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This collection of articles serves to commemorate the legacy of Joseph D''Atri, who passed away on April 29, 1993, a few days after his 55th birthday. Joe D'' Atri is credited with several fundamental discoveries in ge­ ometry. In the beginning of his mathematical career, Joe was interested in the generalization of symmetrical spaces in the E. Cart an sense. Symmetric spaces, differentiated from other homogeneous manifolds by their geomet­ rical richness, allows the development of a deep analysis. Geometers have been constantly interested and challenged by the problem of extending the class of symmetric spaces so as to preserve their geometrical and analytical abundance. The name of D''Atri is tied to one of the most successful gen­ eralizations: Riemann manifolds in which (local) geodesic symmetries are volume-preserving (up to sign). In time, it turned out that the majority of interesting generalizations of symmetrical spaces are D''Atri spaces: natu­ ral reductive homogeneous spaces, Riemann manifolds whose geodesics are orbits of one-parameter subgroups, etc. The central place in D''Atri''s research is occupied by homogeneous bounded domains in en, which are not symmetric. Such domains were discovered by Piatetskii-Shapiro in 1959, and given Joe''s strong interest in the generalization of symmetric spaces, it was very natural for him to direct his research along this path.

AMS packages that replaced AMS-Ib-TEX, version 1.1, but the book still had an AMS-Ib-TEX-centric view. This third edition is about Ib-TEX. Where necessary, I recommend that you use packages to extend Ib-TEX's capabilities. For typesetting mathematics, I strongly recommend that you use the AMS packages. AMS packages, version 2.0 The American Mathematical Society released version 2.0 ofthe AMS packages in 1999. This third edition covers the changes made in this release. Books The first and second editions of this book dealt primarily with the tasks involved in writing artides. In Part V, the third edition addresses the issues that arise when creating longer documents. In addition to chapters on BIBTEX and Makelndex, I have added a new chapter on writing books. Appendix F illustrates the impor­ tance of choosing a well-designed book document dass. Other changes B\1EX Ib-TEX2 has been remarkably stable since its release in 1996, becorning c the standard Ib-TEX (see Section C.1.2). Changeshave been rninor except for ad­ vances in using Ib-TEX with non-English languages (see the new Appendix E) and the widespread use of the PostScript CM and AMS fonts (see the new Section D.l). Reorganization and additions Due to the new emphasis on writing books, a number of sections and subsections have moved from Chapter 2 and Chapters 6-8 to the new Chapter 12.

Intuitively, a foliation corresponds to a decomposition of a manifold into a union of connected, disjoint submanifolds of the same dimension, called leaves, which pile up locally like pages of a book. The theory of foliations, as it is known, began with the work of C. Ehresmann and G. Reeb, in the 1940's; however, as Reeb has himself observed, already in the last century P. Painleve saw the necessity of creating a geometric theory (of foliations) in order to better understand the problems in the study of solutions of holomorphic differential equations in the complex field. The development of the theory of foliations was however provoked by the following question about the topology of manifolds proposed by H. Hopf in the 3 1930's: "Does there exist on the Euclidean sphere S a completely integrable vector field, that is, a field X such that X· curl X • 0?" By Frobenius' theorem, this question is equivalent to the following: "Does there exist on the 3 sphere S a two-dimensional foliation?" This question was answered affirmatively by Reeb in his thesis, where he 3 presents an example of a foliation of S with the following characteristics: There exists one compact leaf homeomorphic to the two-dimensional torus, while the other leaves are homeomorphic to two-dimensional planes which accu­ mulate asymptotically on the compact leaf. Further, the foliation is C"".

In the winter of 1978, Professor George P61ya and I jointly taught Stanford University's introductory combinatorics course. This was a great opportunity for me, as I had known of Professor P61ya since having read his classic book, How to Solve It, as a teenager. Working with P6lya, who ·was over ninety years old at the time, was every bit as rewarding as I had hoped it would be. His creativity, intelligence, warmth and generosity of spirit, and wonderful gift for teaching continue to be an inspiration to me. Combinatorics is one of the branches of mathematics that play a crucial role in computer sCience, since digital computers manipulate discrete, finite objects. Combinatorics impinges on computing in two ways. First, the properties of graphs and other combinatorial objects lead directly to algorithms for solving graph-theoretic problems, which have widespread application in non-numerical as well as in numerical computing. Second, combinatorial methods provide many analytical tools that can be used for determining the worst-case and expected performance of computer algorithms. A knowledge of combinatorics will serve the computer scientist well. Combinatorics can be classified into three types: enumerative, eXistential, and constructive. Enumerative combinatorics deals with the counting of combinatorial objects. Existential combinatorics studies the existence or nonexistence of combinatorial configurations.

During the last ten years a powerful technique for the study of partial differential equations with regular singularities has developed using the theory of hyperfunctions. The technique has had several important applications in harmonic analysis for symmetric spaces.This book gives an introductory exposition of the theory of hyperfunctions and regular singularities, and on this basis it treats two major applications to harmonic analysis. The first is to the proof of Helgason’s conjecture, due to Kashiwara et al., which represents eigenfunctions on Riemannian symmetric spaces as Poisson integrals of their hyperfunction boundary values.A generalization of this result involving the full boundary of the space is also given. The second topic is the construction of discrete series for semisimple symmetric spaces, with an unpublished proof, due to Oshima, of a conjecture of Flensted-Jensen.This first English introduction to hyperfunctions brings readers to the forefront of research in the theory of harmonic analysis on symmetric spaces. A substantial bibliography is also included. This volume is based on a paper which was awarded the 1983 University of Copenhagen Gold Medal Prize.

Thishandbookwasmotivatedinpartbyourexperience(andthatofothers)in performing research and in teaching about networked and embedded control systems (NECS) as well as in implementing such systems. Although NECS— along with the technologies that enable them—have become ubiquitous, there are few, if any, sources where a student, researcher, or developer can gain a su?ciently broad view of the subject. Oftentimes, the needed information is scattered in articles, websites, and speci?cation sheets. Such di?culties are perhaps to be expected, given the relative newness of the subject and the diversity of its constitutive disciplines. From control theory and communi- tions, to computer science and electronics, the variety of approaches, tools, and language used by experts in each ?eld often acts as a barrier to und- standing how ideas ?t within the broader context of networked and embedded control. With the above in mind, we have gathered a collection of articles that provide at least an introduction to the important results, tools, software, and technologies that shape the area of NECS. Our goal was to present the most important knowledge about NECS in a book that would be useful to anyone who wants to learn about any aspect of the subject. We hope that we have succeeded and that every reader will ?nd valuable information in the book. Wethanktheauthorsofeachofthechapters. Theyareallbusypeopleand we are extremely grateful to them for their outstanding work.

In recent years kinetic theory has developed in many areas of the physical sciences and engineering, and has extended the borders of its traditional fields of application. New applications in traffic flow engineering, granular media modeling, and polymer and phase transition physics have resulted in new numerical algorithms which depart from traditional stochastic Monte--Carlo methods. This monograph is a self-contained presentation of such recently developed aspects of kinetic theory, as well as a comprehensive account of the fundamentals of the theory. Emphasizing modeling techniques and numerical methods, the book provides a unified treatment of kinetic equations not found in more focused theoretical or applied works. The book is divided into two parts. Part I is devoted to the most fundamental kinetic model: the Boltzmann equation of rarefied gas dynamics. Additionally, widely used numerical methods for the discretization of the Boltzmann equation are reviewed: the Monte--Carlo method, spectral methods, and finite-difference methods.Part II considers specific applications: plasma kinetic modeling using the Landau--Fokker--Planck equations, traffic flow modeling, granular media modeling, quantum kinetic modeling, and coagulation-fragmentation problems. "Modeling and Computational Methods of Kinetic Equations" will be accessible to readers working in different communities where kinetic theory is important: graduate students, researchers and practitioners in mathematical physics, applied mathematics, and various branches of engineering. The work may be used for self-study, as a reference text, or in graduate-level courses in kinetic theory and its applications.

This book has a dual purpose. One of these is to present material which selec­ tively will be appropriate for a quarter or semester course in time series analysis and which will cover both the finite parameter and spectral approach. The second object is the presentation of topics of current research interest and some open questions. I mention these now. In particular, there is a discussion in Chapter III of the types of limit theorems that will imply asymptotic nor­ mality for covariance estimates and smoothings of the periodogram. This dis­ cussion allows one to get results on the asymptotic distribution of finite para­ meter estimates that are broader than those usually given in the literature in Chapter IV. A derivation of the asymptotic distribution for spectral (second order) estimates is given under an assumption of strong mixing in Chapter V. A discussion of higher order cumulant spectra and their large sample properties under appropriate moment conditions follows in Chapter VI. Probability density, conditional probability density and regression estimates are considered in Chapter VII under conditions of short range dependence. Chapter VIII deals with a number of topics. At first estimates for the structure function of a large class of non-Gaussian linear processes are constructed. One can determine much more about this structure or transfer function in the non-Gaussian case than one can for Gaussian processes. In particular, one can determine almost all the phase information.

One of the mathematical challenges of modern physics lies in the development of new tools to efficiently describe different branches of physics within one mathematical framework. This text introduces precisely such a broad mathematical model, one that gives a clear geometric expression of the symmetry of physical laws and is entirely determined by that symmetry.The first three chapters discuss the occurrence of bounded symmetric domains (BSDs) or homogeneous balls and their algebraic structure in physics. The book further provides a discussion of how to obtain a triple algebraic structure associated to an arbitrary BSD; the relation between the geometry of the domain and the algebraic structure is explored as well. The last chapter contains a classification of BSDs revealing the connection between the classical and the exceptional domains.

Man sollte weniger danach streben, die Grenzen der mathe- matischen Wissenschaften zu erweitern, als vielmehr danach, den bereits vorhandenen Stoff aus umfassenderen Gesichts- punkten zu betrachten - E. Study Today most mathematicians who know about Kronecker's theory of divisors know about it from having read Hermann Weyl's lectures on algebraic number theory [We], and regard it, as Weyl did, as an alternative to Dedekind's theory of ideals. Weyl's axiomatization of what he calls "Kronecker's" theory is built-as Dedekind's theory was built-around unique factor- ization. However, in presenting the theory in this way, Weyl overlooks one of Kronecker's most valuable ideas, namely, the idea that the objective of the theory is to define greatest com- mon divisors, not to achieve factorization into primes. The reason Kronecker gave greatest common divisors the primary role is simple: they are independent of the ambient field while factorization into primes is not.The very notion of primality depends on the field under consideration-a prime in one field may factor in a larger field-so if the theory is founded on factorization into primes, extension of the field entails a completely new theory. Greatest common divisors, on the other hand, can be defined in a manner that does not change at all when the field is extended (see 1.16). Only after he has laid the foundation of the theory of divisors does Kronecker consider factorization of divisors into divisors prime in some specified field.

Control theory represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems. Algebraic geometry may, in an elementary way, be viewed as the study of the structure and properties of the solutions of systems of algebraic equations. The aim of these notes is to provide access to the methods of algebraic geometry for engineers and applied scientists through the motivated context of control theory. I began the development of these notes over fifteen years ago with a series of lectures given to the Control Group at the Lund Institute of Technology in Sweden. Over the following years, I presented the material in courses at Brown several times and must express my appreciation for the feedback (sic!) received from the students. I have attempted throughout to strive for clarity, often making use of constructive methods and giving several proofs of a particular result. Since algebraic geometry draws on so many branches of mathematics and can be dauntingly abstract, it is not easy to convey its beauty and utility to those interested in applications. I hope at least to have stirred the reader to seek a deeper understanding of this beauty and utility in control theory. The first volume dea1s with the simplest control systems (i. e. single input, single output linear time-invariant systems) and with the simplest algebraic geometry (i. e. affine algebraic geometry).

1. The Subject Matter. Consider a complex semisimple Lie group G with Lie algebra g and Weyl group W. In this book, we present a geometric perspective on the following circle of ideas: polynomials The "vertices" of this graph are some of the most important objects in representation theory. Each has a theory in its own right, and each has had its own independent historical development. - A nilpotent orbit is an orbit of the adjoint action of G on g which contains the zero element of g in its closure. (For the special linear group 2 G = SL(n,C), whose Lie algebra 9 is all n x n matrices with trace zero, an adjoint orbit consists of all matrices with a given Jordan canonical form; such an orbit is nilpotent if the Jordan form has only zeros on the diagonal. In this case, the nilpotent orbits are classified by partitions of n, given by the sizes of the Jordan blocks.) The closures of the nilpotent orbits are singular in general, and understanding their singularities is an important problem. - The classification of irreducible Weyl group representations is quite old.

In 1974, the Societal Institute of the Mathematical Sciences (SIMS) initiated.a series of five-day Research Application Conferences (RAC's) at Alta, Utah, for the purpose of probing in depth societal fields in light of their receptivity to mathematical and statistical analysis. The first 10 conferences addressed ecosys­ tems, epidemiology, energy, environmental health, time series and ecological processes, energy and health, energy conversion and fluid mechanics, environ­ mental epidemiology: risk assessment, atomic bomb survivor data: utilization and analysis, and modem statistical methods in chronic disease epidemiology. These Proceedings are a result of the eleventh conference of Scientific Issues in Quantitative Cancer Risk Assessment, which was held in 1989 at Snowbird, Utah. For five days, 45 speakers and observers contributed their expertise in the relevant biology and statistics. The presentations were timely and the discussion was both enlightening and at times spirited. This volume hopefully presents a record that will be useful both now and in the future. Suresh H. Moolgavkar of the Fred Hutchinson Cancer Research Center (Seattle, WA) and the University of Washington (Seattle) chaired the Conference. The Conference was supported by the following organizations: The Dow Chemical Company Haskell Laboratory, E.1. duPont Health and Welfare Canada International Life Sciences Institute Monsanto Company National Cancer Institute Procter & Gamble Shell Oil Company U. S. Department of Energy U.S. Environmental Protection Agency U.S. Food and Drug Administration SIMS is grateful to the above organizations for their support of this Conference. D.L. Thomsen, Jr.

x system Ib-TEX. I wish to thank her for the beautiful work and the numerous discussions on the contents of this book. I am indebted to Peter Fassler, Neu-Technikum Buchs, Switzerland, for drafting the figures, to my students Kurt Rothermann and Stefan Strahl for computer enhancing and labeling the graphics, to Pascal Felder and Markus Wittwer for a simulation program that generated the figures in the stochastics sections. My thanks go to my new colleague at work, Daniel Neuenschwander, for the inspiring discussions related to the section in stochastics and for reading the manuscript to it. I am also grateful to Dacfey Dzung for reading the whole manuscript. Thanks go especially to Professor \Valter Gander of ETH, Zurich, who at the finishing stage and as an expert of 'JEXgenerously invested numerous hours to assist us in solving software as well as hardware problems; thanks go also to Martin Muller, Ingenieurschule Biel, who made the final layout of this book on the NeXT computer.Thanks are also due to Helmut Kopka of the Max Planck Institute, for solving software problems, and to Professor Burchard Kaup of the Uni- versity of Fribourg, Switzerland for adding some useful software; also to Birkhauser Boston Inc. for the pleasant co-operation. Finally, let me be reminiscent of Professor E. Stiefel (deceased 1978) with whom I had many interesting discussions and true co-operation when writing the book in German.

The need for improved mathematics education at the high school and college levels has never been more apparent than in the 1990's. As early as the 1960's, I.M. Gelfand and his colleagues in the USSR thought hard about this same question and developed a style for presenting basic mathematics in a clear and simple form that engaged the curiosity and intellectual interest of thousands of high school and college students. These same ideas, this same content, unchanged by over thirty years of experience and mathematical development, are available in the following books to any student who is willing to read, to be stimulated, and to learn. "Functions and Graphs" provides instruction in transferring formulas and data into geometrical form. Thus, drawing graphs is shown to be one way to "see" formulas and functions and to observe the ways in which they change. This skill is fundamental to the study of calculus and other mathematical topics. Teachers of mathematics will find here a fresh understanding of the subject and a valuable path to the training of students in mathematical concepts and skills.

The need for improved mathematics education at the high school and college levels has never been more apparent than in the 1990s. As early as the 1960s, I. M. Gel'fand and his colleagues in the USSR thought hard about this same question and developed a style for presenting basic mathematics in a clear and simple form that engaged the curiosity and intellectual interest of thousands of high school and college students. These same ideas, this same content, unchanged by over thirty years of experience and mathematical development, are available in the present books to any student who is willing to read, to be stimulated and to learn. "The Method of Coordinates" is a way of transferring geometric images into formulas, a method for describing pictures by numbers and letters denoting constants and variables. It is fundamental to the study of calculus and other mathematical topics. Teachers of mathematics will find here a fresh understanding of the subject and a valuable path to the training of students in mathematical concepts and skills.

This monograph studies the logical aspects of domains as used in de­ notational semantics of programming languages. Frameworks of domain logics are introduced; these serve as foundations for systematic derivations of proof systems from denotational semantics of programming languages. Any proof system so derived is guaranteed to agree with denotational se­ mantics in the sense that the denotation of any program coincides with the set of assertions true of it. The study focuses on two categories for dena­ tational semantics: SFP domains, and the less standard, but important, category of stable domains. The intended readership of this monograph includes researchers and graduate students interested in the relation between semantics of program­ ming languages and formal means of reasoning about programs. A basic knowledge of denotational semantics, mathematical logic, general topology, and category theory is helpful for a full understanding of the material. Part I SFP Domains Chapter 1 Introduction This chapter provides a brief exposition to domain theory, denotational se­ mantics, program logics, and proof systems. It discusses the importance of ideas and results on logic and topology to the understanding of the relation between denotational semantics and program logics. It also describes the motivation for the work presented by this monograph, and how that work fits into a more general program. Finally, it gives a short summary of the results of each chapter. 1. 1 Domain Theory Programming languages are languages with which to perform computa­ tion.

In this monograph we study two generalizations of standard unification, E-unification and higher-order unification, using an abstract approach orig­ inated by Herbrand and developed in the case of standard first-order unifi­ cation by Martelli and Montanari. The formalism presents the unification computation as a set of non-deterministic transformation rules for con­ verting a set of equations to be unified into an explicit representation of a unifier (if such exists). This provides an abstract and mathematically elegant means of analysing the properties of unification in various settings by providing a clean separation of the logical issues from the specification of procedural information, and amounts to a set of 'inference rules' for unification, hence the title of this book. We derive the set of transformations for general E-unification and higher­ order unification from an analysis of the sense in which terms are 'the same' after application of a unifying substitution. In both cases, this results in a simple extension of the set of basic transformations given by Herbrand­ Martelli-Montanari for standard unification, and shows clearly the basic relationships of the fundamental operations necessary in each case, and thus the underlying structure of the most important classes of term unifi­ cation problems.

Arnold Sommerfeld's "Mathematische Theorie der Diffraction" marks a milestone in optical theory, full of insights that are still relevant today. In a stunning tour de force, Sommerfeld derives the first mathematically rigorous solution of an optical diffraction problem. Indeed, his diffraction analysis is a surprisingly rich and complex mix of pure and applied mathematics, and his often-cited diffraction solution is presented only as an application of a much more general set of mathematical results. The body of Sommerfeld's work is devoted to the systematic development of a method for deriving solutions of the wave equation on Riemann surfaces, a fascinating but perhaps underappreciated topic in mathematical physics.

Thismonographdealswiththeexistenceofperiodicmotionsof Lagrangiansystemswith ndegreesoffreedom ij + V'(q) =0, where Visasingularpotential.Aprototypeofsuchaproblem, evenifitisnottheonlyphysicallyinterestingone,istheKepler problem .. q 0 q+yqr= . This,jointlywiththemoregeneralN-bodyproblem,hasalways beentheobjectofagreatdealofresearch.Mostofthoseresults arebasedonperturbationmethods,andmakeuseofthespecific featuresoftheKeplerpotential. OurapproachismoreonthelinesofNonlinearFunctional Analysis:ourmainpurposeistogiveafunctionalframefor systemswithsingularpotentials,includingtheKeplerandthe N-bodyproblemasparticularcases.PreciselyweuseCritical PointTheorytoobtainexistenceresults,qualitativeinnature, whichholdtrueforbroadclassesofpotentials.Thishighlights thatthevariationalmethods,whichhavebeenemployedtoob- tainimportantadvancesinthestudyofregularHamiltonian systems,canbesuccessfallyusedtohandlesingularpotentials aswell. Theresearchonthistopicisstillinevolution,andtherefore theresultswewillpresentarenottobeintendedasthefinal ones. Indeedamajorpurposeofourdiscussionistopresent methodsandtoolswhichhavebeenusedinstudyingsuchprob- lems.Vlll PREFACE Partofthematerialofthisvolumehasbeenpresentedina seriesoflecturesgivenbytheauthorsatSISSA,Trieste,whom wewouldliketothankfortheirhospitalityandsupport. We wishalsotothankUgoBessi,PaoloCaldiroli,FabioGiannoni, LouisJeanjean,LorenzoPisani,EnricoSerra,KazunakaTanaka, EnzoVitillaroforhelpfulsuggestions. May26,1993 Notation n 1.For x, yE IR , x. ydenotestheEuclideanScalarproduct, and IxltheEuclideannorm. 2. meas(A)denotestheLebesguemeasureofthesubset Aof n IR * 3.Wedenoteby ST =[0,T]/{a,T}theunitarycirclepara- metrizedby t E[0,T].Wewillalsowrite SI= ST=I. n 1 n 4.Wewillwrite sn = {xE IR + : Ixl =I}andn = IR \{O}. n 5.Wedenoteby LP([O,T], IR ),1~ p~+00,theLebesgue spaces,equippedwiththestandardnorm lIulip. l n l n 6. H (ST, IR )denotestheSobolevspaceof u E H ,2(0, T; IR ) suchthat u(O) = u(T).Thenormin HIwillbedenoted by lIull2 = lIull~ + lIull~* 7.Wedenoteby(*1*)and11*11respectivelythescalarproduct andthenormoftheHilbertspace E. 8.For uE E, EHilbertorBanachspace,wedenotetheball ofcenter uandradiusrby B(u,r) = {vE E: lIu- vii~ r}.Wewillalsowrite B = B(O, r). r 1 1 9.WesetA (n) = {uE H (St,n)}. k 10.For VE C (1Rxil,IR)wedenoteby V'(t, x)thegradient of Vwithrespectto x. l 11.Given f E C (M,IR), MHilbertmanifold,welet r = {uEM: f(u) ~ a}, f-l(a,b) = {uE E : a~ f(u) ~ b}.x NOTATION 12.Given f E C1(M,JR), MHilbertmanifold,wewilldenote by Zthesetofcriticalpointsof fon Mandby Zctheset Z U f-l(c, c). 13.Givenasequence UnE E, EHilbertspace,by Un ---" Uwe willmeanthatthesequence Unconvergesweaklyto u. 14.With GBP(E)wewilldenotethesetoflinearandcontinuous operatorson E. 15.With Ck"(A,JR)wewilldenotethesetoffunctions ffrom AtoJR, ktimesdifferentiablewhosek-derivativeisHolder continuousofexponent0:. Main Assumptions Wecollecthere,forthereader'sconvenience,themainassump- tionsonthepotential Vusedthroughoutthebook. (VO) VEC1(lRXO,lR),V(t+T,x)=V(t,X) V(t,x)ElRXO, (VI) V(t,x)

This monograph is a slightly revised version of my PhD thesis [86], com­ pleted in the Department of Computer Science at the University of Edin­ burgh in June 1988, with an additional chapter summarising more recent developments. Some of the material has appeared in the form of papers [50,88]. The underlying theme of the monograph is the study of two classical problems: counting the elements of a finite set of combinatorial structures, and generating them uniformly at random. In their exact form, these prob­ lems appear to be intractable for many important structures, so interest has focused on finding efficient randomised algorithms that solve them ap­ proxim~ly, with a small probability of error. For most natural structures the two problems are intimately connected at this level of approximation, so it is natural to study them together. At the heart of the monograph is a single algorithmic paradigm: sim­ ulate a Markov chain whose states are combinatorial structures and which converges to a known probability distribution over them. This technique has applications not only in combinatorial counting and generation, but also in several other areas such as statistical physics and combinatorial optimi­ sation. The efficiency of the technique in any application depends crucially on the rate of convergence of the Markov chain.

Recently, a variety ofresults on the complexitystatusofthegraph isomorphism problem has been obtained. These results belong to the so-called structural part of Complexity Theory. Our idea behind this book is to summarize such results which might otherwise not be easily accessible in the literature, and also, to give the reader an understanding of the aims and topics in Structural Complexity Theory, in general. The text is basically self contained; the only prerequisite for reading it is some elementary knowledge from Complexity Theory and Probability Theory. It can be used to teach a seminar or a monographic graduate course, but also parts of it (especially Chapter 1) provide a source of examples for a standard graduate course on Complexity Theory. Many people have helped us in different ways III the process of writing this book. Especially, we would like to thank V. Arvind, R.V. Book, E. May­ ordomo, and the referee who gave very constructive comments. This book project was especially made possible by a DAAD grant in the "Acciones In­ tegrada" program. The third author has been supported by the ESPRIT project ALCOM-II.

Recent years have witnessed a surge of activity in the field of dynamic games, in both theory and applicati( s. Th well as practical problems in zero-sum and nonzero-sum games, continuous time differential games and discrete time multistage games, and deterministic and stochastic games are currently being investigated by researchers in diverse disciplines, such as engineering, mathematics, biology, economics, management science, and political science. 'Mis surge of interest led, in 1990, to the formation of the International Society of Dynamic Gaines (ISDG), whose primary goal is to foster the development of advanced research and applications in the field of game theory. One important activity of the Society is to hold a biannual, intemational symposium to bring together researchers who contribute to the development of this active field of applied science. In 1992 such a symposium was held in Grimentz, Switzerland. This book, which is the first volume in the new series, Annals of Dynamic Gaines, is based on selected presentations made at this symposium. It is, however, not simply a book of proceedings for a conference.EveTy paper that appears in this volume has passed through a selective refereeing proem, as in an archival technical joumal. This makes this first volume of the Annals of Dynamic Games a quality publication that presents a timely account of the state-of-the-all in this dynamic field. The papers included in this volume attest to the vitality and diversity of ongoing research in dynamic games and applications. The reader wfll fmd here important contributions in the following five areas, which also constitute the groups according to which the twenty-three chapters have been organized: Robust control design and H'-theory; pursuit-evasion games and numerical schemes; numerical solutions of discrete time games, based on mathematical programming techniques; stochastic differential, sequential and Markov games; and applications in ecology, environmental management, and biology. With such diversity of topics, the book should be a valuable resource for researchers in the field of dynamic games and all neighboring disciplines.

Hamilton-Jacobi equations and other types of partial differential equa- tions of the first order are dealt with in many branches of mathematics, mechanics, and physics. These equations are usually nonlinear, and func- tions vital for the considered problems are not smooth enough to satisfy these equations in the classical sense. An example of such a situation can be provided by the value function of a differential game or an optimal control problem. It is known that at the points of differentiability this function satisfies the corresponding Hamilton-Jacobi-Isaacs-Bellman equation. On the other hand, it is well known that the value function is as a rule not everywhere differentiable and therefore is not a classical global solution. Thus in this case, as in many others where first-order PDE's are used, there arises necessity to introduce a notion of generalized solution and to develop theory and methods for constructing these solutions.In the 50s-70s, problems that involve nonsmooth solutions of first- order PDE's were considered by Bakhvalov, Evans, Fleming, Gel'fand, Godunov, Hopf, Kuznetzov, Ladyzhenskaya, Lax, Oleinik, Rozhdestven- ski1, Samarskii, Tikhonov, and other mathematicians. Among the inves- tigations of this period we should mention the results of S.N. Kruzhkov, which were obtained for Hamilton-Jacobi equation with convex Hamilto- nian. A review of the investigations of this period is beyond the limits of the present book. A sufficiently complete bibliography can be found in [58, 126, 128, 141].

This text's key issue is the role of a Riemannian curvature in studies of manifolds and submanifolds with foliations. The results of many geometers are discussed, but this book principally focuses on the Riemannian geometry of manifolds and submanifolds with generators that have non-negative curvature, the main idea being that such manifolds decompose into a direct product when the dimension of leaves is sufficiently large.

Overview This book is intended as a textbook in probability for graduate students in math­ ematics and related areas such as statistics, economics, physics, and operations research. Probability theory is a 'difficult' but productive marriage of mathemat­ ical abstraction and everyday intuition, and we have attempted to exhibit this fact. Thus we may appear at times to be obsessively careful in our presentation of the material, but our experience has shown that many students find them­ selves quite handicapped because they have never properly come to grips with the subtleties of the definitions and mathematical structures that form the foun­ dation of the field. Also, students may find many of the examples and problems to be computationally challenging, but it is our belief that one of the fascinat­ ing aspects of prob ability theory is its ability to say something concrete about the world around us, and we have done our best to coax the student into doing explicit calculations, often in the context of apparently elementary models. The practical applications of probability theory to various scientific fields are far-reaching, and a specialized treatment would be required to do justice to the interrelations between prob ability and any one of these areas. However, to give the reader a taste of the possibilities, we have included some examples, particularly from the field of statistics, such as order statistics, Dirichlet distri­ butions, and minimum variance unbiased estimation.

This book presents a generalized state-space theory for the analysis and synthesis of finite horizon suboptimal H...controllers. Expressions for a suboptimal controller are derived in a general setting and an approximate solution to the H...performance robustness problem is proposed. Chapter 1 treats nonlinear optimal control problems in which the cost functional is of the form of a quotient or a product of powers of definite integrals. Chapter 2 gives a criterion for the evaluation of the infimal H...norm in the finite horizon case. A general suboptimal control problem is then posed, and an expression for a suboptimal H...state feedback controller is derived. Chapter 3 develops expressions for a suboptimal H...output feedback controller in a very general case via the solution of two dynamical Riccati equations. Assuming the adequacy of linear expressions, chapter 4 gives an iterative procedure for the synthesis of a suboptimal H...controller which yields the required performance even under parameter variations. Chapter 5 treats a general minimization problem in which the cost functional is a quotient of definite integrals.The results are useful in computing the actual performance of a given controller. In chapter 6, a design of the F/A-18A automatic carrier landing system is accomplished using finite horizon H...techniques. The book is based mainly on the author's research papers and presents a very practical point of view toward the applications. It can be used as a one semester or one quarter graduate level course in electrical, mechanical or chemical engineering.

"What good is a newborn baby?" Michael Faraday's reputed response when asked, "What good is magnetic induction?" But, it must be admitted that a newborn baby may die in infancy. What about this one- the idea of transfiniteness for graphs, electrical networks, and random walks? At least its bloodline is robust. Those subjects, along with Cantor's transfinite numbers, comprise its ancestry. There seems to be general agreement that the theory of graphs was born when Leonhard Euler published his solution to the "Konigsberg bridge prob­ lem" in 1736 [8]. Similarly, the year of birth for electrical network theory might well be taken to be 184 7, when Gustav Kirchhoff published his volt­ age and current laws [ 14]. Ever since those dates until just a few years ago, all infinite undirected graphs and networks had an inviolate property: Two branches either were connected through a finite path or were not connected at all. The idea of two branches being connected only through transfinite paths,that is, only through paths having infinitely many branches was never invoked, or so it appears from a perusal of various surveys of infinite graphs [17], [20], [29], [32]. Our objective herein is to explore this idea and some of its ramifications. It should be noted however that directed graphs having transfinite paths have appeared in set theory [6, Section 4.