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Calculus of Variations

Science & Math

S. V. Fomin

I.M. Gelfand

Mathematics

PRENTICE/HALL

Lectures on Linear Algebra

Interscience

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)

Dover Publications

The second in a series of systematic studies by a celebrated mathematician I. M. Gelfand and colleagues, this volume presents students with a well-illustrated sequence of problems and exercises designed to illuminate the properties of functions and graphs. Since readers do not have the benefit of a blackboard on which a teacher constructs a graph, the authors abandoned the customary use of diagrams in which only the final form of the graph appears; instead, the book's margins feature step-by-step diagrams for the complete construction of each graph. The first part of the book employs simple functions to analyze the fundamental methods of constructing graphs. The second half deals with more complicated and refined questions concerning linear functions, quadratic trinomials, linear fractional functions, power functions, and rational functions.

Functions and Graphs (Dover Books on Mathematics)

E. G. Glagoleva

E. E. Shnol

Representation Theory and Automorphic Functions

M. I. Graev

I. I. Pyatetskii-Shapiro

W.B.SAUNDERS COMPANY

The Seminar has taken place at Rutgers University in New Brunswick, New Jersey, since 1990 and it has become a tradition, starting in 1992, that the Seminar be held during July at IHES in Bures-sur-Yvette, France. This is the second Gelfand Seminar volume published by Birkhauser, the first having covered the years 1990-1992. Most of the papers in this volume result from Seminar talks at Rutgers, and some from talks at IHES. In the case of a few of the papers the authors did not attend, but the papers are in the spirit of the Seminar. This is true in particular of V. Arnold's paper. He has been connected with the Seminar for so many years that his paper is very natural in this volume, and we are happy to have it included here. We hope that many people will find something of interest to them in the special diversity of topics and the uniqueness of spirit represented here. The publication of this volume would be impossible without the devoted attention of Ann Kostant. We are extremely grateful to her. I. Gelfand J. Lepowsky M. Smirnov Questions and Answers About Geometric Evolution Processes and Crystal Growth Fred Almgren We discuss evolutions of solids driven by boundary curvatures and crystal growth with Gibbs-Thomson curvature effects. Geometric measure theo retic techniques apply both to smooth elliptic surface energies and to non differentiable crystalline surface energies.

The Gelfand Mathematics Seminars, 1993 - 1995

James Lepowsky

Mikhail M. Smirnov

Birkhauser

The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gelfand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory. Volume 1 is devoted to basics of the theory of generalized functions. The first chapter contains main definitions and most important properties of generalized functions as functional on the space of smooth functions with compact support. The second chapter talks about the Fourier transform of generalized functions. In Chapter 3, definitions and properties of some important classes of generalized functions are discussed; in particular, generalized functions supported on submanifolds of lower dimension, generalized functions associated with quadratic forms, and homogeneous generalized functions are studied in detail. Many simple basic examples make this book an excellent place for a novice to get acquainted with the theory of generalized functions. A long appendix presents basics of generalized functions of complex variables.

Generalized Functions: Properties and Operations (AMS Chelsea Publishing)

G. E. Shilov

American Mathematical Society

The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gel'fand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory. Volume 2 is devoted to detailed study of generalized functions as linear functionals on appropriate spaces of smooth test functions. In Chapter 1, the authors introduce and study countable-normed linear topological spaces, laying out a general theoretical foundation for the analysis of spaces of generalized functions. The two most important classes of spaces of test functions are spaces of compactly supported functions and Schwartz spaces of rapidly decreasing functions. In Chapters 2 and 3 of the book, the authors transfer many results presented in Volume 1 to generalized functions corresponding to these more general spaces. Finally, Chapter 4 is devoted to the study of the Fourier transform; in particular, it includes appropriate versions of the Paley-Wiener theorem.

Generalized Functions, Volume 2: Spaces of Fundamental and Generalized Functions

The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gelfand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory. In Volume 3, applications of generalized functions to the Cauchy problem for systems of partial differential equations with constant coefficients are considered. The book includes the study of uniqueness classes of solutions of the Cauchy problem and the study of classes of functions where the Cauchy problem is well posed. The last chapter of this volume presents results related to spectral decomposition of differential operators related to generalized eigenfunctions.

Generalized Functions, Volume 3: Theory of Differential Equations (Ams Chelsea Publishing, 379)

The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gelfand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory. The main goal of Volume 4 is to develop the functional analysis setup for the universe of generalized functions. The main notion introduced in this volume is the notion of rigged Hilbert space (also known as the equipped Hilbert space, or Gelfand triple). Such space is, in fact, a triple of topological vector spaces $E \subset H \subset E'$, where $H$ is a Hilbert space, $E'$ is dual to $E$, and inclusions $E\subset H$ and $H\subset E'$ are nuclear operators. The book is devoted to various applications of this notion, such as the theory of positive definite generalized functions, the theory of generalized stochastic processes, and the study of measures on linear topological spaces.

Generalized Functions: Applications of Harmonic Analysis (AMS Chelsea Publishing)

N. Ya Vilenkin

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