This book is a collection of research articles in algebraic geometry and complex analysis dedicated to Hans Grauert. The authors and editors have made their best efforts in order that these contributions should be adequate to honour the outstanding scientist. The volume contains important new results, solutions to longstanding conjectures, elegant new proofs and new perspectives for future research. The topics range from surface theory and commutative algebra, linear systems, moduli spaces, classification theory, Kähler geometry to holomorphic dynamical systems.

Integral equations have wide applications in various fields, including continuum mechanics, potential theory, geophysics, electricity and magnetism, kinetic theory of gases, hereditary phenomena in physics and biology, renewal theory, quantum mechanics, radiation, optimization, optimal control systems, communication theory, mathematical economics, population genetics, queueing theory, and medicine.

Computational Methods for Linear Integral Equations presents basic theoretical material that deals with numerical analysis, convergence, error estimates, and accuracy. The unique computational aspect leads the reader from theoretical and practical problems all the way through to computation with hands-on guidance for input files and the execution of computer programs.

Features:

* Offers all supporting MathematicaA(R) files related to the book via the Internet at the authors' Web sites: www.math.uno.edu/fac/pkythe.html or www.math.uno.edu/fac/ppuri.html

* Contains identification codes for problems, related methods, and computer programs that are cross-referenced throughout the book to make the connections easy to understand

* Illustrates a how-to approach to computational work in the development of algorithms, construction of input files, timing, and accuracy analysis

* Covers linear integral equations of Fredholm and Volterra types of the first and second kinds as well as associated singular integral equations, integro-differential equations, and eigenvalue problems

* Provides clear, step-by-step guidelines for solving difficult and complex computational problems

This book is an essential reference and authoritative resource for all professionals, graduate students, and researchers in mathematics, physical sciences, and engineering. Researchers interested in the numerical solution of integral equations will find its practical problem-solving style both accessible and useful for their work.

This book is an introduction to the theory of complex manifolds. The authors¿ intent is to familiarize the reader with the most important branches and methods in complex analysis of several variables and to do this as simply as possible. Therefore, the abstract concepts involving sheaves, coherence, and higher-dimensional cohomology have been completely avoided. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used. Nevertheless, deep results can be proved. The book can be used as a first introduction to several complex variables as well as a reference for the expert.

In recent years, algorithmic graph theory has become increasingly important as a link between discrete mathematics and theoretical computer science. This textbook introduces students of mathematics and computer science to the interrelated fields of graphs theory, algorithms and complexity.

The book is an authoritative and up-to-date introduction to the field of analysis and potential theory dealing with the distribution zeros of classical systems of polynomials such as orthogonal polynomials, Chebyshev, Fekete and Bieberbach polynomials, best or near-best approximating polynomials on compact sets and on the real line. The main feature of the book is the combination of potential theory with conformal invariants, such as module of a family of curves and harmonic measure, to derive discrepancy estimates for signed measures if bounds for their logarithmic potentials or energy integrals are known a priori.

Almost Automorphic and Almost Periodic Functions in Abstract Spaces introduces and develops the theory of almost automorphic vector-valued functions in Bochner's sense and the study of almost periodic functions in a locally convex space in a homogenous and unified manner. It also applies the results obtained to study almost automorphic solutions of abstract differential equations, expanding the core topics with a plethora of groundbreaking new results and applications. For the sake of clarity, and to spare the reader unnecessary technical hurdles, the concepts are studied using classical methods of functional analysis.

The book presents the method of difference potentials first proposed by the author in 1969 and contains illustrative examples and new algorithms for solving applied problems of gas dynamics, diffraction, scattering theory, and active noise screening. The fundamentals of the method are described in Parts I-III and its applications in Parts IV-VIII. To get acquainted with the basic ideas of the method, it suffices to study the Introduction. After this, each of the Parts VI-VIII can be read independently. The book is intended for specialists in the field of computational mathematics and the theory of differential and integral equations, as well as for graduate students of related specialities.

The space Q p consists of all holomorphic functions f on the unit disk for which the L^2 area integrals of its derivative against the p-th power of the Green function of the unit disk are uniformly bounded in the variable that survives the integration. It turns out that Q 1 coincides with BMOA, while, for p>1, Q p are just the Bloch space. For p/in (0,1) the Q p furnish an increasing sequence of spaces, each invariant under conformal mappings of the unit disk onto itself, which interpolate between the Dirichlet space and BMOA. This monograph covers a number of important aspects in complex, functional and harmonic analysis. The primary focus is Q p, p/in (0,1), and their equivalent characterizations. Based on the up-to-date results obtained by experts in their respective fields, each of the eight chapters unfolds from the basics to the more complex. The exposition here is rapid-paced and efficient, with proofs and examples.

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 manuscript provides an introduction to the generation theory of nonlinear one-parameter semigroups on a domain of the complex plane in the spirit of the Wolff-Denjoy and Hille-Yoshida theories. Special attention is given to evolution equations reproduced by holomorphic vector fields on the unit disk. A dynamic approach to the study of geometrical properties of univalent functions is emphasized. The book comprises six chapters. The preliminary chapter and chapter 1 give expositions to the theory of functions in the complex plane, and the iteration theory of holomorphic mappings according to Wolff and Denjoy, as well as to Julia and Caratheodory. Chapter 2 deals with elementary hyperbolic geometry on the unit disk, and fixed points of those mappings which are nonexpansive with respect to the PoincarA(c) metric. Chapters 3 and 4 study local and global characteristics of holomorphic and hyperbolically monotone vector-fields, which yield a global description of asymptotic behavior of generated flows. Various boundary and interior flow invariance conditions for such vector-fields and their parametric representations are presented. Applications to univalent starlike and spirallike functions on the unit disk are given in Chapter 5. The approach described may also be useful for higher dimensions. Audience: The book will be of interest to graduate students and research specialists working in the fields of geometrical function theory, iteration theory, fixed point theory, semigroup theory, theory of composition operators and complex dynamical systems.

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.

Singularities arise naturally in a huge number of different areas of mathematics and science. As a consequence, singularity theory lies at the crossroads of paths that connect many of the most important areas of applications of mathematics with some of its most abstract regions. The main goal in most problems of singularity theory is to understand the dependence of some objects of analysis, geometry, physics, or other science (functions, varieties, mappings, vector or tensor fields, differential equations, models, etc.) on parameters. The articles collected here can be grouped under three headings. (A) Singularities of real maps; (B) Singular complex variables; and (C) Singularities of homomorphic maps.

Singularities arise naturally in a huge number of different areas of mathematics and science. As a consequence, singularity theory lies at the crossroads of paths that connect many of the most important areas of applications of mathematics with some of its most abstract regions. The main goal in most problems of singularity theory is to understand the dependence of some objects of analysis, geometry, physics, or other science (functions, varieties, mappings, vector or tensor fields, differential equations, models, etc.) on parameters. The articles collected here can be grouped under three headings. (A) Singularities of real maps; (B) Singular complex variables; and (C) Singularities of homomorphic maps.

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.

The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The book consists of three parts. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle, the Schwarz lemma and hyperbolic geometry, the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics selected include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces. The three geometries, spherical, euclidean, and hyperbolic, are stressed. Exercises range from the very simple to the quite challenging, in all chapters. The book is based on lectures given over the years by the author at several places, particularly the Interuniversity Summer School at Perugia (Italy), and also UCLA, Brown University, Valencia (Spain), and La Plata (Argentina). A native of Minnesota, the author did his undergraduate work at Yale University and his graduate work at UC Berkeley. After spending some time at MIT and at the Universidad Nacional de La Plata (Argentina), he joined the faculty at UCLA in 1968. The author has published a number of research articles and several books on functional analysis and analytic function theory. he is currently involved in the California K-12 education scene.

On the one hand, this monograph serves as a self-contained introduction to Nevanlinna's theory of value distribution because the authors only assume the reader is familiar with the basics of complex analysis. On the other hand, the monograph also serves as a valuable reference for the research specialist because the authors present, for the first time in book form, the most modern and refined versions of the Second Main Theorem with precise error terms, in both the geometric and logarithmic derivative based approaches. A unique feature of the monograph is its "number-theoretic digressions". These special sections assume no background in number theory and explore the exciting interconnections between Nevanlinna theory and the theory of Diophantine approximation.

From the reviews: "Here is a momumental work by Doob, one of the masters, in which Part 1 develops the potential theory associated with Laplace's equation and the heat equation, and Part 2 develops those parts (martingales and Brownian motion) of stochastic process theory which are closely related to Part 1." --G.E.H. Reuter in Short Book Reviews (1985)

This book presents complex analysis in one variable in the context of modern mathematics, with clear connections to several complex variables, de Rham theory, real analysis, and other branches of mathematics. Thus, covering spaces are used explicitly in dealing with Cauchy's theorem, real variable methods are illustrated in the Loman-Menchoff theorem and in the corona theorem, and the algebraic structure of the ring of holomorphic functions is studied.
Using the unique position of complex analysis, a field drawing on many disciplines, the book also illustrates powerful mathematical ideas and tools, and requires minimal background material. Cohomological methods are introduced, both in connection with the existence of primitives and in the study of meromorphic functionas on a compact Riemann surface. The proof of Picard's theorem given here illustrates the strong restrictions on holomorphic mappings imposed by curvature conditions.

From its origins in Newtonian physics, potential theory has developed into a major field of mathematical research. This book provides a comprehensive treatment of classical potential theory: it covers harmonic and subharmonic functions, maximum principles, polynomial expansions, Green functions, potentials and capacity, the Dirichlet problem and boundary integral representations. The first six chapters deal concretely with the basic theory, and include exercises. The final three chapters are more advanced and treat topological ideas specifically created for potential theory, such as the fine topology, the Martin boundary and minimal thinness.
The presentation is largely self-contained and is accessible to graduate students, the only prerequisites being a reasonable grounding in analysis and several variables calculus, and a first course in measure theory. The book will prove an essential reference to all those with an interest in potential theory and its applications.

Nevanlinna theory (or value distribution theory) in complex analysis is so beautiful that one would naturally be interested in determining how such a theory would look in the non Archimedean analysis and Diophantine approximations. There are two "main theorems" and defect relations that occupy a central place in N evanlinna theory. They generate a lot of applications in studying uniqueness of meromorphic functions, global solutions of differential equations, dynamics, and so on. In this book, we will introduce non-Archimedean analogues of Nevanlinna theory and its applications. In value distribution theory, the main problem is that given a holomorphic curve f : C -+ M into a projective variety M of dimension n and a family 01 of hypersurfaces on M, under a proper condition of non-degeneracy on f, find the defect relation. If 01 n is a family of hyperplanes on M = r in general position and if the smallest dimension of linear subspaces containing the image f(C) is k, Cartan conjectured that the bound of defect relation is 2n - k + 1. Generally, if 01 is a family of admissible or normal crossings hypersurfaces, there are respectively Shiffman's conjecture and Griffiths-Lang's conjecture. Here we list the process of this problem: A. Complex analysis: (i) Constant targets: R. Nevanlinna[98] for n = k = 1; H. Cartan [20] for n = k > 1; E. I. Nochka [99], [100],[101] for n > k ~ 1; Shiffman's conjecture partially solved by Hu-Yang [71J; Griffiths-Lang's conjecture (open).

This book focuses on complex analytic dynamics, which dates from 1916 and is currently attracting considerable interest. The text provides a comprehensive, well-organized treatment of the foundations of the theory of iteration of rational functions of a complex variable. The coverage extends from early memoirs of Fatou and Julia to important recent results and methods of Sullivan and Shishikura. Many details of the proofs have not appeared in print before.

There has been fundamental progress in complex differential geometry in the last two decades. For one, The uniformization theory of canonical K hler metrics has been established in higher dimensions, and many applications have been found, including the use of Calabi-Yau spaces in superstring theory. This monograph gives an introduction to the theory of canonical K hler metrics on complex manifolds. It also presents some advanced topics not easily found elsewhere.

Computing all the zeros of an analytic function and their respective multiplicities, locating clusters of zeros and analytic fuctions, computing zeros and poles of meromorphic functions, and solving systems of analytic equations are problems in computational complex analysis that lead to a rich blend of mathematics and numerical analysis. This book treats these four problems in a unified way. It contains not only theoretical results (based on formal orthogonal polynomials or rational interpolation) but also numerical analysis and algorithmic aspects, implementation heuristics, and polished software (the package ZEAL) that is available via the CPC Program Library. Graduate studets and researchers in numerical mathematics will find this book very readable.

This volume consists of papers presented in the special sessions on "Complex and Numerical Analysis," "Value Distribution Theory and Complex Domains," and "Use of Symbolic Computation in Mathematics Education" of the ISAAC'97 Congress held at the University of Delaware, during June 2-7, 1997. The ISAAC Congress coincided with a U.S.-Japan Seminar also held at the University of Delaware. The latter was supported by the National Science Foundation through Grant INT-9603029 and the Japan Society for the Promotion of Science through Grant MTCS-134. It was natural that the participants of both meetings should interact and consequently several persons attending the Congress also presented papers in the Seminar. The success of the ISAAC Congress and the U.S.-Japan Seminar has led to the ISAAC'99 Congress being held in Fukuoka, Japan during August 1999. Many of the same participants will return to this Seminar. Indeed, it appears that the spirit of the U.S.-Japan Seminar will be continued every second year as part of the ISAAC Congresses. We decided to include with the papers presented in the ISAAC Congress and the U.S.-Japan Seminar several very good papers by colleagues from the former Soviet Union. These participants in the ISAAC Congress attended at their own expense.