In this book, Dr. Smithies analyzes the process through which Cauchy created the basic structure of complex analysis, describing first the eighteenth century background before proceeding to examine the stages of Cauchy's own work, culminating in the proof of the residue theorem and his work on expansions in power series. Smithies describes how Cauchy overcame difficulties including false starts and contradictions brought about by over-ambitious assumptions, as well as the improvements that came about as the subject developed in Cauchy's hands. Controversies associated with the birth of complex function theory are described in detail. Throughout, new light is thrown on Cauchy's thinking during this watershed period. This book is the first to make use of the whole spectrum of available original sources and will be recognized as the authoritative work on the creation of complex function theory.

Detailing the main methods in the theory of involutive systems of complex vector fields this book examines the major results from the last twenty five years in the subject. One of the key tools of the subject - the Baouendi-Treves approximation theorem - is proved for many function spaces. This in turn is applied to questions in partial differential equations and several complex variables. Many basic problems such as regularity, unique continuation and boundary behaviour of the solutions are explored. The local solvability of systems of partial differential equations is studied in some detail. The book provides a solid background for others new to the field and also contains a treatment of many recent results which will be of interest to researchers in the subject.

Highly topical and original monograph, introducing the author's work on the Riemann zeta function and its adelic interpretation of interest to a wide range of mathematicians and physicists.

native settlement, in 1950 he graduated - as an extramural studen- from Groznyi Teachers College and in 1957 from Rostov University. He taught mathematics in Novocherkask Polytechnic Institute and its branch in the town of Shachty. That was when his mathematical talent blossomed and he obtained the main results given in the present monograph. In 1969 N. V. Govorov received the degree of Doctor of Mathematics and the aca demic rank of a Professor. From 1970 until his tragic death on 24 April 1981, N. V. Govorov worked as Head of the Department of Mathematical Anal ysis of Kuban' University actively engaged in preparing new courses and teaching young mathematicians. His original mathematical talent, vivid reactions, kindness bordering on self-sacrifice made him highly respected by everybody who knew him. In preparing this book for publication I was given substantial assistance by E. D. Fainberg and A. I. Heifiz, while V. M. Govorova took a significant part of the technical work with the manuscript. Professor C. Prather con tributed substantial assistance in preparing the English translation of the book. I. V. Ostrovskii. PREFACE The classic statement of the Riemann boundary problem consists in finding a function (z) which is analytic and bounded in two domains D+ and D-, with a common boundary - a smooth closed contour L admitting a continuous extension onto L both from D+ and D- and satisfying on L the boundary condition +(t) = G(t)-(t) + g(t).

Based on a series of graduate lectures given by Vladimir Markovic at the University of Warwick in spring 2003, this book is accessible to those with a grounding in complex analysis looking for an introduction to the theory of quasiconformal maps and Teichm ller theory. Assuming some familiarity with Riemann surfaces and hyperbolic geometry, topics covered include the Gr tzch argument, analytical properties of quasiconformal maps, the Beltrami differential equation, holomorphic motions and Teichm ller spaces. Where proofs are omitted, references to where they may be found are always given, and the text is clearly illustrated throughout with diagrams, examples, and exercises for the reader.

The origins of Schur analysis lie in a 1917 article by Issai Schur in which he constructed a numerical sequence to correspond to a holomorphic contractive function on the unit disk. These sequences are now known as Schur parameter sequences. Schur analysis has grown significantly since its beginnings in the early twentieth century and now encompasses a wide variety of problems related to several classes of holomorphic functions and their matricial generalizations. These problems include interpolation and moment problems as well as Schur parametrization of particular classes of contractive or nonnegative Hermitian block matrices. This book is primarily devoted to topics related to matrix versions of classical interpolation and moment problems. The major themes include Schur analysis of nonnegative Hermitian block Hankel matrices and the construction of Schur-type algorithms. This book also covers a number of recent developments in orthogonal rational matrix functions, matrix-valued Caratheodory functions and maximal weight solutions for particular matricial moment problems on the unit circle.

A Kleinian group is a discrete subgroup of the isometry group of hyperbolic 3-space, which is also regarded as a subgroup of Moebius transformations in the complex plane. The present book is a comprehensive guide to theories of Kleinian groups from the viewpoints of hyperbolic geometry and complex analysis. After 1960, Ahlfors and Bers were the leading researchers of Kleinian groups and helped it to become an active area of complex analysis as a branch of Teichmuller theory. Later, Thurston brought a revolution to this area with his profound investigation of hyperbolic manifolds, and at the same time complex dynamical approach was strongly developed by Sullivan. This book provides fundamental results and important theorems which are needed for access to the frontiers of the theory from a modern viewpoint.

This book investigates the convergence and summability of both one-dimensional and multi-dimensional Fourier transforms, as well as the theory of Hardy spaces. To do so, it studies a general summability method known as theta-summation, which encompasses all the well-known summability methods, such as the Fejer, Riesz, Weierstrass, Abel, Picard, Bessel and Rogosinski summations. Following on the classic books by Bary (1964) and Zygmund (1968), this is the first book that considers strong summability introduced by current methodology. A further unique aspect is that the Lebesgue points are also studied in the theory of multi-dimensional summability. In addition to classical results, results from the past 20-30 years - normally only found in scattered research papers - are also gathered and discussed, offering readers a convenient "one-stop" source to support their work. As such, the book will be useful for researchers, graduate and postgraduate students alike.

This book gathers peer-reviewed contributions representing modern trends in the theory of generalized functions and pseudo-differential operators. It is dedicated to Professor Michael Oberguggenberger (Innsbruck University, Austria) in honour of his 60th birthday. The topics covered were suggested by the ISAAC Group in Generalized Functions (GF) and the ISAAC Group in Pseudo-Differential Operators (IGPDO), which met at the 9th ISAAC congress in Krakow, Poland in August 2013. Topics include Columbeau algebras, ultra-distributions, partial differential equations, micro-local analysis, harmonic analysis, global analysis, geometry, quantization, mathematical physics, and time-frequency analysis. Featuring both essays and research articles, the book will be of great interest to graduate students and researchers working in analysis, PDE and mathematical physics, while also offering a valuable complement to the volumes on this topic previously published in the OT series.

Realism and Complexity in Social Science is an argument for a new approach to investigating the social world, that of complex realism. Complex realism brings together a number of strands of thought, in scientific realism, complexity science, probability theory and social research methodology. It proposes that the reality of the social world is that it is probabilistic, yet there exists enough invariance to make the discovery and explanation of social objects and causal mechanisms possible. This forms the basis for the development of a complex realist foundation for social research, that utilises a number of new and novel approaches to investigation, alongside the more traditional corpus of quantitative and qualitative methods. Research examples are drawn from research in sociology, epidemiology, criminology, social policy and human geography. The book assumes no prior knowledge of realism, probability or complexity and in the early chapters, the reader is introduced to these concepts and the arguments against them. Although the book is grounded in philosophical reasoning, this is in a direct and accessible style that will appeal both to social researchers with a methodological interest and philosophers with an interest in social investigation.

This is a collection of surveys on important mathematical ideas, their origin, their evolution and their impact in current research. The authors are mathematicians who are leading experts in their fields. The book is addressed to all mathematicians, from undergraduate students to senior researchers, regardless of the specialty.

The chapters in this volume are based on talks given at the inaugural Aspects of Time-Frequency Analysis conference held in Turin, Italy from July 5-7, 2017, which brought together experts in harmonic analysis and its applications. New connections between different but related areas were presented in the context of time-frequency analysis, encouraging future research and collaborations. Some of the topics covered include: Abstract harmonic analysis, Numerical harmonic analysis, Sampling theory, Compressed sensing, Mathematical signal processing, Pseudodifferential operators, and Applications of harmonic analysis to quantum mechanics. Landscapes of Time-Frequency Analysis will be of particular interest to researchers and advanced students working in time-frequency analysis and other related areas of harmonic analysis.

System state estimation in the presence of noise is critical for control systems, signal processing, and many other applications in a variety of fields. Developed decades ago, the Kalman filter remains an important, powerful tool for estimating the variables in a system in the presence of noise. However, when inundated with theory and vast notations, learning just how the Kalman filter works can be a daunting task.

With its mathematically rigorous, "no frills" approach to the basic discrete-time Kalman filter, A Kalman Filter Primer builds a thorough understanding of the inner workings and basic concepts of Kalman filter recursions from first principles. Instead of the typical Bayesian perspective, the author develops the topic via least-squares and classical matrix methods using the Cholesky decomposition to distill the essence of the Kalman filter and reveal the motivations behind the choice of the initializing state vector. He supplies pseudo-code algorithms for the various recursions, enabling code development to implement the filter in practice. The book thoroughly studies the development of modern smoothing algorithms and methods for determining initial states, along with a comprehensive development of the "diffuse" Kalman filter.

Using a tiered presentation that builds on simple discussions to more complex and thorough treatments, A Kalman Filter Primer is the perfect introduction to quickly and effectively using the Kalman filter in practice.

Complex Analysis and Applications, Second Edition explains complex analysis for students of applied mathematics and engineering. Restructured and completely revised, this textbook first develops the theory of complex analysis, and then examines its geometrical interpretation and application to Dirichlet and Neumann boundary value problems.

A discussion of complex analysis now forms the first three chapters of the book, with a description of conformal mapping and its application to boundary value problems for the two-dimensional Laplace equation forming the final two chapters. This new structure enables students to study theory and applications separately, as needed.

In order to maintain brevity and clarity, the text limits the application of complex analysis to two-dimensional boundary value problems related to temperature distribution, fluid flow, and electrostatics. In each case, in order to show the relevance of complex analysis, each application is preceded by mathematicalbackground that demonstrates how a real valued potential function and its related complex potential can be derived from the mathematics that describes the physical situation.

Spatial data analysis is a fast growing area and Voronoi diagrams provide a means of naturally partitioning space into subregions to facilitate spatial data manipulation, modelling of spatial structures, pattern recognition and locational optimization. With such versatility, the Voronoi diagram and its relative, the Delaunay triangulation, provide valuable tools for the analysis of spatial data. This is a rapidly growing research area and in this fully updated second edition the authors provide an up-to-date and comprehensive unification of all the previous literature on the subject of Voronoi diagrams. Features:&UL; &LI; Expands on the highly acclaimed first edition&LI; Provides an up-to-date and comprehensive survey of the existing literature on Voronoi diagrams&LI; Includes a useful compendium of applications&LI; Contains an extensive bibliography&/UL; The authors guide the reader through all the necessary mathematical background, before introducing a number of generalizations of Voronoi diagrams in Chapter 3. The subsequent chapters cover algorithms, random Voronoi diagrams, spatial interpolation, multivariate data manipulation, spatial process models, point pattern analysis and locational optimization. Emphasis of a particular perspective is deliberately avoided in order to provide a comprehensive and balanced treatment of the topic. A wide range of applications are discussed, enabling this book to serve as an important reference volume on the topic. The text will appeal to students and researchers studying spatial data in a number of areas, in particular applied probability, computational geometry and Geographic Information Science (GIS). This book will appeal equally to those whose interests in Voronoi diagrams are theoretical, practical or both.

This book aims to develop models and modeling techniques that are useful when applied to all complex systems. It adopts both analytic tools and computer simulation. The book is intended for students and researchers with a variety of backgrounds.

Geometric Function Theory is a central part of Complex Analysis (one complex variable). The Handbook of Complex Analysis - Geometric Function Theory deals with this field and its many ramifications and relations to other areas of mathematics and physics. The theory of conformal and quasiconformal mappings plays a central role in this Handbook, for example a priori-estimates for these mappings which arise from solving extremal problems, and constructive methods are considered. As a new field the theory of circle packings which goes back to P. Koebe is included. The Handbook should be useful for experts as well as for mathematicians working in other areas, as well as for physicists and engineers.
- A collection of independent survey articles in the field of GeometricFunction Theory
- Existence theorems and qualitative properties of conformal and quasiconformal mappings
- A bibliography, including many hints to applications in electrostatics, heat conduction, potential flows (in the plane)

The history of mathematics is, to a considerable extent, connected with the study of solutions of the equation f(x)=a=const for functions f(x) of one real or complex variable. Therefore, it is surprising that we know very little about solutions of u(x,y)=A=const. for functions of two real variables. These two solutions, called level of sets, are very important with regard to applications in physics, biology and economics as they make a map of appropriate processes described by the function u(x,y) for given parameters (x,y). In the present volume we study a concept, Gamma-lines which generalizes the concept of levels of sets and, at the same time, the concept of a-points. The aim of the authors is to provide a book on Gamma-lines for the broad specialist and to show the large range of their field of applications. One can expect that the general methods proposed in this volume will be of great use to both physicists and engineers.

Theories, methods and problems in approximation theory and analytic inequalities with a focus on differential and integral inequalities are analyzed in this book. Fundamental and recent developments are presented on the inequalities of Abel, Agarwal, Beckenbach, Bessel, Cauchy-Hadamard, Chebychev, Markov, Euler's constant, Grothendieck, Hilbert, Hardy, Carleman, Landau-Kolmogorov, Carlson, Bernstein-Mordell, Gronwall, Wirtinger, as well as inequalities of functions with their integrals and derivatives. Each inequality is discussed with proven results, examples and various applications. Graduate students and advanced research scientists in mathematical analysis will find this reference essential to their understanding of differential and integral inequalities. Engineers, economists, and physicists will find the highly applicable inequalities practical and useful to their research.

Different aspects of harmonic analysis, complex analysis, sampling theory, approximation theory and related topics are covered in this volume. The topics included are Fourier analysis, Pade approximation, dynamical systems and difference operators, splines, Christoffel functions, best approximation, discrepancy theory and Jackson-type theorems of approximation. The articles of this collection were originated from the International Conference in Approximation Theory, held in Savannah, GA in 2017, and organized by the editors of this volume.

This book takes an in-depth look at abelian relations of codimension one webs in the complex analytic setting. In its classical form, web geometry consists in the study of webs up to local diffeomorphisms. A significant part of the theory revolves around the concept of abelian relation, a particular kind of functional relation among the first integrals of the foliations of a web. Two main focuses of the book include how many abelian relations can a web carry and which webs are carrying the maximal possible number of abelian relations. The book offers complete proofs of both Chern's bound and Trepreau's algebraization theorem, including all the necessary prerequisites that go beyond elementary complex analysis or basic algebraic geometry. Most of the examples known up to date of non-algebraizable planar webs of maximal rank are discussed in detail. A historical account of the algebraization problem for maximal rank webs of codimension one is also presented.

Discrete Fourier Analysis.

Cauchy Integrals.

Potential Theory in the Plane.

Construction of Conformal Maps: Simply Connected Regions.

Construction of Conformal Maps for Multiply Connected Regions.

Polynomial Expansions and Conformal Maps.

Univalent Functions.

Bibliography.

Index.

This monograph provides a concise, accessible snapshot of key topics in several complex variables, including the Cauchy Integral Formula, sequences of holomorphic functions, plurisubharmonic functions, the Dirichlet problem, and meromorphic functions. Based on a course given at Universite de Montreal, this brief introduction covers areas of contemporary importance that are not mentioned in most treatments of the subject, such as modular forms, which are essential for Wiles' theorem and the unification of quantum theory and general relativity. Also covered is the Riemann manifold of a function, which generalizes the Riemann surface of a function of a single complex variable and is a topic that is well-known in one complex variable, but rarely treated in several variables. Many details, which are intentionally left out, as well as many theorems are stated as problems, providing students with carefully structured instructive exercises. Prerequisites for use of this book are functions of one complex variable, functions of several real variables, and topology, all at the undergraduate level. Lectures on Several Complex Variables will be of interest to advanced undergraduate and beginning undergraduate students, as well as mathematical researchers and professors.

This book gives a systematic presentation of real algebraic varieties. Real algebraic varieties are ubiquitous.They are the first objects encountered when learning of coordinates, then equations, but the systematic study of these objects, however elementary they may be, is formidable. This book is intended for two kinds of audiences: it accompanies the reader, familiar with algebra and geometry at the masters level, in learning the basics of this rich theory, as much as it brings to the most advanced reader many fundamental results often missing from the available literature, the "folklore". In particular, the introduction of topological methods of the theory to non-specialists is one of the original features of the book. The first three chapters introduce the basis and classical methods of real and complex algebraic geometry. The last three chapters each focus on one more specific aspect of real algebraic varieties. A panorama of classical knowledge is presented, as well as major developments of the last twenty years in the topology and geometry of varieties of dimension two and three, without forgetting curves, the central subject of Hilbert's famous sixteenth problem. Various levels of exercises are given, and the solutions of many of them are provided at the end of each chapter.