This book is devoted to algebraic-geometric codes - the recently emerged area which combines algebraic geometry and coding theory. Its interdisciplinary nature - which makes it so beautiful - also leads to some difficulties for those who want to master it. We try to help all those coming in touch with this field, from students to specialists. starting from essentially first principles, the text leads the reader to the most recent published and unpublished results. We do not assume any initial knowledge either of coding theory or of algebraic geometry.

I first learned the theory of distributions from Professor Ebbe Thue Poulsen in an undergraduate course at Aarhus University. Both his lectures and the textbook, Topological Vector Spaces, Distributions and Kernels by F. Treves, used in the course, opened my eyes to the beauty and abstract simplicity of the theory. However my incomplete study of many branches of classical analysis left me with the question: Why is the theory of distributions important? In my continued studies this question was gradually answered, but my growing interest in the history of mathematics caused me to alter my question to other questions such as: For what purpose, if any, was the theory of distributions originally created? Who invented distributions and when? I quickly found answers to the last two questions: distributions were invented by S. Sobolev and L. Schwartz around 1936 and 1950, respectively. Knowing this answer, however, only created a new question: Did Sobolev and Schwartz construct distributions from scratch or were there earlier trends and, if so, what were they? It is this question, concerning the pre­ history of the theory of distributions, which I attempt to answer in this book. Most of my research took place at the History of Science Department of Aarhus University. I wish to thank this department for its financial and intellectual support. I am especially grateful to Lektors Kirsti Andersen from the History of Science Department and Lars Mejlbo from the Mathematics Department, for their kindness, constructive criticism, and encouragement.

Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non-
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.

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

In the preface to Volume One I promised a second volume which would contain the theory of linear mappings and special classes of spaces im­ portant in analysis. It took me nearly twenty years to fulfill this promise, at least to some extent. To the six chapters of Volume One I added two new chapters, one on linear mappings and duality (Chapter Seven), the second on spaces of linear mappings (Chapter Eight). A glance at the Contents and the short introductions to the two new chapters will give a fair impression of the material included in this volume. I regret that I had to give up my intention to write a third chapter on nuclear spaces. It seemed impossible to include the recent deep results in this field without creating a great further delay. A substantial part of this book grew out of lectures I held at the Mathematics Department of the University of Maryland· during the academic years 1963-1964, 1967-1968, and 1971-1972. I would like to express my gratitude to my colleagues J. BRACE, S. GOLDBERG, J. HORVATH, and G. MALTESE for many stimulating and helpful discussions during these years. I am particularly indebted to H. JARCHOW (Ziirich) and D. KEIM (Frankfurt) for many suggestions and corrections. Both have read the whole manuscript. N. ADASCH (Frankfurt), V. EBERHARDT (Miinchen), H. MEISE (Diisseldorf), and R. HOLLSTEIN (Paderborn) helped with important observations.

It has been my experience in teaching graduate and undergraduate courses that if the students are conversant with the pertinent mathematical proce­ dures, and can "think mathematically," there is almost no limit to their comprehension. Most courses that are considered difficult by students are either poorly taught or require a degree of mathematical sophistication that the students do not possess. In Transport Analysis, J have culled some basic momentum transport (fluid flow) and mass transport phenomena and explicitly revealed the derivation of the governing equations. There is no mystery, no omitted steps or "it can be shown" phrases that are usually the bane of the student. There are chapters that review basic calculus, vector and matrix concepts, Laplace transform operations, and finite difference calculus. Ordinary dif­ ferential and partial differential equations are derived and solved. This book is intended for undergraduates and graduate students in engineering, chemistry, physics, and even biology and medicine. It is also intended for my non-engineering colleagues with whom I have collaborated during our cooperative research in the life sciences. If they knew what is contained in Transport Analysis, they probably wouldn't need me. v Acknowledgments To Barbara and Michael, who helped keep me alert, happy, and ful­ filled. To Barbara, who deserves belated thanks for doing the drawings in E1'eryday Science. To Anne Hagedorn, thanks for doing some of the typing. To Gerry Denterlein, thanks for keeping tabs on the drawings.

Portrait de Joseph, ouvrier agricole dans une ferme du Cantal, âgé d'une soixantaine d'années. Seul depuis que, à la mort de son père, son frère a emmené leur mère vivre ailleurs, Joseph est spectateur de la vie de ceux qui l'entourent. Cet être doux et silencieux boit pour oublier Sylvie, qu'il a aimée il y a trente ans, partie au milieu de l'hiver avec un autre. …

Turbulence takes place in practically all flow situations that occur naturally or in modern technological systems. Therefore, considerable effort is being expended in an attempt to understand this very complex physical phenome­ non and to develop both empirical and mathematical models for its description. Such numerical and analytical computational schemes would allow the reliable prediction and design of turbulent flow processes to be carried out. The purpose of this book is to bring together, in a usable form, some of the fundamental concepts of turbulence along with turbulence models and experimental techniques. It is hoped that these have "general applicability" in current engineering design. The phrase "general applicabil­ ity" is highlighted because the theory of turbulence is still so much in a formative stage that completely general analyses are not available now, nor will they be available in the immediate future. The concepts and models described herein represent the state-of-the­ art methods that are now being used to give answers to turbulent flow problems. As in all turbulent flow analysis, the methods are a blend of analytical and empirical input, and the reader should be cognizant of the simplification and restrictions imposed upon the methods when applyingthem to physical situations different from those for which they have been developed.

The study of paleocurrents, since 1963, is now a very routine part of sedimentology, and more and more such studies are finding use in other fields. Thus it seemed appropriate for us to review post-1963 developments and present them in a compact manner for the interested reader. Instead of rewriting a second edition, which thirteen years later we would organize in a completely dif­ ferent way, we have brought each chapter up to date with new material up to 1976. A new update supplement has in this edition been inserted after each one of the original chapters. We have stayed close to the original theme of paleocurrents-how to measure them and how to use them to solve geological problems ranging in scale from the hand specimen to the sedimentary basin and beyond. We have used many annotated references and tables to help pre­ sent this information to the reader. The reader will note that we have cited a few 1962 references - pUblications that appeared too late to be cited in the original 1963 edition. A few times we have also cited a reference which was included in the first edition. These are marked with an asterisk and hence do not appear in the new lists of references. We have been aided by many. In Cincinnati, WANDA OSBORNE and JEAN CARROL did typing and RICHARD SPOHN, the University's geological librarian, was very helpful in obtaining many references to the literature.

It is an incontestable fact that numerical analysis techniques are used rou­ tinely (although not always effectively) in virtually every quantitative field of scientific endeavor. In this book, which is directed toward upper-division and graduate level students in engineering and mathematics, we have selected for discussion subjects that are traditionally found in numerical analysis texts. But our choice of methodology rejects the traditional where analysis and experience clearly warrant such a departure, and one of our primary aspirations in this work is to equip the reader with the wherewithal to apply numerical analysis thinking to nontraditional subjects. For there is a plethora of computer-oriented sciences such as optimization, statistics, and system analysis and identification that are sorely in need of methods comparable to those related here for classical numerical analysis problems. Toward uncovering for the reader the structure of numerical methods we have, for example, devoted a chapter to a metric space theory for iter­ ative application of operators. In this chapter, we have collected those definitions and concepts of real and functional analysis that are requisite to a modern intermediate-level exposition of the principles of numerical anal­ ysis. Further, we derive the abstract theory (most notably, the contraction mapping theorem) for iteration processes.

The theory of controlled processes is one of the most recent mathematical theories to show very important applications in modern engineering, parti­ cularly for constructing automatic control systems, as well as for problems of economic control. However, actual systems subject to control do not admit a strictly deterministic analysis in view of random factors of various kinds which influence their behavior. Such factors include, for example, random noise occurring in the electrical system, variations in the supply and demand of commodities, fluctuations in the labor force in economics, and random failures of components on an automated line. The theory of con­ trolled processes takes the random nature of the behavior of a system into account. In such cases it is natural, when choosing a control strategy, to proceed from the average expected result, taking note of all the possible variants of the behavior of a controlled system. An extensive literature is devoted to various economic and engineering systems of control (some of these works are listed in the Bibliography). is no text which adequately covers the general However, as of now there mathematical theory of controlled processes. The authors ofthis monograph have attempted to fill this gap. In this volume the general theory of discrete-parameter (time) controlled processes (Chapter 1) and those with continuous-time (Chapter 2), as well as the theory of controlled stochastic differential equations (Chapter 3), are presented.