In classical analysis, there is a vast difference between the class of problems that may be handled by means of the methods of calculus and the class of problems requiring combinatorial techniques. With the advent of the digital computer, the distinction begins to blur, and with the increasing emphasis on problems involving optimization over structures, tIlE' distinction vanishes. What is necessary for the analytic and computational treatment of significant questions arising in modern control theory, mathematical economics, scheduling theory, operations research, bioengineering, and so forth is a new and more flexible mathematical theory which subsumes both the cla8sical continuous and discrete t 19orithms. The work by HAMMER (IVANESCU) and RUDEANU on Boolean methods represents an important step in this dnectlOn, and it is thus a great pleasure to welcome it into print. It will certainly stimulate a great deal of additional research in both theory and application. RICHARD BELLMAN University of Southern California FOf(,WOl'

A few years aga the authors started a project of a book on the theory of systems of one-dimensional singular integral equa tions which was planned as a continuation of the monograph by one of the authors and N. Ya. Krupnik ~~ concerning scalar equa tions. This set of notes was initiated as a chapter dealing with problems of factorization of matrix functions vis-a-vis appli cations to systems of singular integral equations. Working systematically onthischapter and adding along the way new points of view, new proofs and results, we finally saw that the material connected with factorizations is of independent interest and we decided to publish this chapter as aseparate volume. In fact, because of recent activity, the amount of material was quite large and we quickly learned that we cannot cover all of the results in complete detail. We have tried to include a represen tative variety of all kinds of methods, techniques,results and applications. Apart of the current work exposes results from the Russian literature which have never appeared in English translation. We have also decided to reflect some of the recent results which make interesting connections between factorization of matrix functions and systems theory. The field remains very active and many results and connec tions are still not weIl understood. These notes should be viewed as a stepping stone to further development. The authors hope that sometime they will return to complete their original plan.

This paper is a largely expository account of the theory of p x p matrix polyno mials associated with Hermitian block Toeplitz matrices and some related problems of interpolation and extension. Perhaps the main novelty is the use of reproducing kernel Pontryagin spaces to develop parts of the theory in what hopefully the reader will regard as a reasonably lucid way. The topics under discussion are presented in a series of short sections, the headings of which give a pretty good idea of the overall contents of the paper. The theory is a rich one and the present paper in spite of its length is far from complete. The author hopes to fill in some of the gaps in future publications. The story begins with a given sequence h_n" ... , hn of p x p matrices with h-i = hj for j = 0, ... , n. We let k = O, ... ,n, (1.1) denote the Hermitian block Toeplitz matrix based on ho, ... , hk and shall denote its 1 inverse H k by (k)] k [ r = .. k = O, ... ,n, (1.2) k II} . '-0 ' I- whenever Hk is invertible.

In this volume three important papers of M.G. Krein appear for the first time in English translation. Each of them is a short self-contained monograph, each a masterpiece of exposition. Although two of them were written more than twenty years ago, the passage of time has not decreased their value. They are as fresh and vital as if they had been written only yesterday. These papers contain a wealth of ideas, and will serve as a source of stimulation and inspiration for experts and beginners alike. The first paper is dedicated to the theory of canonical linear differential equations, with periodic coefficients. It focuses on the study of linear Hamiltonian systems with bounded solutions which stay bounded under small perturbations of the system. The paper uses methods from operator theory in finite and infinite dimensional spaces and complex analysis. For an account of more recent literature which was generated by this paper see AMS Translations (2), Volume 93, 1970, pages 103-176 and Integral Equations and Operator Theory, Volume 5, Number 5, 1982, pages 718-757.

Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis. This self-contained textbook gives a complete course on all these topics. It is written by a leading specialist who is also a noted expositor. This book provides a thorough introduction to functional analysis and includes many novel elements as well as the standard topics. A short course on nonsmooth analysis and geometry completes the first half of the book whilst the second half concerns the calculus of variations and optimal control. The author provides a comprehensive course on these subjects, from their inception through to the present. A notable feature is the inclusion of recent, unifying developments on regularity, multiplier rules, and the Pontryagin maximum principle, which appear here for the first time in a textbook. Other major themes include existence and Hamilton-Jacobi methods. The many substantial examples, and the more than three hundred exercises, treat such topics as viscosity solutions, nonsmooth Lagrangians, the logarithmic Sobolev inequality, periodic trajectories, and systems theory. They also touch lightly upon several fields of application: mechanics, economics, resources, finance, control engineering. Functional Analysis, Calculus of Variations and Optimal Control is intended to support several different courses at the first-year or second-year graduate level, on functional analysis, on the calculus of variations and optimal control, or on some combination. For this reason, it has been organized with customization in mind. The text also has considerable value as a reference. Besides its advanced results in the calculus of variations and optimal control, its polished presentation of certain other topics (for example convex analysis, measurable selections, metric regularity, and nonsmooth analysis) will be appreciated by researchers in these and related fields.

The book examines in some depth two important classes of point processes, determinantal processes and 'Gaussian zeros', i.e., zeros of random analytic functions with Gaussian coefficients. These processes share a property of 'point-repulsion', where distinct points are less likely to fall close to each other than in processes, such as the Poisson process, that arise from independent sampling. Nevertheless, the treatment in the book emphasizes the use of independence: for random power series, the independence of coefficients is key; for determinantal processes, the number of points in a domain is a sum of independent indicators, and this yields a satisfying explanation of the central limit theorem (CLT) for this point count. Another unifying theme of the book is invariance of considered point processes under natural transformation groups. The book strives for balance between general theory and concrete examples. On the one hand, it presents a primer on modern techniques on the interface of probability and analysis. On the other hand, a wealth of determinantal processes of intrinsic interest are analyzed; these arise from random spanning trees and eigenvalues of random matrices, as well as from special power series with determinantal zeros. The material in the book formed the basis of a graduate course given at the IAS-Park City Summer School in 2007; the only background knowledge assumed can be acquired in first-year graduate courses in analysis and probability.

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.

The text is based on an established graduate course given at MIT that provides an introduction to the theory of the dynamical Yang-Baxter equation and its applications, which is an important area in representation theory and quantum groups. The book, which contains many detailed proofs and explicit calculations, will be accessible to graduate students of mathematics, who are familiar with the basics of representation theory of semi-simple Lie algebras.

The present volume contains a collection of original research articles and expository contributions on recent developments in operator theory and its multifaceted applications. They cover a wide range of themes from the IWOTA 2010 conference held at the TU Berlin, Germany, including spectral theory, function spaces, mathematical system theory, evolution equations and semigroups, and differential and difference operators. The book encompasses new trends and various modern topics in operator theory, and serves as a useful source of information to mathematicians, scientists and engineers.

This book explains and examines the theoretical underpinnings of the Complex Variable Boundary Element Method (CVBEM) as applied to higher dimensions, providing the reader with the tools for extending and using the CVBEM in various applications. Relevant mathematics and principles are assembled and the reader is guided through the key topics necessary for an understanding of the development of the CVBEM in both the usual two as well as three or higher dimensions. In addition to this, problems are provided that build upon the material presented. The Complex Variable Boundary Element Method (CVBEM) is an approximation method useful for solving problems involving the Laplace equation in two dimensions. It has been shown to be a useful modelling technique for solving two-dimensional problems involving the Laplace or Poisson equations on arbitrary domains. The CVBEM has recently been extended to 3 or higher spatial dimensions, which enables the precision of the CVBEM in solving the Laplace equation to be now available for multiple dimensions. The mathematical underpinnings of the CVBEM, as well as the extension to higher dimensions, involve several areas of applied and pure mathematics including Banach Spaces, Hilbert Spaces, among other topics. This book is intended for applied mathematics graduate students, engineering students or practitioners, developers of industrial applications involving the Laplace or Poisson equations and developers of computer modelling applications.

Leon Ehrenpreis has been one of the leading mathematicians in the twentieth century. His contributions to the theory of partial differential equations were part of the golden era of PDEs, and led him to what is maybe his most important contribution, the Fundamental Principle, which he announced in 1960, and fully demonstrated in 1970. His most recent work, on the other hand, focused on a novel and far reaching understanding of the Radon transform, and offered new insights in integral geometry. Leon Ehrenpreis died in 2010, and this volume collects writings in his honor by a cadre of distinguished mathematicians, many of which were his collaborators.

This book focuses on the constructive and practical aspects of spectral methods. It rigorously examines the most important qualities as well as drawbacks of spectral methods in the context of numerical methods devoted to solve non-standard eigenvalue problems. In addition, the book also considers some nonlinear singularly perturbed boundary value problems along with eigenproblems obtained by their linearization around constant solutions.

The book is mathematical, poising problems in their proper function spaces, but its emphasis is on algorithms and practical difficulties. The range of applications is quite large. High order eigenvalue problems are frequently beset with numerical ill conditioning problems. The book describes a wide variety of successful modifications to standard algorithms that greatly mitigate these problems.

In addition, the book makes heavy use of the concept of pseudospectrum, which is highly relevant to understanding when disaster is imminent in solving eigenvalue problems. It also envisions two classes of applications, the stability of some elastic structures and the hydrodynamic stability of some parallel shear flows.

This book is an ideal reference text for professionals (researchers) in applied mathematics, computational physics and engineering. It will be very useful to numerically sophisticated engineers, physicists and chemists. The book can also be used as a textbook in review courses such as numerical analysis, computational methods in various engineering branches or physics and computational methods in analysis.

This textbook is a thorough, accessible introduction to digital Fourier analysis for undergraduate students in the sciences. Beginning with the principles of sine/cosine decomposition, the reader walks through the principles of discrete Fourier analysis before reaching the cornerstone of signal processing: the Fast Fourier Transform.

Saturated with clear, coherent illustrations, "Digital Fourier Analysis" includes practice problems and thorough Appendices for the advanced reader. As a special feature, the book includes interactive applets (available online) that mirror the illustrations. These user-friendly applets animate concepts interactively, allowing the user to experiment with the underlying mathematics.

For example, a real sine signal can be treated as a sum of clockwise and counter-clockwise rotating vectors. The applet illustration included with the book animates the rotating vectors and the resulting sine signal. By changing parameters such as amplitude and frequency, the reader can test various cases and view the results until they fully understand the principle.

Additionally, the applet source code in Visual Basic is provided online, allowing this book to be used for teaching simple programming techniques.

A complete, intuitive guide to the basics, "Digital Fourier Analysis - Fundamentals" is an essential reference for undergraduate students in science and engineering.

The idea of optimization runs through most parts of control theory. The simplest optimal controls are preplanned (programmed) ones. The problem of constructing optimal preplanned controls has been extensively worked out in literature (see, e. g., the Pontrjagin maximum principle giving necessary conditions of preplanned control optimality). However, the concept of op timality itself has a restrictive character: it is limited by what one means under optimality in each separate case. The internal contradictoriness of the preplanned control optimality ("the better is the enemy of the good") yields that the practical significance of optimal preplanned controls proves to be not great: such controls are usually sensitive to unregistered disturbances (includ ing the round-off errors which are inevitable when computer devices are used for forming controls), as there is the effect of disturbance accumulation in the control process which makes controls to be of little use on large time inter vals. This gap is mainly provoked by oversimplified settings of optimization problems. The outstanding result of control theory established in the end of the first half of our century is that controls in feedback form ensure the weak sensitivity of closed loop systems with respect to "small" unregistered internal and external disturbances acting in them (here we do not need to discuss performance indexes, since the considered phenomenon is of general nature). But by far not all optimal preplanned controls can be represented in a feedback form."

Since the publication of Banach's treatise on the theory of linear operators, the literature on the theory of bases in topological vector spaces has grown enormously. Much of this literature has for its origin a question raised in Banach's book, the question whether every sepa rable Banach space possesses a basis or not. The notion of a basis employed here is a generalization of that of a Hamel basis for a finite dimensional vector space. For a vector space X of infinite dimension, the concept of a basis is closely related to the convergence of the series which uniquely correspond to each point of X. Thus there are different types of bases for X, according to the topology imposed on X and the chosen type of convergence for the series. Although almost four decades have elapsed since Banach's query, the conjectured existence of a basis for every separable Banach space is not yet proved. On the other hand, no counter examples have been found to show the existence of a special Banach space having no basis. However, as a result of the apparent overconfidence of a group of mathematicians, who it is assumed tried to solve the problem, we have many elegant works which show the tight connection between the theory of bases and structure of linear spaces."

The volume contains selected papers of the Spectral Function Theory seminar, Leningrad Branch of Steklov Mathematical Institute. The papers are mostly devoted to the theory of Toeplitz and model operators. These subjects are considered here from various points of view. Several papers concern the relationships of Toeplitz operators to weighted polynomial approximation. Namely, two papers by B. Solomyak and A. Volberg intensively treat the problem of spectra! multiplicity f~r analytic Toeplitz operators (which are, in fact, multiplication operators) and my paper can serve as an introduction to the problem. This theme of multiplicities is continued in a paper by V. Vasyunin where the multiplicity of the spectrum is computed for Hilbert space contractions with finite defect indices. V. Peller's paper deals with a perturbation theory problem for Toeplitz operators. In a paper by D. Yakubovich a new similarity model for a class of Toeplitz operators is constructed. S. Treil' presents a survey of a part of spectral function theory for vector valued function (Szego-Kolmogorov extreme prob!ems for operator weights, bases of vector rational functions, estimations of Hilbert transform with respect to operator weights, the operator corona problem). As a concluding remark I dare only note that the whole collection convinces us once more without a doubt of the fruitfullness of the natural union of operator theory and complex analysis (if at all the union of these fields is at all different from their intersection).

Spectral analysis of linear operators has always been one of the more active and important fields of operator theory, and of extensive interest to many operator theorists. Its devel opments usually are closely related to certain important problems in contemporary mathematics and physics. In the last 20 years, many new theories and interesting results have been discovered. Now, in this direction, the fields are perhaps wider and deeper than ever. This book is devoted to the study of hyponormal and semi-hyponormal operators. The main results we shall present are those of the author and his collaborators and colleagues, as well as some concerning related topics. To some extent, hyponormal and semi-hyponormal opera tors are "close" to normal ones. Although those two classes of operators contain normal operators as a subclass, what we are interested in are, naturally, nonnormal operators in those classes. With the well-studied normal operators in hand, we cer tainly wish to know the properties of hyponormal and semi-hypo normal operators which resemble those of normal operators. But more important than that, the investigations should be concen trated on the phenomena which only occur in the nonnormal cases."

This volume contains twenty-one solicited articles by speakers at the IWOTA 2009 workshop, ranging from expository surveys to original research papers, each carefully refereed.The contributions reflect recent developments in operator theory and its applications. Consistent with the topics of recent IWOTA meetings, IWOTA 2009 was designed as a comprehensive, inclusive conference covering all aspects of theoretical and applied operator theory, ranging from classical analysis, differential and integral equations, complex and harmonic analysis to mathematical physics, mathematical systems and control theory, signal processing and numerical analysis. The conference brought together international experts for a week-long stay at Hotel Real de Minas, in an atmosphere conducive to fruitful professional interactions. These Proceedings reflect the high quality of the papers presented at the conference. "

The aim of this book is to give a systematic and self-contained presentation of the Mathematical Scattering Theory within the framework of operator theory in Hilbert space. The term Mathematical Scattering Theory denotes that theory which is on the one hand the common mathematical foundation of several physical scattering theories (scattering of quantum objects, of classical waves and particles) and on the other hand a branch of operator theory devoted to the study of the behavior of the continuous part of perturbed operators (some authors also use the term Abstract Scattering Theory). EBBential contributions to the development of this theory are due to K. FRIEDRICHS, J. CooK, T. KATo, J. M. JAuCH, S. T. KURODA, M.S. BmMAN, M.G. KREiN, L. D. FAD DEEV, R. LAVINE, W. 0. AMREIN, B. SIMoN, D. PEARSON, V. ENss, and others. It seems to the authors that the theory has now reached a sufficiently developed state that a self-contained presentation of the topic is justified."

Blaschke Products and Their Applications presents a collection of survey articles that examine Blaschke products and several of its applications to fields such as approximation theory, differential equations, dynamical systems, harmonic analysis, to name a few. Additionally, this volume illustrates the historical roots of Blaschke products and highlights key research on this topic. For nearly a century, Blaschke products have been researched. Their boundary behaviour, the asymptomatic growth of various integral means and their derivatives, their applications within several branches of mathematics, and their membership in different function spaces and their dynamics, are a few examples of where Blaschke products have shown to be important. The contributions written by experts from various fields of mathematical research will engage graduate students and researches alike, bringing the reader to the forefront of research in the topic. The readers will also discover the various open problems, enabling them to better pursue their own research.

This book provides the foundations for a rigorous theory of functional analysis with bicomplex scalars. It begins with a detailed study of bicomplex and hyperbolic numbers and then defines the notion of bicomplex modules. After introducing a number of norms and inner products on such modules (some of which appear in this volume for the first time), the authors develop the theory of linear functionals and linear operators on bicomplex modules. All of this may serve for many different developments, just like the usual functional analysis with complex scalars and in this book it serves as the foundational material for the construction and study of a bicomplex version of the well known Schur analysis.

The purpose of this book is to provide an integrated course in real and complex analysis for those who have already taken a preliminary course in real analysis. It particularly emphasises the interplay between analysis and topology. Beginning with the theory of the Riemann integral (and its improper extension) on the real line, the fundamentals of metric spaces are then developed, with special attention being paid to connectedness, simple connectedness and various forms of homotopy. The final chapter develops the theory of complex analysis, in which emphasis is placed on the argument, the winding number, and a general (homology) version of Cauchy's theorem which is proved using the approach due to Dixon. Special features are the inclusion of proofs of Montel's theorem, the Riemann mapping theorem and the Jordan curve theorem that arise naturally from the earlier development. Extensive exercises are included in each of the chapters, detailed solutions of the majority of which are given at the end. From Real to Complex Analysis is aimed at senior undergraduates and beginning graduate students in mathematics. It offers a sound grounding in analysis; in particular, it gives a solid base in complex analysis from which progress to more advanced topics may be made.

Nonlinear functional analysis is an important branch of contemporary mathematics. It's related to topology, ordinary differential equations, partial differential equations, groups, dynamical systems, differential geometry, measure theory, and more. In this book, the author presents some new and interesting results on fundamental methods in nonlinear functional analysis, namely variational, topological and partial order methods, which have been used extensively to solve existence of solutions for elliptic equations, wave equations, Schroedinger equations, Hamiltonian systems etc., and are also used to study the existence of multiple solutions and properties of solutions. This book is useful for researchers and graduate students in the field of nonlinear functional analysis.