Bøger udgivet af Birkhauser Boston

To commemorate properly the 70th birthday of a man who, by his very nature, is too busy to pause for any kind of ceremonial event unless it has a concomitant functional output was a difficult problem for the Staff and Associates of the Neurosciences Research Program. Frank (F. O. S. ) has always dreaded the prospect that sometime it might be appropriate for his colleagues to present him a Fest­ schrift. In fact, "Fest me no Schriften" became his battle cry, expressing his feeling that the idea of testimonials clustered into a book was anathema. So the "break­ through" idea for the planners was to organize a symposium around the theme of discovery in neuroscience that would be valuable scientifically and, in its demon­ stration of interdisciplinary interaction, would support that emphasis in Frank's career. After much planning a program was developed, beginning with a birthday party the evening before, followed by the two-day symposium, and closing with the first F. O. Schmitt Lecture in Neuroscience. We hope that publication of the scientific proceedings in this volume will be of interest not only to the neuroscience community, but also to a broad general readership interested in discovery, under­ standing, and the creative processes in scientific work. An organizing committee, chaired by Fred Worden, collected advice and guidance leading to the selection of speakers whose scientific careers have played an important part in the recent history of modern neuroscience.

The 1990 Seminar on Stochastic Processes was held at the University of British Columbia from May 10 through May 12, 1990. This was the tenth in a series of annual meetings which provide researchers with the opportunity to discuss current work on stochastic processes in an informal and enjoyable atmosphere. Previous seminars were held at Northwestern University, Princeton University, the Univer- sity of Florida, the University of Virginia and the University of California, San Diego. Following the successful format of previous years, there were five invited lectures, delivered by M. Marcus, M. Vor, D. Nualart, M. Freidlin and L. C. G. Rogers, with the remainder of the time being devoted to informal communications and workshops on current work and problems. The enthusiasm and interest of the participants created a lively and stimulating atmosphere for the seminar. A sample of the research discussed there is contained in this volume. The 1990 Seminar was made possible by the support of the Natural Sciences and Engin~ring Research Council of Canada, the Southwest University Mathematics Society of British Columbia, and the University of British Columbia. To these entities and the organizers of this year's conference, Ed Perkins and John Walsh, we extend oul' thanks. Finally, we acknowledge the support and assistance of the staff at Birkhauser Boston.

A so-called "effective" algorithm may require arbitrarily large finite amounts of time and space resources, and hence may not be practical in the real world. A "feasible" algorithm is one which only requires a limited amount of space and/or time for execution; the general idea is that a feasible algorithm is one which may be practical on today's or at least tomorrow's computers. There is no definitive analogue of Church's thesis giving a mathematical definition of feasibility; however, the most widely studied mathematical model of feasible computability is polynomial-time computability. Feasible Mathematics includes both the study of feasible computation from a mathematical and logical point of view and the reworking of traditional mathematics from the point of view of feasible computation. The diversity of Feasible Mathematics is illustrated by the. contents of this volume which includes papers on weak fragments of arithmetic, on higher type functionals, on bounded linear logic, on sub recursive definitions of complexity classes, on finite model theory, on models of feasible computation for real numbers, on vector spaces and on recursion theory. The vVorkshop on Feasible Mathematics was sponsored by the Mathematical Sciences Institute and was held at Cornell University, June 26-28, 1989.

1. 1 The problem and the approach The model developed here, which is actually more a collection of com- ponents than a single monolithic structure, traces a path from relatively low-level neural/connectionistic structures and processes to relatively high-level animal/artificial intelligence behaviors. Incremental extension of this initial path permits increasingly sophisticated representation and processing strategies, and consequently increasingly sophisticated behavior. The initial chapters develop the basic components of the sys- tem at the node and network level, with the general goal of efficient category learning and representation. The later chapters are more con- cerned with the problems of assembling sequences of actions in order to achieve a given goal state. The model is referred to as connectionistic rather than neural, be- cause, while the basic components are neuron-like, there is only limited commitment to physiological realism. Consequently the neuron-like ele- ments are referred to as "e;nodes"e; rather than "e;neurons"e;. The model is directed more at the behavioral level, and at that level, numerous con- cepts from animal learning theory are directly applicable to connectionis- tic modeling. An attempt to actually implement these behavioral theories in a computer simulation can be quite informative, as most are only partially specified, and the gaps may be apparent only when actual- ly building a functioning system. In addition, a computer implementa- tion provides an improved capability to explore the strengths and limita- tions of the different approaches as well as their various interactions.

Dr. A. D. Morris had a long interest in, and great familiarity with, the life and times of James Parkinson (1755-1824). He was an avid collector of material related to Parkinson, some of which he communicated to medi· cal and historical groups, and which he also incorporated into publica· tions, especially his admirable work, The Hoxton Madhouses. When Dr. Morris died, in 1980, he left behind a large typescript devoted to Parkinson's life. It was single·minded in its dedication to primary texts, quoting liberally from the whole range of Parkinson's writings. This was particularly valuable since so many of Parkinson's publications were tracts, pamphlets, or occasional pieces which are now very scarce. A copy of the entire manuscript has been deposited in the Library of the Well· come Institute for the History of Medicine in London, where it may be consulted. The length of the manuscript made publication of the whole impossible, especially since it would have had to include the facsimile reproduction of Morris's The Hoxton Madhouses.

The five stories in this book are tales about human beings and the human condition in which they find themselves. They are stories of scientists - but not of white-coated laboratory figures, happy to leave to others the practical application of their discoveries. In the circumstances I recount, the scientists were brought face to face, sometimes in dramatic confrontations, with the very people whose problems their work might help solve. As I came to realize, there now exists an international network of unusual scientists whose members are concerned individuals, deter- mined that their scientific work should help alleviate the human condition. This book was conceived as an account of some exciting epi- sodes in contemporary biomedicine. But during the four years it took to complete, several other themes emerged. First, each story illustrates aspects of the relation between Western science and technology and those major health problems which are often of dreadful significance for the Third World. In this relation the fruits of Western research are not simply applied to global health problems. Rather the relation is reciprocal, for scientific research, whether prompted by the medical problems of the Third World or actually conducted there, is yielding vital clues to many fundamental aspects of human biology, as well as pointing toward possible therapies for the serious diseases of Western society, such as cancer or the dementi as of old age. After I had written the first draft a second theme emerged.

STANISLAW MARCIN ULAM, or Stan as his friends called him, was one of those great creative mathematicians whose interests ranged not only over all fields of mathematics, but over the physical and biological sciences as well. Like his good friend "e;Johnny"e; von Neumann, and unlike so many of his peers, Ulam is unclassifiable as a pure or applied mathematician. He never ceased to find as much beauty and excitement in the applications of mathematics as in working in those rarefied regions where there is a total un- concern with practical problems. In his Adventures of a Mathematician Ulam recalls playing on an oriental carpet when he was four. The curious patterns fascinated him. When his father smiled, Ulam remembers thinking: "e;He smiles because he thinks I am childish, but I know these are curious patterns. I know something my father does not know."e; The incident goes to the heart of Ulam's genius. He could see quickly, in flashes of brilliant insight, curious patterns that other mathematicians could not see. "e;I am the type that likes to start new things rather than improve or elaborate,"e; he wrote. "e;I cannot claim that I know much of the technical material of mathematics.

. . . And indeed, these latest centuries merit praise because it is during them that the arts and sciences, discovered Iry the ancients, have been reduced to so great and constantlY increasing perfec­ tion through the investigations and experiments of clear-seeing minds. This development is particularlY evident in the case of the mathematical sciences. Here, without mentioning various men who have achieved success, we must without hesitation and with the unanimous approval of scholars assign the first place to Gali­ leo Galilei, Member of the Acadmry of the Lincei. This he de­ serves not onlY because he has effectivelY demonstrated faUacies in many of our current conclusions, as is amplY shown Iry his published works, but also Iry means of the telescope (invented in this country but greatlY perfected Iry him) he has discovered the four sateOites of Jupiter, has shown us the true character of the Milky wa~ and has made us acquainted with spots on the Sun, with the rough and cloudy portions of the lunar surface, with the threefold nature of Saturn, with the phases of Venus and with the physical character of comets. These matters were entirelY unknown to the ancient astronomers and philosophers; so that we may trulY say that he has restored to the world the science of astronomy and has presented it in a new light.

by Gian-Carlo Rota The subjects of mathematics, like the subjects of mankind, have finite lifespans, which the historian will record as he freezes history at one instant of time. There are the old subjects, loaded with distinctions and honors. As their problems are solved away and the applications reaped by engineers and other moneymen, ponderous treatises gather dust in library basements, awaiting the day when a generation as yet unborn will rediscover the lost paradise in awe. Then there are the middle-aged subjects. You can tell which they are by roaming the halls of Ivy League universities or the Institute for Advanced Studies. Their high priests haughtily refuse fabulous offers from eager provin­ cial universities while receiving special permission from the President of France to lecture in English at the College de France. Little do they know that the load of technicalities is already critical, about to crack and submerge their theorems in the dust of oblivion that once enveloped the dinosaurs. Finally, there are the young subjects-combinatorics, for instance. Wild­ eyed individuals gingerly pick from a mountain of intractable problems, chil­ dishly babbling the first words of what will soon be a new language. Child­ hood will end with the first Seminaire Bourbaki. It could be impossible to find a more fitting example than matroid theory of a subject now in its infancy. The telltale signs, for an unfailing diagnosis, are the abundance of deep theorems, going together with a paucity of theories.

The theory of General Relativity, after its invention by Albert Einstein, remained for many years a monument of mathemati- cal speculation, striking in its ambition and its formal beauty, but quite separated from the main stream of modern Physics, which had centered, after the early twenties, on quantum mechanics and its applications. In the last ten or fifteen years, however, the situation has changed radically. First, a great deal of significant exper~en- tal data became available. Then important contributions were made to the incorporation of general relativity into the framework of quantum theory. Finally, in the last three years, exciting devel- opments took place which have placed general relativity, and all the concepts behind it, at the center of our understanding of par- ticle physics and quantum field theory. Firstly, this is due to the fact that general relativity is really the "e;original non-abe- lian gauge theory,"e; and that our description of quantum field in- teractions makes extensive use of the concept of gauge invariance. Secondly, the ideas of supersymmetry have enabled theoreticians to combine gravity with other elementary particle interactions, and to construct what is perhaps the first approach to a more finite quantum theory of gravitation, which is known as super- gravity.

This volume consists of about half of the papers presented during a three-day seminar on stochastic processes held at Northwestern University in March 1982. This was the second of such yearly seminars aimed at bringing together a small group of researchers to discuss their current work in an informal atmosphere. The invited participants in this year's seminar were B. ATKINSON, R. BASS, K. BICHTELER, D. BURKHOLDER, K.L. CHUNG, J.L. DOOB, C. DOLEANS-DADE, H. FOLLMER, R.K. GETOOR, J. GLOVER, J. MITRO, D. MONRAD, E. PERKINS, J. PITMAN, Z. POP-STOJANOVIC, M.J. SHARPE, and J. WALSH. We thank them and the other participants for the lively atmosphere of the seminar. As mentioned above, the present volume is only a fragment of the work discussed at the seminar, the other work having been committed to other publications. The seminar was made possible through the enlightened support of the Air Force Office of Scientific Research, Grant No. 80-0252A. We are grateful to them as well as the publisher, Birkhauser, Boston, for their support and encouragement. E.C. , Evanston, 1983 Seminar on stochastic Processes, 1982 Birkhauser, Boston, 1983 GERM FIELDS AND A CONVERSE TO THE STRONG MARKOV PROPERTY by BRUCE W. ATKINSON 1. Introduction The purpose of this paper is to give an intrinsic characterization of optional (i.e., stopping) times for the general germ Markov process, which includes the general right process as a special case. We proceed from the general to the specific.

In many scientific or engineering applications, where ordinary differen- tial equation (OOE),partial differential equation (POE), or integral equation (IE) models are involved, numerical simulation is in common use for prediction, monitoring, or control purposes. In many cases, however, successful simulation of a process must be preceded by the solution of the so-called inverse problem, which is usually more complex: given meas- ured data and an associated theoretical model, determine unknown para- meters in that model (or unknown functions to be parametrized) in such a way that some measure of the "e;discrepancy"e; between data and model is minimal. The present volume deals with the numerical treatment of such inverse probelms in fields of application like chemistry (Chap. 2,3,4, 7,9), molecular biology (Chap. 22), physics (Chap. 8,11,20), geophysics (Chap. 10,19), astronomy (Chap. 5), reservoir simulation (Chap. 15,16), elctrocardiology (Chap. 14), computer tomography (Chap. 21), and control system design (Chap. 12,13). In the actual computational solution of inverse problems in these fields, the following typical difficulties arise: (1) The evaluation of the sen- sitivity coefficients for the model. may be rather time and storage con- suming. Nevertheless these coefficients are needed (a) to ensure (local) uniqueness of the solution, (b) to estimate the accuracy of the obtained approximation of the solution, (c) to speed up the iterative solution of nonlinear problems. (2) Often the inverse problems are ill-posed. To cope with this fact in the presence of noisy or incomplete data or inev- itable discretization errors, regularization techniques are necessary.

Quite apart from the fact that percolation theory had its orlgln in an honest applied problem (see Hammersley and Welsh (1980)), it is a source of fascinating problems of the best kind a mathematician can wish for: problems which are easy to state with a minimum of preparation, but whose solutions are (apparently) difficult and require new methods. At the same time many of the problems are of interest to or proposed by statistical physicists and not dreamt up merely to demons~te ingenuity. Progress in the field has been slow. Relatively few results have been established rigorously, despite the rapidly growing literature with variations and extensions of the basic model, conjectures, plausibility arguments and results of simulations. It is my aim to treat here some basic results with rigorous proofs. This is in the first place a research monograph, but there are few prerequisites; one term of any standard graduate course in probability should be more than enough. Much of the material is quite recent or new, and many of the proofs are still clumsy. Especially the attempt to give proofs valid for as many graphs as possible led to more complications than expected. I hope that the Applications and Examples provide justifi- cation for going to this level of generality.

Branching processes form one of the classical fields of applied probability and are still an active area of research. The field has by now grown so large and diverse that a complete and unified treat- ment is hardly possible anymore, let alone in one volume. So, our aim here has been to single out some of the more recent developments and to present them with sufficient background material to obtain a largely self-contained treatment intended to supplement previous mo- nographs rather than to overlap them. The body of the text is divided into four parts, each of its own flavor. Part A is a short introduction, stressing examples and applications. In Part B we give a self-contained and up-to-date pre- sentation of the classical limit theory of simple branching processes, viz. the Gal ton-Watson ( Bienayme-G-W) process and i ts continuous time analogue. Part C deals with the limit theory of Il!arkov branching processes with a general set of types under conditions tailored to (multigroup) branching diffusions on bounded domains, a setting which also covers the ordinary multitype case. Whereas the point of view in Parts A and B is quite pedagogical, the aim of Part C is to treat a large subfield to the highest degree of generality and completeness possi"e;ble. Thus the exposition there is at times quite technical.

In a remarkably short time, the field of inequality problems has seen considerable development in mathematics and theoretical mechanics. Applied mechanics and the engineering sciences have also benefitted from these developments in that open problems have been treated and entirely new classes of problems have been formulated and solved. This book is an outgrowth of seven years of seminars and courses on inequality problems in mechanics for a variety of audiences in the Technical University of Aachen, the Aristotle University of Thessaloniki, the University of Hamburg and the Technical University of Milan. The book is intended for a variety of readers, mathematicians and engineers alike, as is detailed in the Guidelines for the Reader. It goes without saying that the work of G. Fichera, J. L. Lions, G. Maier, J. J. Moreau in originating and developing the theory of inequality problems has considerably influenced the present book. I also wish to acknowledge the helpful comments received from C. Bisbos, J. Haslinger, B. Kawohl, H. Matthies, H. O. May, D. Talaslidis and B. Werner. Credit is also due to G. Kyriakopoulos and T. Mandopoulou for their exceptionally diligent work in the preparation of the fmal figures. Many thanks are also due to T. Finnegan and J. Gateley for their friendly assistance from the linguistic standpoint. I would also like to thank my editors in Birkhiiuser Verlag for their cooperation, and all those who helped in the preparation of the manuscript.

~his Monograph has two objectives : to analyze a f inite e l e m en t m e th o d useful for solving a large class of t hi n shell prob l e ms, and to show in practice how to use this method to simulate an arch dam prob lem. The first objective is developed in Part I. We record the defi- tion of a general thin shell model corresponding to the W.T. KOlTER linear equations and we show the existence and the uniqueness for a solution. By using a co nform ing fi nite e l e m ent me t hod , we associate a family of discrete problems to the continuous problem ; prove the convergence of the method ; and obtain error estimates between exact and approximate solutions. We then describe the impl em enta t ion of some specific conforming methods. The second objective is developed in Part 2. It consists of applying these finite element methods in the case of a representative practical situation that is an arc h dam pro b le m. This kind of problem is still of great interest, since hydroelectric plants permit the rapid increase of electricity production during the day hours of heavy consumption. This regulation requires construction of new hydroelectric plants on suitable sites, as well as permanent control of existing dams that may be enlightened by numerical stress analysis .

The material discussed in this monograph should be accessible to upper level undergraduates in the mathemati­ cal sciences. Formal prerequisites include a solid intro­ duction to calculus and one semester of probability. Although differential equations are employed, these are all linear, constant coefficient, ordinary differential equa­ tions which are solved either by separation of variables or by introduction of an integrating factor. These techniques can be taught in a few minutes to students who have studied calculus. The models developed to describe an epidemic outbreak of smallpox are standard stochastic processes (birth-death, random walk and branching processes). While it would be helpful for students to have seen these prior to their introduction in this monograph, it is certainly not necessary. The stochastic processes are developed from first principles and then solved using elementary tech­ niques. Since all that turns out to be necessary are ex­ pected values of random variables, the differential-differ­ ence equatlon descriptions of the stochastic processes are reduced to ordinary differential equations before being solved. Students who have studied stochastic processes are generally pleased to learn that different formulations are possible for the same set of conditions. The choice of which formulation to employ depends upon what one wishes to calculate. Specifically, in Section 6 a birth-death pro­ cess is replaced by a random walk and in Section 7 a prob­ lem is formulated both as a multi-birth-death process and as a branching process.

During the last few decades historians of science have shown a growing interest in science as a cultural activity and have regarded science more and more as part of the gene- ral developments that have occurred in society. This trend has been less evident arnong historians of mathematics, who traditionally concentrate primarily on tracing the develop- ment of mathematical knowledge itself. To some degree this restriction is connected with the special role of mathematics compared with the other sciences; mathematics typifies the most objective, most coercive type of knowledge, and there- fore seems to be least affected by social influences. Nevertheless, biography, institutional history and his- tory of national developments have long been elements in the historiography of mathematics. This interest in the social aspects of mathematics has widened recently through the stu- dy of other themes, such as the relation of mathematics to the development of the educational system. Some scholars have begun to apply the methods of historical sociology of knowledge to mathematics; others have attempted to give a ix x Marxist analysis of the connection between mathematics and productive forces, and there have been philosophical studies about the communication processes involved in the production of mathematical knowledge. An interest in causal analyses of historical processes has led to the study of other factors influencing the development of mathematics, such as the f- mation of mathematical schools, the changes in the profes- onal situation of the mathematician and the general cultural milieu of the mathematical scientist.