànktheit.- Beschränktheit.- Monotonie.- Monotone beschrankte Folgen.- § 3 Konvergenz und Divergenz.- Konvergenz.- Divergenz.- Rechenregeln für konvergente Folgen.- Beispiele.- Rekursiv definierte Folgen.- § 4 Komplexe Folgen.- Zusammenfassung.- 7. Einführung in die Integralrechnung.- § 1 Beispiele.- § 2 Obersumme und Untersurame.- § 3 Die Definition des Integrals.- § 4 Das Riemannsche Integrabilitäts-kriterium.- Integrierbarkeit monotoner Funktionen.- § 5 Integral als Grenzwert einer Folge.- Das Riemannsche Summen-Kriterium.- § 6 Numerische Integration.- Die Rechteckregel.- Die Trapezregel.- Die Simpsonregel.- § 7 Eigenschaften des Integrals.- Eigenschaften des Integrals bezüg-lich des Integrationsintervalls.- Eigenschaften bezüglich des Inte-granden.- Ungleichungen für Integrale.- Zusammenfassung.- 8. Reihen.- (Zenon’s Paradoxon).- § 1 Beispiele.- § 2 Konvergente Reihen.- Geometrische Reihen.- Die „Schneeflockenkurve“.- Rechenregeln für konvergente Reihen.- Notwendiges Konvergenzkriterium.- § 3 Konvergenzkriterien.- Vergleichskriterien.- Wurzelkriterium.- Quotientenkriterium.- Alternierende Reihen.- § 4 Absolut konvergente Reihen.- Zusammenfassung.- 9. Potenzreihen und spezielle Funktionen.- § 1 Potenzreihen.- Konvergenz von Potenzreihen.- Zusammenfassung: Potenzreihen als Funktionen.- § 2 Exponentialfunktion.- Definition der Exponentialfunktion.- Eigenschaften der Exponentialfunktion.- § 3 Sinus und Cosinus.- § 4 Hyperbelfunktionen.- Zusammenfassung.- 10. Stetige Funktionen.- § 1 Stetigkeit.- Grenzwerte von Funktionen.- Einseitige und uneigentliche Grenzwerte.- Stetige Funktionen.- Trigonometrische Funktionen und Exponentialfunktion sind stetig.- Stetig auf [a,b]: Drei Sät6ze.- § 2 Anwendung auf spezielle Funktionen.- Exponentialfunktion, Logarithmus und allgemeine Potenz.- Trigonometrische Funktionen.- § 3 Die ?-?-Definition der Stetigkeit und die Lipschitz-Stegigkeit.- § 4 Stetigkeit und Integration.- Zusammenfassung.- 11. Differentialrechnung.- § 1 Lineare Approximation.- § 2 Definition der Differenzierbarkeit.- § 3 Differenzierbare Funktionen.- § 4 Rechenregeln für differenzierbare Funktionen.- Summe, Produkt, Quotient.- Die Kettenregel.- Die Ableitung der Umkehrfunktion.- Differenzierbarkeit von Potenzreihen.- § 5 Die Ableitung komplexer Funktionen.- § 6 Höhere Ableitungen.- Aufgaben zum Einuben der Diffe-rentiationstechniken.- § 7 Beispiele von Differential-gleichungen und Lösungen.- Losung der Schwingungsgleichung durch Potenzreihenansatz.- § 8 Der erste Mittelwertsatz.- Lokale Extrema.- Der erste Mittelwertsatz der Differentialrechnung.- Anwendungen des ersten Mittel-wertSät6zes.- § 9 Die Regeln von de L’Hôpital.- Zusammenfassung.- 12. Integralrechnung-Integrationstechnik.- § 1 Der Hauptsatz der Differential-und Integralrechnung.- § 2 Die Stammfunktion.- § 3 Eine andere Formulierung des HauptSät6zes.- § 4 Integration zur Lösung einfachster Differentialgleichungen.- § 5 Das unbestimmte Integral.- § 6 Die Integration komplexer Funktionen.- § 7 Integrationsmethoden.- Integranden der Form % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaatC % vAUfKttLearyqr1ngBPrgaiuaacuWFMbGzgaqbaaqaaiab-zgaMbaa % aaa!3D98! $$ \frac{{f'}}{f} $$.- Partielle Integration.- Substitution.- Eine Umformulierung der Substitu-tionsregel.- Substitution bei bestimmten Inte-gralen.- § 8 Separable Differentialgleichungen.- Lösungsmethode.- Merkregel.- Anfangswertprobleme.- § 9 Integration rationaler Funktionen.- 1. Schritt: Polynomdivision.- 2. Schritt: Polynomzerlegung.- 3. Schritt: Partialbruchzerlegung.- 4. Schritt: Integration rationaler Funktionen.- Kurze Merkregelsammlung.- Zusammenfassung.- 13. Uneigentliche Integrale.- § 1 Unbeschränktes Integrationsintervall.- Integrationsintervall ]- ?,? [.- Konvergenzkriterien.- § 2 Unbeschränkter Integrand.- Konvergenzkriterien.- § 3 Die Gammafunktion.- § 4 Die Laplace-Transformation.- Linearität und elementare Laplace-Transformationen.- Bemerkungen zum Umkehrproblem.- Transformation von Ableitungen.- Transformation von f(at±b).- Verschiebung des Arguments in der Bildfunktion.- Kurze Übersicht.- Zusammenfassung.- 14. Taylorpolynome und Taylorreihen.- § 1 Approximation durch Polynome.- Approximation.- Taylorpolynome.- § 2 Restglied.- Restglied nach Taylor.- Anwendung: Funktionswerte berechnen.- Restglied nach Lagrange.- Restglied abschätzen.- Anwendung: Lokale Extrema 2.- § 3 Taylorreihen.- Definition.- Ein Gegenbeispiel.- Konvergenz der Taylorreihe.- Beispiel Logarithmus.- Beispiel Arcus-Tangens.- Beispiel Binomische Reihe.- Zusammenfassung.- Lösungen der Aufgaben.

Read more less

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

Read more less

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.

Read more less

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.

Read more less

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

Read more less

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.

Read more less

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.

Read more less

Book SynopsisThe subject of partial differential equations holds an exciting and special position in mathematics. Partial differential equations were not consciously created as a subject but emerged in the 18th century as ordinary differential equations failed to describe the physical principles being studied. The subject was originally developed by the major names of mathematics, in particular, Leonard Euler and Joseph-Louis Lagrange who studied waves on strings; Daniel Bernoulli and Euler who considered potential theory, with later developments by Adrien-Marie Legendre and Pierre-Simon Laplace; and Joseph Fourier's famous work on series expansions for the heat equation. Many of the greatest advances in modern science have been based on discovering the underlying partial differential equation for the process in question. James Clerk Maxwell, for example, put electricity and magnetism into a unified theory by establishing Maxwell's equations for electromagnetic theory, which gave solutions for prob­ lems in radio wave propagation, the diffraction of light and X-ray developments. Schrodinger's equation for quantum mechanical processes at the atomic level leads to experimentally verifiable results which have changed the face of atomic physics and chemistry in the 20th century. In fluid mechanics, the Navier­ Stokes' equations form a basis for huge number-crunching activities associated with such widely disparate topics as weather forecasting and the design of supersonic aircraft. Inevitably the study of partial differential equations is a large undertaking, and falls into several areas of mathematics.Table of Contents1. Background Mathematics.- 1.1 Introduction.- 1.2 Vector and Matrix Norms.- 1.3 Gerschgorin’s Theorems.- 1.4 Iterative Solution of Linear Algebraic Equations.- 1.5 Further Results on Eigenvalues and Eigenvectors.- 1.6 Classification of Second Order Partial Differential Equations.- 2. Finite Differences and Parabolic Equations.- 2.1 Finite Difference Approximations to Derivatives.- 2.2 Parabolic Equations.- 2.3 Local Truncation Error.- 2.4 Consistency.- 2.5 Convergence.- 2.6 Stability.- 2.7 The Crank-Nicolson Implicit Method.- 2.8 Parabolic Equations in Cylindrical and Spherical Polar Coordinates.- 3. Hyperbolic Equations and Characteristics.- 3.1 First Order Quasi-linear Equations.- 3.2 Lax-Wendroff and Wendroff Methods.- 3.3 Second Order Quasi-linear Hyperbolic Equations.- 3.4 Reetangular Nets and Finite Difference Methods for Second Order Hyperbolic Equations.- 4. Elliptic Equations.- 4.1 Laplace’s Equation.- 4.2 Curved Boundaries.- 4.3 Solution of Sparse Systems of Linear Equations.- 5. Finite Element Method for Ordinary Differential Equations.- 5.1 Introduction.- 5.2 The Collocation Method.- 5.3 The Least Squares Method.- 5.4 The Galerkin Method.- 5.5 Symmetrie Variational Forrnulation.- 5.6 Finite Element Method.- 5.7 Some Worked Examples.- 6. Finite Elements for Partial Differential Equations.- 6.1 Introduction.- 6.2 Variational Methods.- 6.3 Some Specific Elements.- 6.4 Assembly of the Elements.- 6.5 Worked Example.- 6.6 A General Variational Principle.- 6.7 Assembly and Solution.- 6.8 Solution of the Worked Example.- 6.9 Further Interpolation Functions.- 6.10 Quadrature Methods and Storage Considerations.- 6.11 Boundary Element Method.- A. Solutions to Exercises.- References and Further Reading.

Read more less