Calculus and mathematical analysis Books

Book SynopsisPresents an understanding of scattering theory, starting with basic principles and extending to research. This work presents a comprehensive exposition of the theory, covering different methods (of trace class and smooth perturbations) and approaches (time dependent and stationary) and discussing the relationships among them.Table of ContentsPreliminary facts Basic concepts of scattering theory Further properties of the WO Scattering for relatively smooth perturbations The general setup in stationary scattering theory Scattering for perturbations of trace class type Properties of the scattering matrix (SM) The spectral shift function (SSF) and the trace formula.

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Book SynopsisThe study of group actions on manifolds is the meeting ground of a variety of mathematical areas. In particular, interesting geometric insights can be obtained by applying measure-theoretic techniques. This book provides an introduction to some of the important methods, major developments, and open problems on the subject.

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Book SynopsisIn 1972, G D Birkhoff wrote a treatise on the theory of dynamical systems that would inspire many later mathematicians to do great work. To a large extent, Birkhoff was writing about his own work on the subject, which was itself strongly influenced by Poincare's approach to dynamical systems. This volume is the revised 1966 reprinting of the book.Table of ContentsPhysical aspects of dynamical systems Variational principles and applications Formal aspects of dynamics Stability of periodic motions Existence of periodic motions Application of Poincare's geometric theorem General theory of dynamical systems The case of two degrees of freedom The problem of three bodies Addendum Footnotes Bibliography Index.

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Book SynopsisMorse theory is a study of deep connections between analysis and topology. In its classical form, it provides a relationship between the critical points of certain smooth functions on a manifold and the topology of the manifold. This book is Morse's own exposition of his ideas.Table of ContentsThe fixed end point problem in non-parametric form General end conditions The index form Self-adjoint systems The functional on a Riemannian space The critical sets of functions The boundary problem in the large Closed extremals Solution of the Poincare continuation problem.

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Book SynopsisPart of ""Colloquium Series"", this book presents systematic treatment of orthogonal polynomials.Trade ReviewThis is the first detailed systematic treatment of ... (a) the asymptotic behaviour of orthogonal polynomials, by various methods, with applications, in particular, to the `classical' polynomials of Legendre, Jacobi, Laguerre and Hermite; (b) a detailed study of expansions in series of orthogonal polynomials, regarding convergence and summability; (c) a detailed study of orthogonal polynomials in the complex domain; (d) a study of the zeros of orthogonal polynomials, particularly of the classical ones, based upon an extension of Sturm's theorem for differential equations. The book presents many new results; many results already known are presented in generalized or more precise form, with new simplified proofs." -- Mathematical ReviewsTable of Contents•Preliminaries •Definition of orthogonal polynomials; principal examples •General properties of orthogonal polynomials •Jacobi polynomials •Laguerre and Hermite polynomials •Zeros of orthogonal polynomials •Inequalities •Asymptotic properties of the classical polynomials •Expansion problems associated with the classical polynomials •Representation of positive functions •Polynomials orthogonal on the unit circle •Asymptotic properties of general orthogonal polynomials •Expansion problems associated with general orthogonal polynomials •Interpolation •Mechanical quadrature •Polynomials orthogonal on an arbitrary curve •Problems and exercises •Further problems and exercises •Appendix •List of references •Further references •Index

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Book SynopsisTable of ContentsPart One. Functional Analysis: Abstract spaces Linear transformations Vector-valued functions Banach algebras General properties Analysis in a Banach algebra Laplace integrals and binomial series Part Two. Basic Properties of Semi-Groups: Subadditive functions Semi-modules Addition theorem in a Banach algebra Semi-groups in the strong topology Generator and resolvent Generation of semi-groups Part Three. Advanced Analytical Theory of Semi-Groups: Perturbation theory Adjoint theory Operational calculus Spectral theory Holomorphic semi-groups Applications to ergodic theory Part Four. Special Semi-groups and Applications: Translations and powers Trigonometric semi-groups Semi-groups in $L_p(-\infty,\infty)$ Semi-groups in Hilbert space Miscellaneous applications Part Five. Extensions of the theory: Notes on Banach algebras Lie semi-groups Functions on vectors to vectors Bibliography Index.

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Book SynopsisA collection of some of the work of Frederick J Almgren, Jr, the man most noted for defining the shape of geometric variational problems and for his role in founding The Geometry Center. It includes a summary by Sheldon Chang of the famous 1,700 page paper on singular sets of area-minimizing $m$-dimensional surfaces in $R^n$.Table of ContentsThe mathematics of F. J. Almgren, Jr. by B. White On Almgren's regularity result by S. X. Chang The homotopy groups of the integral cycle groups by F. J. Almgren, Jr. An isoperimetric inequality by F. J. Almgren, Jr. Three theorems on manifolds with bounded mean curvature by F. J. Almgren, Jr. Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure by F. J. Almgren, Jr. Measure theoretic geometry and elliptic variational problems by F. J. Almgren, Jr. The structure of limit varifolds associated with minimizing sequences of mappings by F. J. Almgren, Jr. Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints by F. J. Almgren, Jr. The structure of stationary one dimensional varifolds with positive density by W. K. Allard and F. J. Almgren, Jr. The geometry of soap films and soap bubbles by F. J. Almgren, Jr. and J. E. Taylor Examples of unknotted curves which bound only surfaces of high genus within their convex hulls by F. J. Almgren, Jr. and W. P. Thurston Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals by R. Schoen, L. Simon, and F. J. Almgren, Jr. Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents by F. J. Almgren, Jr. Liquid crystals and geodesics by R. N. Thurston and F. J. Almgren $\mathbf{Q}$ valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two by F. J. Almgren, Jr. Optimal isoperimetric inequalities by F. Almgren Co-area, liquid crystals, and minimal surfaces by F. Almgren, W. Browder, and E. Lieb Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds by F. J. Almgren, Jr. and E. H. Lieb Symmetric decreasing rearrangement is sometimes continuous by F. J. Almgren, Jr. and E. H. Lieb Questions and answers about area-minimizing surfaces and geometric measure theory by F. Almgren Curvature-driven flows: A variational approach by F. Almgren, J. E. Taylor, and L. Wang Questions and answers about geometric evolution processes and crystal growth by F. Almgren.

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Book SynopsisPresents a comprehensive introduction to free probability theory, a highly noncommutative probability theory with independence based on free products instead of tensor products. This title provides basic examples of this kind of theory by convolution operators on free groups and by the asymptotic behavior of large Gaussian random matrices.Table of ContentsFree products Free random variables in noncommutative probability theory Free harmonic analysis Random matrices and asymptotic freeness Free product factors.

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Book SynopsisPresents the proceedings of the Joint Summer Research Conference on $q$-series, combinatorics, and computer algebra held at Mount Holyoke College (South Hadley, MA). This book includes articles which reflect the diversity of areas that overlap with $q$-series, as well as the usefulness of $q$-series across the mathematical sciences.Table of ContentsThe mathematical contributions of Richard Askey by G. Gasper, M. E.-H. Ismail, T. Koornwinder, P. Nevai, and D. Stanton Curriculum vitae of Richard A. Askey by M. E. H. Ismail and D. W. Stanton Reformulations of a partition theorem of Gollnitz and $q$-series identities by K. Alladi Schur's theorem, partitions with odd parts and the Al-Salam-Carlitz polynomials by G. E. Andrews Singularity and monodromy of quasi-hypergeometric functions by K. Aomoto and K. Iguchi Incomplete elliptic integrals in Ramanujan's lost notebook by B. C. Berndt, H. H. Chan, and S.-S. Huang Measure algebras associated with orthogonal polynomials by W. C. Connett and A. L. Schwartz Word straightening and $q$-Eulerian calculus by D. Foata and G. Han Path generating transforms by O. Foda, K. S. M. Lee, Y. Pugai, and T. A. Welsh $q$-extensions of Erdelyi's fractional integral representations for hypergeometric functions and some summation formulas for double $q$-Kampe de Feriet series by G. Gasper Numerical investigation of basic Fourier series by R. Wm. Gosper, Jr. and S. K. Suslov An identity of Ramanujan, and applications by M. D. Hirschhorn Addition theorems for the $q$-exponential function by M. E. H. Ismail and D. W. Stanton The Schur functions for partitions with complex parts by K. W. J. Kadell On Forrester's generalization of Morris constant term identity by J. Kaneko New combinatorial formula for modified Hall-Littlewood polynomials by A. N. Kirillov Schur function identities and the number of perfect matchings of Holey Aztec rectangles by C. Krattenthaler A new $U(n)$ generalization of the Jacobi triple product identity by S. C. Milne A new quantum algebraic interpretation of the Askey-Wilson polynomials by H. Rosengren Some properties of Koornwinder polynomials by S. Sahi A new multidimensional matrix inversion in $A_r$ by M. Schlosser.

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Book SynopsisContains historical background and discussion of results for each chapter, References, and an Index.Table of ContentsIntroduction: 1 Examples and prospectus 2 Metric spaces 3 Normed linear spaces 4 Inner-product spaces 5 Convexity 6 Existence and unicity of best approximations 7 Convex functions The Tchebycheff Solution of Inconsistent Linear Equations: 1 Introduction 2 Systems of equations with one unknown 3 Characterization of the solution 4 The special case 5 Polya's algorithm 6 The ascent algorithm 7 The descent algorithm 8 Convex programming Tchebycheff Approximation by Polynomials and Other Linear Families: 1 Introduction 2 Interpolation 3 The Weierstrass theorem 4 General linear families 5 the unicity problem 6 Discretization errors: General theory 7 Discretization: Algebraic polynomials. The inequalities of Markoff and Bernstein 8 Algorithms Least-squares Approximation and Related Topics: 1 Introduction 2 Orthogonal systems of polynomials 3 Convergence of orthogonal expansions 4 Approximation by series of Tchebycheff polynomials 5 Discrete least-squares approximation 6 The Jackson theorems Rational Approximation: 1 Introduction 2 Existence of best rational approximations 3 The characterization of best approximations 4 Unicity; Continuity of best-approximation operators 5 Algorithms 6 Pade Approximation and its generalizations 7 Continued fractions Some Additional Topics: 1 The Stone approximation theorem 2 The Muntz theorem 3 The converses of the Jackson theorems 4 Polygonal approximation and bases in $C[a, b]$ 5 The Kharshiladze-Lozinski theorems 6 Approximation in the mean Notes References Index.

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Book SynopsisPresents the analysis of optimal control problems for systems described by partial differential equations. This book covers cases where the controlled system corresponds to well-posed or ill-posed boundary value problems, which can be linear or nonlinear.Table of ContentsThe existence of solutions to optimal control problems Optimality system for optimal control problems The solvability of boundary value problems for a dense set of data The problem of work minimization in accelerating still fluid to a prescribed velocity Optimal boundary control for nonstationary problems of fluid flow and nonhomogeneous boundary value problems for the Navier-Stokes equations The Cauchy problem for elliptic equations in a conditionally well-posed formulation The local exact controllability of the flow of incompressible viscous fluid Bibliography Index.

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Book SynopsisMuch has been written on the theory of discontinuous groups and automorphic functions since 1880, when the subject received its first formulation. This book intends to bring together in one place both the classical and modern aspects of the theory, and to present them clearly and in a modern language and notation.

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Book SynopsisExamines the theory of measures having values in Banach spaces. This book deals with countably additive vector measures finitely additive vector measures, the Orlicz-Pettis theorem and its relatives. It also concentrates on measurable vector valued functions and the Bochner integral.

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Book SynopsisSymplectic geometry and the theory of Fourier integral operators are modern manifestations of themes that have occupied a central position in mathematical thought. This book intends to develop these themes, and present some of the advances, using the language of differential geometry as a unifying influence.

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Book SynopsisIn addition to the standard topics, this volume contains many topics not often found in an algebra book, such as inequalities, and the elements of substitution theory. Especially extensive is Chrystal's treatment of the infinite series, infinite products, and (finite and infinite) continued fractions.

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Book SynopsisContains many topics in algebra such as inequalities, and the elements of substitution theory. This volume includes over 2,400 exercises with solutions. It also gives a treatment of the infinite series, infinite products, and (finite and infinite) continued fractions.Table of ContentsVolume I: Fundamental laws and processes of algebra Monomials--Laws of indices--Degree Theory of quotients--First principles of theory of numbers Distribution of products--Elements of the theory of rational integral functions Transformation of the quotient of two integral functions Greatest common measure and least common multiple Factorisation of integral functions Rational fractions Further applications to the theory of numbers Irrational functions Arithmetical theory of surds Complex numbers Ratio and proportion On conditional equations in general Variation of a function Equations and functions of first degree Equations of the second degree General theory of integral functions Solution of problems by means of equations Arithmetic, geometric, and allied series Logarithms Theory of interest and annuities Appendix Results of exercises Index for parts I and II Volume II: Permutations and combinations General theory of inequalities Limits Convergence of infinite series and of infinite products Binomial and multinomial series for any index Exponential and logarithmic series Summation of the fundamental power-series for complex values of the variable General theorems regarding the expansion of functions in infinite forms Summation and transformation of series in general Simple continued fractions On recurring continued fractions General continued fractions General properties of integral numbers Probability, or the theory of averages Results of exercises Index of proper names for parts I and II Index for parts I and II.

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Book SynopsisThe laws of thermodynamics happen to be an extremely powerful tool for understanding the objects of holomorphic dynamical systems. This book develops a 'thermodynamic formalism', bringing together notions that are a priori unrelated. It describes this formalism both from the physical and mathematical point of view.Table of ContentsIntroduction The ergodic hypothesis The concept of entropy Entropy in ergodic theory The Perron-Frobenius-Ruelle theorem Conformal repellers Iteration of quadratic polynomials Phase transitions Hausdorff measures and dimension Bibliography.

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Book SynopsisThe Teichmuller space $T(X)$ is the space of marked conformal structures on a given quasi conformal surface $X$. This book uses quasi conformal mapping to give a unified and up-to-date treatment of $T(X)$. It offers a treatment of deformations of complex structures on infinite Riemann surfaces.Table of ContentsQuasiconformal mapping Riemann surfaces Quadratic differentials, Part I Quadratic differentials, Part II Teichmuller equivalence The Bers embedding Kobayashi's metric on Teichmuller space Isomorphisms and automorphisms Teichmuller uniqueness The mapping class group Jenkins-Strebel differentials Measured foliations Obstacle problems Asymptotic Teichmuller space Asymptotically extremal maps Universal Teichmuller space Substantial boundary points Earthquake mappings Bibliography Index.

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Book SynopsisThere is a deep and fundamental relationship between the differential equations that occur in the calculus of variations and partial differential equations of the first order: in particular, to each such partial differential equation there correspond variational problems. This book discusses this basic fact.Table of ContentsContinuous convergence, implicit functions, ordinary differential equations Fields of curves and multidimensional surfaces, complete systems Partial differential equations of the first order, theory of characteristics Poisson brackets, systems of partial differential equations of the first order Elements of tensor calculus Canonical transformations Contact transformations The Pfaff problem Function groups The integration theories of Lagrange, Jacobi, Adolph Mayer and Lie Ordinary maxima and minima. Quadratic forms Simple variational problems in the small Variational problems in parametric representation Positive-definite variational problems Quadratic variational problems. Theory of the second variation The boundary-value problem and the question of the absolute minimum Closed extremals. Periodic variational problems The problem of Lagrange Guide to the literature Index.

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Book SynopsisA monograph that develops the spectral theory of an $n$th order non-self-adjoint two-point differential operator $L$ in the Hilbert space $L^2[0,1]$. It lays the mathematical foundation in the first part, where the spectral theory is developed for closed linear operators and Fredholm operators.Table of ContentsUnbounded linear operators Fredholm operators Introduction to the spectral theory of differential operators Principal part of a differential operator Projections and generalized eigenfunction expansions Spectral theory for general differential operators Bibliography Index.

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Book SynopsisFor a Riemannian manifold $M$, the geometry, topology and analysis are interrelated in ways that are widely explored in modern mathematics. This book examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold.Table of ContentsClifford algebras and spin representation Spin structures Dirac operators Analytical properties of Dirac operators Eigenvalue estimates for the Dirac operator and twistor spinors Seiberg-Witten invariants Principal bundles and connections Bibliography Index.

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Book SynopsisOperator theory is a significant part of many important areas of modern mathematics: functional analysis, differential equations, index theory, representation theory, and mathematical physics. This text covers the central themes of operator theory. It is suitable for graduate students who have had a standard course in functional analysis.Trade Reviewrecommended it to anyone wishing to gain a better understanding of operator theory. Bulletin of the London Mathematical SocietyTable of ContentsIntroduction to C*-algebras Normal operators Compact operators Some non-normal operators More on C*-algebras Compact perturbations Introduction to von Neumann algebras Reflexivity Bibliography Index List of symbols.

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Book SynopsisBased on lectures delivered at the AMS meeting in 1901, this book describes the progress in calculus of variations. This title describes the landmark results of Weierstrass on sufficient conditions for the extremum of a functional in terms of the second variation. It helps in learning about classical calculus of variations.Table of ContentsThe first variation of the integral $\int_{x_0}^{x_1}F(x,y,y')dx$ The second variation of the integral $\int_{x_0}^{x_1}F(x,y,y')dx$ Sufficient conditions for an extremum of the integral $\int_{x_0}^{x_1}F(x,y,y')dx$ Weierstrass's theory of the problem in parameter representation Kneser's theory Weierstrass's theory of the isoperimetric problems Hilbert's existence theorem Index.

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Book SynopsisPresents a comprehensive exposition of the theory of linear operators on Banach spaces and Banach lattices. This book offers a presentation that includes various developments in operator theory and draws together results that are spread over the vast literature. It contains over 600 exercises to help students master the material developed.Table of ContentsOdds and ends Basic operator theory Operators on $AL$- and $AM$-spaces Special classes of operators Integral operators Spectral properties Some special spectra Positive matrices Irreducible operators Invariant subspaces The Daugavet equation Bibliography Index.

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Book SynopsisExpands on a set of lectures held at the Courant Institute on Riemann-Hilbert problems, orthogonal polynomials, and random matrix theory. The goal of the course was to prove universality for a variety of statistical quantities arising in the theory of random matrix models.Table of ContentsRiemann-Hilbert problems Jacobi operators Orthogonal polynomials Continued fractions Random matrix theory Equilibrium measures Asymptotics for orthogonal polynomials Universality Bibliography.

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Book SynopsisSuitable for second-year graduate students, this title is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. It explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. It also deals with vector fields and differential forms.Table of ContentsBackground material Differentiable manifolds Tangent space Integration of vector fields and differential forms Linear connections Riemannian manifolds The Yamabe problem-An introduction to research Bibliography Subject index Notation.

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Book SynopsisEmphasizing integral formulas, the geometric theory of pseudoconvexity, estimates, partial differential equations, approximation theory, inner functions, invariant metrics, and mapping theory, this title is intended for the student with a background in real and complex variable theory, harmonic analysis, and differential equations.Table of ContentsAn introduction to the subject Some integral formulas Subharmonicity and its applications Convexity Hormander's solution of the $\bar\partial$ equation Solution of the Levi problem and other applications of $\bar\partial$ techniques Cousin problems, cohomology, and sheaves The zero set of a holomorphic function Some harmonic analysis Constructive methods Integral formulas for solutions to the $\bar\partial$ problem and norm estimates Holomorphic mappings and invariant metrics Manifolds Area measures Exterior algebra Vectors, covectors, and differential forms List of notation Bibliography Index.

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Book SynopsisSuitable as a text for both Riemannian geometry and for the analysis and geometry of symmetric spaces, this title features chapters on differential geometry and Lie groups. It also includes a chapter on functions on symmetric spaces, that gives an introduction to the study of spherical functions, and the theory of invariant differential operators.Table of ContentsElementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces On the classification of symmetric spaces Functions on symmetric spaces Bibliography List of notational conventions Symbols frequently used Author index Subject index Reviews for the first edition.

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Book SynopsisThere have been many wonderful developments in the theory of minimal surfaces and geometric measure theory. This book covers variational geometry. It focuses on Plateau's Problem, which is concerned with surfaces that model the behavior of soap films.Table of ContentsThe phenomena of least area problems Integration of differential forms over rectifiable sets Varifolds Variational problems involving varifolds References Additional references Index.

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Book SynopsisPresents an introduction to the basic ideas of the theory of large deviations and makes a suitable package on which to base a semester-length course for advanced graduate students with a background in analysis and some probability theory. This book also covers various non-uniform results.Table of ContentsSome examples Some generalities General Cramer theory Uniform large deviations Non-uniform results Analytic considerations Historical notes and references Name index Bibliography Frequently used notation Index.

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Book SynopsisEmphasizes on functions rather than on geometric or physical applications.Trade ReviewAnd what a book it is! The marks of Landau's thoroughness and elegance, and his undoubted authority, impress themselves on the reader at every turn, from the opening of the preface ... to the closing of the final chapter. It is a book that all analysts ... should possess ... to see how a master of his craft like landau presented the calculus when he was at the height of his power and reputation Mathematical GazetteTable of ContentsPart One. Differential Calculus: Limits as $n=\infty$ Logarithms, powers, and roots Functions and continuity Limits as $x=\xi$ Definition of the derivative General theorems on the formation of the derivative Increase, decrease, maximum, minimum General properties of continuous functions on closed intervals Rolle's theorem and the theorem of the mean Derivatives of higher order; Taylor's theorem "0/0" and similar matters Infinite series Uniform convergence Power series Exponential series and binomial series The trigonometric functions Functions of two variables and partial derivatives Inverse functions and implicit functions The inverse trigonometric functions Some necessary algebraic theorems Part Two. Integral Calculus: Definition of the integral Basic formulas of the integral calculus The integration of rational functions The integration of certain non-rational functions Concept of the definite integral Theorems on the definite integral The integration of infinite series The improper integral The integral with infinite limits The gamma function Fourier series Index of definitions Subject index.

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Book SynopsisAlgebraic curves have many special properties that make their study rewarding. This work presents the various aspects of curves and emphasizes connections with algebra. It covers topics in the geometry of algebraic curves, such as line and vector bundles, the Riemann-Roch Theorem, divisors, coherent sheaves, and zeroth and first cohomology groups.Table of ContentsAlgebraic Preliminaries From algebra to geometry Geometry of dimension one Divisors and line bundles Vector bundles, coherent sheaves, and cohomology Vector bundles on $\mathbb{P}^{1}$ General theory of curves Elliptic curves The Riemann-Roch theorem Curves over arithmetic fields Bibliography Index.

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Book SynopsisFocuses on normed vector spaces and their important examples, Banach spaces and Hilbert spaces. This work covers topics including perturbation classes, measures of noncompactness, strictly singular operators and operator constants.Table of ContentsBasic notions Duality Linear operators The Riesz theory for compact operators Fredholm operators Spectral theory Unbounded operators Reflexive Banach spaces Banach algebras Semigroups Hilbert space Bilinear forms Selfadjoint operators Measures of operators Examples and applications Glossary Major Theorems Bibliography Index.

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Book SynopsisCovers the key topics of numerical methods. This work covers topics including interpolation, the fast Fourier transform, iterative methods for solving systems of linear and nonlinear equations, numerical methods for solving ODEs, numerical methods for matrix eigenvalue problems, approximation theory and computer arithmetic.Table of ContentsInterpolation by polynomials Spline functions The discrete Fourier transform and its applications Solution of linear systems of equations Nonlinear systems of equations The numerical integration of functions Explicit one-step methods for initial value problems in ordinary differential equations Multistep methods for initial value problems of ordinary differential equations Boundary value problems for ordinary differential equations Jacobi, Gauss-Seidel and relaxation methods for the solution of linear systems of equations The conjugate gradient and GMRES methods Eigenvalue problems Numerical methods for eigenvalue problems Peano's error representation Approximation theory Computer arithmetic Bibliography Index.

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Book SynopsisPlaces Lebesgue's early work on integration theory within in proper historical context by relating it to the developments during the nineteenth century that motivated it and gave it significance and also to the contributions made in this field by Lebesgue's contemporaries.Trade Review... the epitome of what a mathematical history should be. ScienceTable of ContentsRiemann's theory of integration The development of riemann's ideas: 1870-80 Set theory and the theory of integration The end of the century: A period of transition The creation of modern integration theory Pioneering applications of the Lebesgue integral Epilogue: The Lebesgue-Stieltjes integral Appendix: Dini's theorem on the differentiability of continuous functions Glossary Special symbols List of abbreviations Bibliography Index.

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Book SynopsisConvexity is a simple idea that manifests itself in a surprising variety of places. This work demonstrates that simplicity, intuitive appeal, and the universality of applications make teaching (and learning) convexity a gratifying experience. It demonstrates the power of a few simple unifying principles in a variety of pure and applied problems.Table of ContentsConvex sets at large Faces and extreme points Convex sets in topological vector spaces Polarity, duality and linear programming Convex bodies and ellipsoids Faces of polytopes Lattices and convex bodies Lattice points and polyhedra Bibliography Index.

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Book SynopsisContains papers written by participants of the special session on value distribution theory and complex dynamics held in Hong Kong at the First Joint International Meeting of the AMS and the Hong Kong Mathematical Society in December 2000. This book discusses non-Archimedean analogs of Picard's theorems.Table of ContentsOn proper analytic maps with one critical point by W. Bergweiler Non-Archimedean analytic maps to algebraic curves by W. Cherry and J. T.-Y. Wang Some progress in non-Archimedean analysis by P.-C. Hu and C.-C. Yang On period doubling phenomena and Sharkovskii type ordering for the family $\lambda\tan(z)$ by L. Keen and J. Kotus The parabolic implosion: Lavaurs maps and strong convergence for rational maps by R. Oudkerk Synthetic deformation space of an entire function by M. Taniguchi.

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Book SynopsisCovers the classical theory of abstract Riemann surfaces. This book presents the requisite function theory and topology for Riemann surfaces. It also covers differentials and uniformization. For compact Riemann surfaces, it features topics such as divisors, Weierstrass points, and the Riemann-Roch theorem.Table of ContentsIntroduction:; 1-1 Algebraic functions and Riemann surfaces; 1-2 Plane fluid flows; 1-3 Fluid flows on surfaces; 1-4 Regular potentials; 1-5 Meromorphic functions; 1-6 Function theory on a torus General Topology:; 2-1 Topological spaces; 2-2 Functions and mappings; 2-3 Manifolds Riemann Surface of an Analytic Function:; 3-1 The complete analytic function; 3-2 The analytic configuration Covering Manifolds:; 4-1 Covering manifolds; 4-2 Monodromy theorem; 4-3 Fundamental group; 4-4 Covering transformations Combinatorial Topology:; 5-1 Triangulation; 5-2 Barycentric coordinates and subdivision; 5-3 Orientability; 5-4 Differentiable and analytic curves; 5-5 Normal forms of compact orientable surfaces; 5-6 Homology groups and Betti numbers; 5-7 Invariance of the homology groups; 5-8 Fundamental group and first homology group; 5-9 Homology on compact surfaces Differentials and Integrals:; 6-1 Second-order differentials and surface integrals; 6-2 First-order differentials and line integrals; 6-3 Stokes' theorem; 6-4 The exterior differential calculus; 6-5 Harmonic and analytic differentials The Hilbert Space of Differentials:; 7-1 Definition and properties of Hilbert space; 7-2 Smoothing operators; 7-3 Weyl's lemma and orthogonal projections Existence of Harmonic and Analytic Differentials:; 8-1 Existence theorems; 8-2 Countability of a Riemann surface Uniformization:; 9-1 Schlichtartig surfaces; 9-2 Universal covering surfaces; 9-3 Triangulation of a Riemann surface; 9-4 Mappings of a Riemann surface onto itself Compact Riemann Surfaces:; 10-1 Regular harmonic differentials; 10-2 The bilinear relations of Riemann; 10-3 Bilinear relations for differentials with singularities; 10-4 Divisors; 10-5 The Riemann-Roch theorem; 10-6 Weierstrass points; 10-7 Abel's theorem; 10-8 Jacobi inversion problem; 10-9 The field of algebraic functions; 10-10 The hyperelliptic case References Index.

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