In recent years the development of new classification and regression algorithms based on deep learning has led to a revolution in the fields of artificial intelligence, machine learning, and data analysis. The development of a theoretical foundation to guarantee the success of these algorithms constitutes one of the most active and exciting research topics in applied mathematics. This book presents the current mathematical understanding of deep learning methods from the point of view of the leading experts in the field. It serves both as a starting point for researchers and graduate students in computer science, mathematics, and statistics trying to get into the field and as an invaluable reference for future research.

This IMA Volume in Mathematics and its Applications PATTERN FORMATION IN CONTINUOUS AND COUPLED SYSTEMS is based on the proceedings of a workshop with the same title, but goes be yond the proceedings by presenting a series of mini-review articles that sur vey, and provide an introduction to, interesting problems in the field. The workshop was an integral part of the 1997-98 IMA program on "EMERG ING APPLICATIONS OF DYNAMICAL SYSTEMS." I would like to thank Martin Golubitsky, University of Houston (Math ematics) Dan Luss, University of Houston (Chemical Engineering), and Steven H. Strogatz, Cornell University (Theoretical and Applied Mechan ics) for their excellent work as organizers of the meeting and for editing the proceedings. I also take this opportunity to thank the National Science Foundation (NSF), and the Army Research Office (ARO), whose financial support made the workshop possible. Willard Miller, Jr., Professor and Director v PREFACE Pattern formation has been studied intensively for most of this cen tury by both experimentalists and theoreticians, and there have been many workshops and conferences devoted to the subject. In the IMA workshop on Pattern Formation in Continuous and Coupled Systems held May 11-15, 1998 we attempted to focus on new directions in the patterns literature."

Most topics dealt with here deal with complex analysis of both one and several complex variables. Several contributions come from elasticity theory. Areas covered include the theory of p-adic analysis, mappings of bounded mean oscillations, quasiconformal mappings of Klein surfaces, complex dynamics of inverse functions of rational or transcendental entire functions, the nonlinear Riemann-Hilbert problem for analytic functions with nonsmooth target manifolds, the Carleman-Bers-Vekua system, the logarithmic derivative of meromorphic functions, G-lines, computing the number of points in an arbitrary finite semi-algebraic subset, linear differential operators, explicit solution of first and second order systems in bounded domains degenerating at the boundary, the Cauchy-Pompeiu representation in L2 space, strongly singular operators of Calderon-Zygmund type, quadrature solutions to initial and boundary-value problems, the Dirichlet problem, operator theory, tomography, elastic displacements and stresses, quantum chaos, and periodic wavelets.

The third edition of this well known text continues to provide a solid foundation in mathematical analysis for undergraduate and first-year graduate students. The text begins with a discussion of the real number system as a complete ordered field. (Dedekind's construction is now treated in an appendix to Chapter I.) The topological background needed for the development of convergence, continuity, differentiation and integration is provided in Chapter 2. There is a new section on the gamma function, and many new and interesting exercises are included. This text is part of the Walter Rudin Student Series in Advanced Mathematics.

This book provides the main results and ideas in the theories of completely bounded maps, operator spaces, and operator algebras, along with some of their main applications. It requires only a basic background in functional analysis to read through the book. The descriptions and discussions of the topics are self-explained. It is appropriate for graduate students new to the subject and the field. The book starts with the basic representation theorems for abstract operator spaces and their mappings, followed by a discussion of tensor products and the analogue of Grothendieck's approximation property. Next, the operator space analogues of the nuclear, integral, and absolutely summing mappings are discussed. In what is perhaps the deepest part of the book, the authors present the remarkable ""non-classical"" phenomena that occur when one considers local reflexivity and exactness for operator spaces. This is an area of great beauty and depth, and it represents one of the triumphs of the subject. In the final part of the book, the authors consider applications to non-commutative harmonic analysis and non-self-adjoint operator algebra theory. Operator space theory provides a synthesis of Banach space theory with the non-commuting variables of operator algebra theory, and it has led to exciting new approaches in both disciplines. This book is an indispensable introduction to the theory of operator spaces.

Almost a century ago, harmonic analysis entered a (still continuing) Golden Age, with the emergence of many great masters throughout Europe. They created a wealth of profound analytic methods, to be successfully exploited and further developed by succeeding generations. This flourishing of harmonic analysis is today as lively as ever, as the papers presented here demonstrate. In addition to its own ongoing internal development and its basic role in other areas of mathematics, physics and chemistry, financial analysis, medicine, and biological signal processing, harmonic analysis has made fundamental contributions to essentially all twentieth century technology-based human endeavours, including telephone, radio, television, radar, sonar, satellite communications, medical imaging, the Internet, and multimedia. This ubiquitous nature of the subject is amply illustrated.

The book not only promotes the infusion of new mathematical tools into applied harmonic analysis, but also to fuel the development of applied mathematics by providing opportunities for young engineers, mathematicians and other scientists to learn more about problem areas in today's technology that might benefit from new mathematical insights.

Generalized Measure Theory examines the relatively new mathematical area of generalized measure theory. The exposition unfolds systematically, beginning with preliminaries and new concepts, followed by a detailed treatment of important new results regarding various types of nonadditive measures and the associated integration theory. The latter involves several types of integrals: Sugeno integrals, Choquet integrals, pan-integrals, and lower and upper integrals. All of the topics are motivated by numerous examples, culminating in a final chapter on applications of generalized measure theory. Some key features of the book include: many exercises at the end of each chapter along with relevant historical and bibliographical notes, an extensive bibliography, and name and subject indices. The work is suitable for a classroom setting at the graduate level in courses or seminars in applied mathematics, computer science, engineering, and some areas of science. A sound background in mathematical analysis is required. Since the book contains many original results by the authors, it will also appeal to researchers working in the emerging area of generalized measure theory.

This book presents an extensive overview of logarithmic integral operators with kernels depending on one or several complex parameters. Solvability of corresponding boundary value problems and determination of characteristic numbers are analyzed by considering these operators as operator-value functions of appropriate complex (spectral) parameters. Therefore, the method serves as a useful addition to classical approaches. Special attention is given to the analysis of finite-meromorphic operator-valued functions, and explicit formulas for some inverse operators and characteristic numbers are developed, as well as the perturbation technique for the approximate solution of logarithmic integral equations. All essential properties of the generalized single- and double-layer potentials with logarithmic kernels and Green's potentials are considered. Fundamentals of the theory of infinite-matrix summation operators and operator-valued functions are presented, including applications to the solution of logarithmic integral equations. Many boundary value problems for the two-dimensional Helmholtz equation are discussed and explicit formulas for Green's function of canonical domains with separated logarithmic singularities are presented.

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

This book aims to provide a comprehensive study of the mathematical theory of the vortex method, from its origins in the 1930s, through the developments of the '70s when the use of computers made advanced research possible, to current work on this subject in China and elsewhere. The five chapters treat vortex methods for the Euler and Navier-Stokes equations; mathematical theory for incompressible flows; convergence of vortex methods for the Euler equations; convergence of viscosity splitting; and convergence of the random vortex method. Audience: This volume will be of interest to researchers and graduate students of applied mathematics, scientists in fluid dynamics, and aviation engineers.

As is known, the book named "Multivariate spline functions and their applications" has been published by the Science Press in 1994. This book is an English edition based on the original book mentioned 1 above with many changes, including that of the structure of a cubic - interpolation in n-dimensional spline spaces, and more detail on triangu- lations have been added in this book. Special cases of multivariate spline functions (such as step functions, polygonal functions, and piecewise polynomials) have been examined math- ematically for a long time. I. J. Schoenberg (Contribution to the problem of application of equidistant data by analytic functions, Quart. Appl. Math., 4(1946), 45 - 99; 112 - 141) and W. Quade & L. Collatz (Zur Interpo- lations theories der reellen periodischen function, Press. Akad. Wiss. (PhysMath. KL), 30(1938), 383- 429) systematically established the the- ory of the spline functions. W. Quade & L. Collatz mainly discussed the periodic functions, while I. J. Schoenberg's work was systematic and com- plete. I. J. Schoenberg outlined three viewpoints for studing univariate splines: Fourier transformations, truncated polynomials and Taylor ex- pansions. Based on the first two viewpoints, I. J. Schoenberg deduced the B-spline function and its basic properties, especially the basis func- tions. Based on the latter viewpoint, he represented the spline functions in terms of truncated polynomials. These viewpoints and methods had significantly effected on the development of the spline functions.

This book presents applications of Newton-like and other similar methods to solve abstract functional equations involving fractional derivatives. It focuses on Banach space-valued functions of a real domain - studied for the first time in the literature. Various issues related to the modeling and analysis of fractional order systems continue to grow in popularity, and the book provides a deeper and more formal analysis of selected issues that are relevant to many areas - including decision-making, complex processes, systems modeling and control - and deeply embedded in the fields of engineering, computer science, physics, economics, and the social and life sciences. The book offers a valuable resource for researchers and graduate students, and can also be used as a textbook for seminars on the above-mentioned subjects. All chapters are self-contained and can be read independently. Further, each chapter includes an extensive list of references.

It isn't that they can't see the solution. It is Approach your problems from the right end and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father The Hermit Gad in Crane Feathers' in R. Brown The point of a Pin'. van GuIik's The Chinese Maze Murders. Growing speciaIization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

The aim of this book is to construct categories of spaces which contain all the C?-manifolds, but in addition infinitesimal spaces and arbitrary function spaces. To this end, the techniques of Grothendieck toposes (and the logic inherent to them) are explained at a leisurely pace and applied. By discussing topics such as integration, cohomology and vector bundles in the new context, the adequacy of these new spaces for analysis and geometry will be illustrated and the connection to the classical approach to C?-manifolds will be explained.

Statistical inferential methods are widely used in the study of various physical, biological, social, and other phenomena. Parametric estimation is one such method. Although there are many books which consider problems of statistical point estimation, this volume is the first to be devoted solely to the problem of unbiased estimation. It contains three chapters dealing, respectively, with the theory of point statistical estimation, techniques for constructing unbiased estimators, and applications of unbiased estimation theory. These chapters are followed by a comprehensive appendix which classifies and lists, in the form of tables, all known results relating to unbiased estimators of parameters for univariate distributions. About one thousand minimum variance unbiased estimators are listed. The volume also contains numerous examples and exercises. This volume will serve as a handbook on point unbiased estimation for researchers whose work involves statistics. It can also be recommended as a supplementary text for graduate students.

This volume contains a collection of papers in Analytic and Elementary Number Theory in memory of Professor Paul Erd s, one of the greatest mathematicians of this century. Written by many leading researchers, the papers deal with the most recent advances in a wide variety of topics, including arithmetical functions, prime numbers, the Riemann zeta function, probabilistic number theory, properties of integer sequences, modular forms, partitions, and q-series. Audience: Researchers and students of number theory, analysis, combinatorics and modular forms will find this volume to be stimulating.

The history of continued fractions is certainly one of the longest among those of mathematical concepts, since it begins with Euclid's algorithm for the great est common divisor at least three centuries B.C. As it is often the case and like Monsieur Jourdain in Moliere's "Ie bourgeois gentilhomme" (who was speak ing in prose though he did not know he was doing so), continued fractions were used for many centuries before their real discovery. The history of continued fractions and Pade approximants is also quite im portant, since they played a leading role in the development of some branches of mathematics. For example, they were the basis for the proof of the tran scendence of 11' in 1882, an open problem for more than two thousand years, and also for our modern spectral theory of operators. Actually they still are of great interest in many fields of pure and applied mathematics and in numerical analysis, where they provide computer approximations to special functions and are connected to some convergence acceleration methods. Con tinued fractions are also used in number theory, computer science, automata, electronics, etc ..."

Partition functions arise in combinatorics and related problems of statistical physics as they encode in a succinct way the combinatorial structure of complicated systems. The main focus of the book is on efficient ways to compute (approximate) various partition functions, such as permanents, hafnians and their higher-dimensional versions, graph and hypergraph matching polynomials, the independence polynomial of a graph and partition functions enumerating 0-1 and integer points in polyhedra, which allows one to make algorithmic advances in otherwise intractable problems. The book unifies various, often quite recent, results scattered in the literature, concentrating on the three main approaches: scaling, interpolation and correlation decay. The prerequisites include moderate amounts of real and complex analysis and linear algebra, making the book accessible to advanced math and physics undergraduates.

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.

The 1985 Castel vecchio-Pas coli NATO Advanced Study Institute is aimed to complete the trilogy with the two former institutes I organized: "Boundary Value Problem for Evolution Partial Differential Operators," Liege, 1976 and "Singularities in Boundary Value Problems," Maratea, 1980. It was indeed necessary to record the considerable progress realized in the field of the propagation of singularities of Schwartz Distri butions which led recently to the birth of a new branch of Mathema tical Analysis called Microlocal Analysis. Most of this theory was mainly built to be applied to distribution solutions of linear partial differential problems. A large part of this institute still went in this direction. But, on the other hand, it was also time to explore the new trend to use microlocal analysis In non linear differential problems. I hope that the Castelvecchio NATO ASI reached its purposes with the help of the more famous authorities in the field. The meeting was held in Tuscany (Italy) at Castelvecchio-Pascoli, little village in the mountains north of Lucca on September 2-12, 1985. It was hosted by "11 Ciocco" an international vacation Center, In a comfortable hotel located in magnificent mountain surroundings and provided with all conference and sport facilities."

Karl Menger, one of the founders of dimension theory, belongs to the most original mathematicians and thinkers of the twentieth century. He was a member of the Vienna Circle and the founder of its mathematical equivalent, the Viennese Mathematical Colloquium. Both during his early years in Vienna and, after his emigration, in the United States, Karl Menger made significant contributions to a wide variety of mathematical fields, and greatly influenced some of his colleagues. The Selecta Mathematica contain Menger's major mathematical papers, based on his own selection from his extensive writings. They deal with topics as diverse as topology, geometry, analysis and algebra, as well as writings on economics, sociology, logic, philosophy and mathematical results. The two volumes are a monument to the diversity and originality of Menger's ideas.

This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory.

In April of 1996 an array of mathematicians converged on Cambridge, Massachusetts, for the Rotafest and Umbral Calculus Workshop, two con ferences celebrating Gian-Carlo Rota's 64th birthday. It seemed appropriate when feting one of the world's great combinatorialists to have the anniversary be a power of 2 rather than the more mundane 65. The over seventy-five par ticipants included Rota's doctoral students, coauthors, and other colleagues from more than a dozen countries. As a further testament to the breadth and depth of his influence, the lectures ranged over a wide variety of topics from invariant theory to algebraic topology. This volume is a collection of articles written in Rota's honor. Some of them were presented at the Rotafest and Umbral Workshop while others were written especially for this Festschrift. We will say a little about each paper and point out how they are connected with the mathematical contributions of Rota himself."

Wavelets and wavelet packets provide a theory analogous to Fourier analysis and tools analogous to coherent state methods. Among their numerous applications, wavelets have been used to data compression in both image and sound processing. They are intimately related to splines, and wavelet applications in spline theory are significant. Wavelets have become a tool in analyzing fractals and iterative schemes associated with dynamical systems. Signal processing methods such as quadrature mirror filters go hand in hand with wavelet techniques in studying a host of communcations problems. The profound issues of classical turbulence are being studied using wavelet packets. Both wavelet packet software and wavelet transform microchips are now available. There are also applications of wavelet theory in theoretical physics, oil exploration, irregular sampling, and singular integral operators. Many of the world's experts in the field of wavelets were principal speakers at the ASI, and their papers appear in this volume. Furthermore, these renowned scientists addressed their talks to an audience which consisted of a broad spectrum of pure and applied mathematicians, as well as a diverse group of engineers and scientists. Thus, the reader has the opportunity to both learn or reinforce the fundamental concepts from the individuals who have created and developed the blossoming field of wavelets, and to see them discuss in accessible terms their profound contributions and ideas for future research.