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Calculus of Variations

Science & Math

S. V. Fomin

I.M. Gelfand

Mathematics

PRENTICE/HALL

This book was written in 1966 as a memorial tribute to Nicholai Alexandrovich Bernstein, the eminent Russian pioneer in cybernetics. Representing the work of a number of contributors, it is made up of a collection of papers presenting models of various physiological systems. Together, the papers summarize a great deal of work not conveniently available elsewhere. The collection includes only that sort of model for which some real physiological system is the subject; constructions not having well-grounded roots in biology are omitted, and so are those that mathematicians might find simplistic and unlikely to evolve fruitfully. Since from the editors' point of view, a model description and physiological experiment are consecutive stages of research, some of these chapters are purely theoretical, while others are of a traditionally experimental character. In their Introduction, the editors have attempted to draw together the various presentations and diversities of subject matter; in his Foreword, Peter H. Green has provided not only current information on the topics discussed in the articles, but also a bibliography of current works and articles. Individual papers on the functioning of specific systems discuss the construction of models that correctly describe the functioning system. The postulates employed correspond to the real properties of the elements of the modeled system and to the interrelationship of the elements. The model is then refined to a level at which conclusions can be made that allow experimental verification. The ultimate goal in each case is, of course, to render predictable certain properties of the system not previously obvious, and to allow the corresponding model to be utilized in gaining an understanding of the function of the system under investigation. Introduction: Mathematical Modeling of Mechanisms of the Central Nervous System. Part I: The Spreading of Excitation in Continuous Media and the Electrical Properties of Branching Structures—An Analysis of the Functional Properties of Dendrites in Relation to Their Structure; The Electrical Behavior of the Myocardium as a System and the Characteristics of the Cellular Membrane of the Heart; The Problem of the Electrical Properties of Syncytia; The Periodic Blocking of Impulses in Excitable Tissues. Part II: The Organization of Certain Parts of the Central Nervous System and the Modeling of their Operation—Characteristics of the Respiratory Neurons of Different Levels of the Central Nervous System; A Model of a System of Neurons with Periodic Volley Activity (Resistant to Random Afferent Influences); Some Special Features of Organization of the Cerebellar Cortex. Part III: The Regulation of Movements—Some Problems in the Analysis of Movements; An Analysis of the Physiological Tremor by Means of a General-Purpose Computer; Concerning Tuning before Movement; The Control of Movements of the Joints and Kinesthetic Afferentation; The Compensation of Respiratory Disturbances of the Erect Posture of Man as an Example of the Organization of Interarticular Interaction.

Models of the Structural-Functional Organization of Certain Biological Systems

V. S. Gurfinkel

M. L. Tsetlin

Biological Sciences

Mit Pr

Ergodic theory is one of the few branches of mathematics which has changed radically during the last two decades. Before this period, with a small number of exceptions, ergodic theory dealt primarily with averaging problems and general qualitative questions, while now it is a powerful amalgam of methods used for the analysis of statistical properties of dyna mical systems. For this reason, the problems of ergodic theory now interest not only the mathematician, but also the research worker in physics, biology, chemistry, etc. The outline of this book became clear to us nearly ten years ago but, for various reasons, its writing demanded a long period of time. The main principle, which we adhered to from the beginning, was to develop the approaches and methods or ergodic theory in the study of numerous concrete examples. Because of this, Part I of the book contains the description of various classes of dynamical systems, and their elementary analysis on the basis of the fundamental notions of ergodicity, mixing, and spectra of dynamical systems. Here, as in many other cases, the adjective" elementary" i~ not synonymous with "simple. " Part II is devoted to "abstract ergodic theory. " It includes the construc tion of direct and skew products of dynamical systems, the Rohlin-Halmos lemma, and the theory of special representations of dynamical systems with continuous time. A considerable part deals with entropy.

Ergodic Theory (Grundlehren der mathematischen Wissenschaften)

I. P. Kornfeld

Springer

Originally published in two volumes, this advanced-level text is based on courses and lectures given by the authors at Moscow State University and the University of Moscow. Reprinted here in one volume, the first part is devoted to metric and normal spaces. Beginning with a brief introduction to set theory and mappings, the authors offer a clear presentation of the theory of metric and complete metric spaces. The principle of contraction mappings and its applications to the proof of existence theorems in the theory of differential and integral equations receives detailed analysis, as do continuous curves in metric spaces — a topic seldom discussed in textbooks. Part One also includes discussions of other subjects, such as elements of the theory of normed linear spaces, weak sequential convergence of elements and linear functionals, adjoint operators, and linear operator equations. Part Two focuses on an exposition of measure theory, the Lebesque interval and Hilbert Space. Both parts feature numerous exercises at the end of each section and include helpful lists of symbols, definitions, and theorems.

Elements of the Theory of Functions and Functional Analysis (Dover Books on Mathematics)

A. N. Kolmogorov

Dover Publications

Based on a series of lectures given by I. M. Gelfand at Moscow State University, this book actually goes considerably beyond the material presented in the lectures. The aim is to give a treatment of the elements of the calculus of variations in a form both easily understandable and sufficiently modern. Considerable attention is devoted to physical applications of variational methods, e.g., canonical equations, variational principles of mechanics, and conservation laws. The reader who merely wishes to become familiar with the most basic concepts and methods of the calculus of variations need only study the first chapter. Students wishing a more extensive treatment, however, will find the first six chapters comprise a complete university-level course in the subject, including the theory of fields and sufficient conditions for weak and strong extrema. Chapter 7 considers the application of variational methods to the study of systems with infinite degrees of freedom, and Chapter 8 deals with direct methods in the calculus of variations. The problems following each chapter were made specially for this English-language edition, and many of them comment further on corresponding parts of the text. Two appendices and suggestions for supplementary reading round out the text. Substantially revised and corrected by the translator, this inexpensive new edition will be welcomed by advanced undergraduate and graduate students of mathematics and physics.

Calculus of Variations (Dover Books on Mathematics)

This volume in Richard Silverman's exceptional series of translations of Russian works in the mathematical science is a comprehensive, elementary introduction to real and functional analysis by two faculty members from Moscow University. It is self-contained, evenly paced, eminently readable, and readily accessible to those with adequate preparation in advanced calculus. The first four chapters present basic concepts and introductory principles in set theory, metric spaces, topological spaces, and linear spaces. The next two chapters consider linear functionals and linear operators, with detailed discussions of continuous linear functionals, the conjugate space, the weak topology and weak convergence, generalized functions, basic concepts of linear operators, inverse and adjoint operators, and completely continuous operators. The final four chapters cover measure, integration, differentiation, and more on integration. Special attention is here given to the Lebesque integral, Fubini's theorem, and the Stieltjes integral. Each individual section — there are 37 in all — is equipped with a problem set, making a total of some 350 problems, all carefully selected and matched. With these problems and the clear exposition, this book is useful for self-study or for the classroom — it is basic one-year course in real analysis. Dr. Silverman is a former member of the Institute of Mathematical Sciences of New York University and the Lincoln Library of M.I.T. Along with his translation, he has revised the text with numerous pedagogical and mathematical improvements and restyled the language so that it is even more readable.

Introductory Real Analysis (Dover Books on Mathematics)

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