The problems of conditional optimization of the uniform (or C-) norm for polynomials and rational functions arise in various branches of science and technology. Their numerical solution is notoriously difficult in case of high degree functions. The book develops the classical Chebyshev's approach which gives analytical representation for the solution in terms of Riemann surfaces. The techniques born in the remote (at the first glance) branches of mathematics such as complex analysis, Riemann surfaces and Teichmuller theory, foliations, braids, topology are applied to approximation problems. The key feature of this book is the usage of beautiful ideas of contemporary mathematics for the solution of applied problems and their effective numerical realization. This is one of the few books where the computational aspects of the higher genus Riemann surfaces are illuminated. Effective work with the moduli spaces of algebraic curves provides wide opportunities for numerical experiments in mathematics and theoretical physics.

Since the publication of our first book [80], there has been a real resiu-gence of interest in the study of almost automorphic functions and their applications ([16, 17, 28, 29, 30, 31, 32, 40, 41, 42, 46, 51, 58, 74, 75, 77, 78, 79]). New methods (method of invariant s- spaces, uniform spectrum), and new concepts (almost periodicity and almost automorphy in fuzzy settings) have been introduced in the literature. The range of applications include at present linear and nonlinear evolution equations, integro-differential and functional-differential equations, dynamical systems, etc...It has become imperative to take a bearing of the main steps of the the ory. That is the main purpose of this monograph. It is intended to inform the reader and pave the road to more research in the field. It is not a self contained book. In fact, [80] remains the basic reference and fimdamental source of information on these topics. Chapter 1 is an introductory one. However, it contains also some recent contributions to the theory of almost automorphic functions in abstract spaces. VIII Preface Chapter 2 is devoted to the existence of almost automorphic solutions to some Unear and nonUnear evolution equations. It con tains many new results. Chapter 3 introduces to almost periodicity in fuzzy settings with applications to differential equations in fuzzy settings. It is based on a work by B. Bede and S. G. Gal [40].

This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable, the level being gauged for graduate students. It treats several topics in geometric function theory as well as potential theory in the plane, covering in particular: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. A knowledge of integration theory and functional analysis is assumed.

Featuring the work of twenty-three internationally-recognized experts, this volume explores the trace formula, spectra of locally symmetric spaces, p-adic families, and other recent techniques from harmonic analysis and representation theory. Each peer-reviewed submission in this volume, based on the Simons Foundation symposium on families of automorphic forms and the trace formula held in Puerto Rico in January-February 2014, is the product of intensive research collaboration by the participants over the course of the seven-day workshop. The goal of each session in the symposium was to bring together researchers with diverse specialties in order to identify key difficulties as well as fruitful approaches being explored in the field. The respective themes were counting cohomological forms, p-adic trace formulas, Hecke fields, slopes of modular forms, and orbital integrals.

Analytic Extension is a mysteriously beautiful property of analytic functions. With this point of view in mind the related survey papers were gathered from various fields in analysis such as integral transforms, reproducing kernels, operator inequalities, Cauchy transform, partial differential equations, inverse problems, Riemann surfaces, Euler-Maclaurin summation formulas, several complex variables, scattering theory, sampling theory, and analytic number theory, to name a few. Audience: Researchers and graduate students in complex analysis, partial differential equations, analytic number theory, operator theory and inverse problems.

We study in Part I of this monograph the computational aspect of almost all moduli of continuity over wide classes of functions exploiting some of their convexity properties. To our knowledge it is the first time the entire calculus of moduli of smoothness has been included in a book. We then present numerous applications of Approximation Theory, giving exact val ues of errors in explicit forms. The K-functional method is systematically avoided since it produces nonexplicit constants. All other related books so far have allocated very little space to the computational aspect of moduli of smoothness. In Part II, we study/examine the Global Smoothness Preservation Prop erty (GSPP) for almost all known linear approximation operators of ap proximation theory including: trigonometric operators and algebraic in terpolation operators of Lagrange, Hermite-Fejer and Shepard type, also operators of stochastic type, convolution type, wavelet type integral opera tors and singular integral operators, etc. We present also a sufficient general theory for GSPP to hold true. We provide a great variety of applications of GSPP to Approximation Theory and many other fields of mathemat ics such as Functional analysis, and outside of mathematics, fields such as computer-aided geometric design (CAGD). Most of the time GSPP meth ods are optimal. Various moduli of smoothness are intensively involved in Part II. Therefore, methods from Part I can be used to calculate exactly the error of global smoothness preservation. It is the first time in the literature that a book has studied GSPP."

The subject matter of this book formed the substance of a mathematical se am which was worked by many of the great mathematicians of the last century. The mining metaphor is here very appropriate, for the analytical tools perfected by Cauchy permitted the mathematical argument to penetra te to unprecedented depths over a restricted region of its domain and enabled mathematicians like Abel, Jacobi, and Weierstrass to uncover a treasurehouse of results whose variety, aesthetic appeal, and capacity for arousing our astonishment have not since been equaled by research in any other area. But the circumstance that this theory can be applied to solve problems arising in many departments of science and engineering graces the topic with an additional aura and provides a powerful argument for including it in university courses for students who are expected to use mathematics as a tool for technological investigations in later life. Unfortunately, since the status of university staff is almost wholly determined by their effectiveness as research workers rather than as teachers, the content of undergraduate courses tends to reflect those academic research topics which are currently popular and bears little relationship to the future needs of students who are themselves not destined to become university teachers. Thus, having been comprehensively explored in the last century and being undoubtedly difficult .

The first survey of its kind, written by internationally known, outstanding experts who developed substantial parts of the field. The book contains an introduction written by Remmert, describing the history of the subject, and is very useful to graduate students and researchers in complex analysis, algebraic geometry and differential geometry.

The book presents an introduction to the geometry of Hilbert spaces and operator theory, targeting graduate and senior undergraduate students of mathematics. Major topics discussed in the book are inner product spaces, linear operators, spectral theory and special classes of operators, and Banach spaces. On vector spaces, the structure of inner product is imposed. After discussing geometry of Hilbert spaces, its applications to diverse branches of mathematics have been studied. Along the way are introduced orthogonal polynomials and their use in Fourier series and approximations. Spectrum of an operator is the key to the understanding of the operator. Properties of the spectrum of different classes of operators, such as normal operators, self-adjoint operators, unitaries, isometries and compact operators have been discussed. A large number of examples of operators, along with their spectrum and its splitting into point spectrum, continuous spectrum, residual spectrum, approximate point spectrum and compression spectrum, have been worked out. Spectral theorems for self-adjoint operators, and normal operators, follow the spectral theorem for compact normal operators. The book also discusses invariant subspaces with special attention to the Volterra operator and unbounded operators. In order to make the text as accessible as possible, motivation for the topics is introduced and a greater amount of explanation than is usually found in standard texts on the subject is provided. The abstract theory in the book is supplemented with concrete examples. It is expected that these features will help the reader get a good grasp of the topics discussed. Hints and solutions to all the problems are collected at the end of the book. Additional features are introduced in the book when it becomes imperative. This spirit is kept alive throughout the book.

A development of some of the principal applications of function theory in several complex variables to Banach algebras. The authors do not presuppose any knowledge of several complex variables on the part of the reader, and all relevant material is developed within the text. Furthermore, the book deals with problems of uniform approximation on compact subsets of the space of n complex variables. This third edition contains new material on maximum modulus algebras and subharmonicity, the hull of a smooth curve, integral kernels, perturbations of the Stone-Weierstrass Theorem, boundaries of analytic varieties, polynomial hulls of sets over the circle, areas, and the topology of hulls. The authors have also included a new chapter commenting on history and recent developments, as well as an updated and expanded reading list.

This volume is a collection of up-to-date research and expository papers on different aspects of complex analysis, including relations to operator theory and hypercomplex analysis. The articles cover many important and essential subjects, such as the SchrAdinger equation, subelliptic operators, Lie algebras and superalgebras, Toeplitz and Hankel operators, reproducing kernels and Qp spaces, among others. Most of the papers were presented at the International Symposium on Complex Analysis and Related Topics held in Cuernavaca (Morelos), Mexico, in November 1996, which was attended by approximately 50 experts in the field. The book can be used as a reference work on recent research in the subjects covered. It is one of the few books stressing the relation between operator theory and complex and hypercomplex analyses. The book is addressed to researchers and postgraduate students in the fields named here and in related ones.

The monograph is devoted to the systematic presentation of the so called dressing method for solving differential equations (both linear and nonlinear) of mathematical physics. The essence of the dressing method consists in a generation of new non-trivial solutions of a given equation from (maybe trivial) solution of the same or related equation. The Moutard and Darboux transformations discovered in XIX century as applied to linear equations, the Backlund transformation in differential geometry of surfaces, the factorization method, the Riemann-Hilbert problem in the form proposed by Shabat and Zakharov for soliton equations and its extension in terms of the d-bar formalism comprise the main objects of the book. Throughout the text, a generally sufficient linear experience of readers is exploited, with a special attention to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions.

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

Althoughsubmanifoldscomplexmanifoldshasbeenanactive?eldofstudyfor many years, in some sense this area is not su?ciently covered in the current literature. This text deals with the CR submanifolds of complex manifolds, with particular emphasis on CR submanifolds of complex projective space, and it covers the topics which are necessary for learning the basic properties of these manifolds. We are aware that it is impossible to give a complete overview of these submanifolds, but we hope that these notes can serve as an introduction to their study. We present the fundamental de?nitions and results necessary for reaching the frontiers of research in this ?eld. There are many monographs dealing with some current interesting topics in di?erential geometry, but most of these are written as encyclopedias, or research monographs, gathering recent results and giving the readers ample usefulinformationaboutthetopics. Therefore, thesekindsofmonographsare attractive to specialists in di?erential geometry and related ?elds and acce- able to professional di?erential geometers. However, for graduate students who are less advanced in di?erential geometry, these texts might be hard to read without assistance from their instructors. By contrast, the general philosophy of this book is to begin with the elementary facts about complex manifolds and their submanifolds, give some details and proofs, and introduce the reader to the study of CR submanifolds of complex manifolds; especially complex projective space. It includes only a few original results with precise proofs, while the others are cited in the reference list.

This book features original research and survey articles on the topics of function spaces and inequalities. It focuses on (variable/grand/small) Lebesgue spaces, Orlicz spaces, Lorentz spaces, and Morrey spaces and deals with mapping properties of operators, (weighted) inequalities, pointwise multipliers and interpolation. Moreover, it considers Sobolev-Besov and Triebel-Lizorkin type smoothness spaces. The book includes papers by leading international researchers, presented at the International Conference on Function Spaces and Inequalities, held at the South Asian University, New Delhi, India, on 11-15 December 2015, which focused on recent developments in the theory of spaces with variable exponents. It also offers further investigations concerning Sobolev-type embeddings, discrete inequalities and harmonic analysis. Each chapter is dedicated to a specific topic and written by leading experts, providing an overview of the subject and stimulating future research.

This completely revised and updated edition of the one variable part of the author's classic older book "Iteration Theory of Holomorphic Maps on Taut Manifolds" presents the theory of holomorphic dynamical systems on hyperbolic Riemann surfaces from the very beginning of the subject up to the most recent developments. It is intended both as a reference book for the experts and as an accessible gateway to this beautiful theory for Master and Ph.D. students. It also contains extensive historical notes and references for further readings.

In its traditional form, Clifford analysis provides the function theory for solutions of the Dirac equation. From the beginning, however, the theory was used and applied to problems in other fields of mathematics, numerical analysis, and mathematical physics. recently, the theory has enlarged its scope considerably by incorporating geometrical methods from global analysis on manifolds and methods from representation theory. New, interesting branches of the theory are based on conformally invariant, first-order systems other than the Dirac equation, or systems that are invariant with respect to a group other than the conformal group. This book represents an up-to-date review of Clifford analysis in its present form, its applications, and directions for future research. Readership: Mathematicians and theoretical physicists interested in Clifford analysis itself, or in its applications to other fields.

This two-volume work introduces the theory and applications of Schur-convex functions. The first volume introduces concepts and properties of Schur-convex functions, including Schur-geometrically convex functions, Schur-harmonically convex functions, Schur-power convex functions, etc. and also discusses applications of Schur-convex functions in symmetric function inequalities.

In the spectrum of mathematics, graph theory which studies a mathe matical structure on a set of elements with a binary relation, as a recognized discipline, is a relative newcomer. In recent three decades the exciting and rapidly growing area of the subject abounds with new mathematical devel opments and significant applications to real-world problems. More and more colleges and universities have made it a required course for the senior or the beginning postgraduate students who are majoring in mathematics, computer science, electronics, scientific management and others. This book provides an introduction to graph theory for these students. The richness of theory and the wideness of applications make it impossi ble to include all topics in graph theory in a textbook for one semester. All materials presented in this book, however, I believe, are the most classical, fundamental, interesting and important. The method we deal with the mate rials is to particularly lay stress on digraphs, regarding undirected graphs as their special cases. My own experience from teaching out of the subject more than ten years at University of Science and Technology of China (USTC) shows that this treatment makes hardly the course di: fficult, but much more accords with the essence and the development trend of the subject."

Starting with the fundamentals of Q spaces and their relationships to Besov spaces, this book presents all major results around Q spaces obtained in the past 16 years. The applications of Q spaces in the study of the incompressible Navier-Stokes system and its stationary form are also discussed. This self-contained book can be used as an essential reference for researchers and graduates in analysis and partial differential equations.

There has been a flurry of activity in recent years in the loosely defined area of holomorphic spaces. This book discusses the most well-known and widely used spaces of holomorphic functions in the unit ball of Cn. Spaces discussed include the Bergman spaces, the Hardy spaces, the Bloch space, BMOA, the Dirichlet space, the Besov spaces, and the Lipschitz spaces. Most proofs in the book are new and simpler than the existing ones in the literature. The central idea in almost all these proofs is based on integral representations of holomorphic functions and elementary properties of the Bergman kernel, the Bergman metric, and the automorphism group.

The unit ball was chosen as the setting since most results can be achieved there using straightforward formulas without much fuss. The book can be read comfortably by anyone familiar with single variable complex analysis; no prerequisite on several complex variables is required. The author has included exercises at the end of each chapter that vary greatly in the level of difficulty.

We study the boundary behaviour of a conformal map of the unit disk onto an arbitrary simply connected plane domain. A principal aim of the theory is to obtain a one-to-one correspondence between analytic properties of the function and geometrie properties of the domain. In the classical applications of conformal mapping, the domain is bounded by a piecewise smooth curve. In many recent applications however, the domain has a very bad boundary. It may have nowhere a tangent as is the case for Julia sets. Then the conformal map has many unexpected properties, for instance almost all the boundary is mapped onto almost nothing and vice versa. The book is meant for two groups of users. (1) Graduate students and others who, at various levels, want to learn about conformal mapping. Most sections contain exercises to test the understand ing. They tend to be fairly simple and only a few contain new material. Pre requisites are general real and complex analyis including the basic facts about conformal mapping (e.g. AhI66a). (2) Non-experts who want to get an idea of a particular aspect of confor mal mapping in order to find something useful for their work. Most chapters therefore begin with an overview that states some key results avoiding tech nicalities. The book is not meant as an exhaustive survey of conformal mapping. Several important aspects had to be omitted, e.g. numerical methods (see e.g."

This volume is dedicated to Tsuyoshi Ando, a- foremost expert in operator theory, matrix theory, complex analysis, and their applications, on the occasion of his 60th birthday. The book opens with his biography and list of publications. It contains a selection of papers covering a broad spectrum of topics ranging from abstract operator theory to various concrete problems and applications. The majority of the papers deal with topics in modern operator theory and its applications. This volume also contains papers on interpolation and completion problems, factorisation problems and problems connected with complex analysis. The book will appeal to a wide audience of pure and applied mathematicians.

An intensive development of the theory of generalized analytic functions started when methods of Complex Analysis were combined with methods of Functional Analysis, especially with the concept of distributional solutions to partial differential equations. The power of these interactions is far from being exhausted. In order to promote the further development of the theory of generalized analytic functions and applications of partial differential equations to Mechanics, the Technical University of Graz organized a conference whose Proceedings are contained in the present volume. The contributions on generalized analytic functions (Part One) deal not only with problems in the complex plane (boundary value and initial value problems), but also related problems in higher dimensions are investigated where both several complex variables and the technique of Clifford Analysis are used. Part Two of the Proceedings is devoted to applications to Mechanics. It contains contributions to a variety of general methods such as L p-methods, boundary elements and asymptotic methods, and hemivariational inequalities. A substantial number of the papers of Part Two, however, deals with problems in Ocean Acoustics. The papers of both parts of the Proceedings can be recommended to mathematicians, physicists, and engineers working in the fields mentioned above, as well as for further reading within graduate studies.

This EMS volume gives an overview of the modern theory of elliptic boundary value problems, with contributions focusing on differential elliptic boundary problems and their spectral properties, elliptic pseudodifferential operators, and general differential elliptic boundary value problems in domains with singularities.