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This Seminar began in Moscow in November 1943 and has continued
without interruption up to the present. We are happy that with this
vol ume, Birkhiiuser has begun to publish papers of talks from the
Seminar. It was, unfortunately, difficult to organize their
publication before 1990. Since 1990, most of the talks have taken
place at Rutgers University in New Brunswick, New Jersey. Parallel
seminars were also held in Moscow, and during July, 1992, at IRES
in Bures-sur-Yvette, France. Speakers were invited to submit papers
in their own style, and to elaborate on what they discussed in the
Seminar. We hope that readers will find the diversity of styles
appealing, and recognize that to some extent this reflects the
diversity of styles in a mathematical society. The principal aim
was to have interesting talks, even if the topic was not especially
popular at the time. The papers listed in the Table of Contents
reflect some of the rich variety of ideas presented in the Seminar.
Not all the speakers submit ted papers. Among the interesting talks
that influenced the seminar in an important way, let us mention,
for example, that of R. Langlands on per colation theory and those
of J. Conway and J. McKay on sporadic groups. In addition, there
were many extemporaneous talks as well as short discus sions."
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."
After reading the manuscript, some biologists inquired why, on the
basis of the broad experimental material presented in this book, we
had not come up with a model describing the operation of the
cerebellum. To answer this question, we decided to write a preface
to our book. How the nervous system copes with the complexity of
the world is one of the central problems of neurophys iology. The
question was clearly formulated for the frrst time by N. A.
Bernstein. Considering the problem of motor control, he pointed out
that the main objective of motor coordination is to overcome the
redundant number of degrees of freedom of the motor apparatus or,
in other words, to diminish the number of independent variables
which control the movement (Bernstein 1967). These I. M. Gelfand
and M. L. Zetlin ideas were further developed by (Gelfand and
Zetlin 1966). They proposed, in particular, the
"non-individualized" ("non-addressed") mode of control in complex
systems, where only the highest levels of the system have the full
notion about the fmal task while the main "effectors" act on the
basis of very limited information. These propositions were made by
Gelfand and Zetlin in a very general form, but, nevertheless,
proved to be fruitful in determining the direction of experimental
research. For instance, the discovery of the "locomotory region" of
the brain stem (Shik et al."
Irene Dorfman died in Moscow on April 6, 1994, shortly after seeing
her beautiful book on Dirac structures [I]. The present volume
contains a collection of papers aiming at celebrating her
outstanding contributions to mathematics. Her most important
discoveries are associated with the algebraic structures arising in
the study of integrable equations. Most of the articles contained
in this volume are in the same spirit. Irene, working as a student
of Israel Gel'fand made the fundamental dis- covery that
integrability is closely related to the existence of bi-Hamiltonian
structures [2], [3]. These structures were discovered
independently, and al- most simultaneously, by Magri [4] (see also
[5]). Several papers in this book are based on this remarkable
discovery. In particular Fokas, Olver, Rosenau construct large
classes on integrable equations using bi-Hamiltonian struc- tures,
Fordy, Harris derive such structures by considering the restriction
of isospectral flows to stationary manifolds and Fuchssteiner
discusses similar structures in a rather abstract setting.
It is very tempting but a little bit dangerous to compare the style
of two great mathematicians or of their schools. I think that it
would be better to compare papers from both schools dedicated to
one area, geometry and to leave conclusions to a reader of this
volume. The collaboration of these two schools is not new. One of
the best mathematics journals Functional Analysis and its
Applications had I.M. Gelfand as its chief editor and V.I. Arnold
as vice-chief editor. Appearances in one issue of the journal
presenting remarkable papers from seminars of Arnold and Gelfand
always left a strong impact on all of mathematics. We hope that
this volume will have a similar impact. Papers from Arnold's
seminar are devoted to three important directions developed by his
school: Symplectic Geometry (F. Lalonde and D. McDuff), Theory of
Singularities and its applications (F. Aicardi, I. Bogaevski, M.
Kazarian), Geometry of Curves and Manifolds (S. Anisov, V.
Chekanov, L. Guieu, E. Mourre and V. Ovsienko, S. Gusein-Zade and
S. Natanzon). A little bit outside of these areas is a very
interesting paper by M. Karoubi Produit cyclique d'espaces et
operations de Steenrod.
It is very tempting but a little bit dangerous to compare the style
of two great mathematicians or of their schools. I think that it
would be better to compare papers from both schools dedicated to
one area, geometry and to leave conclusions to a reader of this
volume. The collaboration of these two schools is not new. One of
the best mathematics journals Functional Analysis and its
Applications had I.M. Gelfand as its chief editor and V.I. Arnold
as vice-chief editor. Appearances in one issue of the journal
presenting remarkable papers from seminars of Arnold and Gelfand
always left a strong impact on all of mathematics. We hope that
this volume will have a similar impact. Papers from Arnold's
seminar are devoted to three important directions developed by his
school: Symplectic Geometry (F. Lalonde and D. McDuff), Theory of
Singularities and its applications (F. Aicardi, I. Bogaevski, M.
Kazarian), Geometry of Curves and Manifolds (S. Anisov, V.
Chekanov, L. Guieu, E. Mourre and V. Ovsienko, S. Gusein-Zade and
S. Natanzon). A little bit outside of these areas is a very
interesting paper by M. Karoubi Produit cyclique d'espaces et
operations de Steenrod.
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."
This Seminar began in Moscow in November 1943 and has continued
without interruption up to the present. We are happy that with this
vol ume, Birkhiiuser has begun to publish papers of talks from the
Seminar. It was, unfortunately, difficult to organize their
publication before 1990. Since 1990, most of the talks have taken
place at Rutgers University in New Brunswick, New Jersey. Parallel
seminars were also held in Moscow, and during July, 1992, at IRES
in Bures-sur-Yvette, France. Speakers were invited to submit papers
in their own style, and to elaborate on what they discussed in the
Seminar. We hope that readers will find the diversity of styles
appealing, and recognize that to some extent this reflects the
diversity of styles in a mathematical society. The principal aim
was to have interesting talks, even if the topic was not especially
popular at the time. The papers listed in the Table of Contents
reflect some of the rich variety of ideas presented in the Seminar.
Not all the speakers submit ted papers. Among the interesting talks
that influenced the seminar in an important way, let us mention,
for example, that of R. Langlands on per colation theory and those
of J. Conway and J. McKay on sporadic groups. In addition, there
were many extemporaneous talks as well as short discus sions."
"All through both volumes [Functions & Graphs and The Methods
of Coordinates], one finds a careful description of the
step-by-step thinking process that leads up to the correct
definition of a concept or to an argument that clinches in the
proof of a theorem. We are ... very fortunate that an account of
this caliber has finally made it to printed pages... Anyone who has
taken this guided tour will never be intimidated by n ever again...
High school students (or teachers) reading through these two books
would learn an enormous amount of good mathematics. More
importantly, they would also get a glimpse of how mathematics is
done." -- H. Wu, The Mathematical Intelligencer 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 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.Contents Preface
Foreword Introduction Chapter 1 Examples Chapter 2 The Linear
Function Chapter 3 The Function y =] x ] Chapter 4 The Quadratic
Equation Chapter 5 The Linear Fractional Function Chapter 6 Power
Functions Chapter 7 Rational Functions Problems for Independent
Solution Answers and Hints to Problems Marked with the Sign
"All through both volumes [Functions & Graphs and The Methods
of Coordinates], one finds a careful description of the
step-by-step thinking process that leads up to the correct
definition of a concept or to an argument that clinches in the
proof of a theorem. We are ... very fortunate that an account of
this caliber has finally made it to printed pages... Anyone who has
taken this guided tour will never be intimidated by n ever again...
High school students (or teachers) reading through these two books
would learn an enormous amount of good mathematics. More
importantly, they would also get a glimpse of how mathematics is
done." -- H. Wu, The Mathematical Intelligencer 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 development, are available
in the following 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. Contents Preface
Foreword Introduction PART I Chapter 1 TheCoordinates of Points on
a Line 1. The Number Axis 2. The Absolute Value of Number 3. The
Distance Between Two Points Chapter 2 The Coordinates of Points in
the Plane 4. The Coordinate Plane 5. Relations Connecting
Coordinates 6. The Distance Between Two Points 7. Defining Figures
8. We Begin to Solve Problems 9. Other Systems of Coordinates
Chapter 3 The Coordinates of a Point in Space 10. Coordinate Axes
and Planes 11. Defining Figures in Space PART II Chapter 1
Introduction 1. Some General Considerations 2. Geometry as an Aid
in Calculation 3. The Need for Introducing Four-Dimensional Space
4. The Peculiarities of Four-Dimensional Space 5. Some Physics
Chapter 2 Four-Dimensional Space 6. Coordinate Axes and Planes 7.
Some Problems Chapter 3 The Four-Dimensional Cube 8. The Definition
of the Sphere and the Cube 9. The Structure of the Four-Dimensional
Cube 10. Problems on the Cube
This book is about algebra. This is a very old science and its gems
have lost their charm for us through everyday use. We have tried in
this book to refresh them for you. The main part of the book is
made up of problems. The best way to deal with them is: Solve the
problem by yourself - compare your solution with the solution in
the book (if it exists) - go to the next problem. However, if you
have difficulties solving a problem (and some of them are quite
difficult), you may read the hint or start to read the solution. If
there is no solution in the book for some problem, you may skip it
(it is not heavily used in the sequel) and return to it later. The
book is divided into sections devoted to different topics. Some of
them are very short, others are rather long. Of course, you know
arithmetic pretty well. However, we shall go through it once more,
starting with easy things. 2 Exchange of terms in addition Let's
add 3 and 5: 3]5=8. And now change the order: 5+3=8. We get the
same result. Adding three apples to five apples is the same as
adding five apples to three - apples do not disappear and we get
eight of them in both cases. 3 Exchange of terms in multiplication
Multiplication has a similar property. But let us first agree on
notation.
In a sense, trigonometry sits at the center of high school
mathematics. It originates in the study of geometry when we
investigate the ratios of sides in similar right triangles, or when
we look at the relationship between a chord of a circle and its
arc. It leads to a much deeper study of periodic functions, and of
the so-called transcendental functions, which cannot be described
using finite algebraic processes. It also has many applications to
physics, astronomy, and other branches of science. It is a very old
subject. Many of the geometric results that we now state in
trigonometric terms were given a purely geometric exposition by
Euclid. Ptolemy, an early astronomer, began to go beyond Euclid,
using the geometry of the time to construct what we now call tables
of values of trigonometric functions. Trigonometry is an important
introduction to calculus, where one stud ies what mathematicians
call analytic properties of functions. One of the goals of this
book is to prepare you for a course in calculus by directing your
attention away from particular values of a function to a study of
the function as an object in itself. This way of thinking is useful
not just in calculus, but in many mathematical situations. So
trigonometry is a part of pre-calculus, and is related to other
pre-calculus topics, such as exponential and logarithmic functions,
and complex numbers."
2012 Reprint of 1963 Edition. Exact facsimile of the original
edition, not reproduced with Optical Recognition Software. Gelfand
was a Soviet mathematician who made major contributions to many
branches of mathematics, including group theory, representation
theory and functional analysis. The recipient of numerous awards
and honors, including the Order of Lenin and the Wolf Prize, he was
a Fellow of the Royal Society and a lifelong academic, serving
decades as a professor at Moscow State University and, after
immigrating to the United States shortly before his 76th birthday,
at the Busch Campus of New Jersey's Rutgers University. He is known
for having educated and inspired generations of students through
his legendary seminar at Moscow State University. This treatise is
devoted to the description and detailed study of the
representations of the rotation group of three dimensional space
and of the Lorentz group. These groups are of fundamental
importance in theoretical physics. The book is also designed for
mathematicians studying the representations of Lie groups. For them
the book can serve as in introduction to the general theory of
representations.
Dedicated to the memory of Chih-Han Sah, this volume continues a
long tradition of one of the most influential mathematical seminars
of this century. A number of topics are covered, including
combinatorial geometry, connections between logic and geometry, Lie
groups, algebras and their representations. An additional area of
importance is noncommutative algebra and geometry, and its
relations to modern physics. Distinguished mathematicians
contributing to this work: T.V. Alekseevskaya V. Kac A.V. Borovik
A. Kazarnovsky-Krol C.-H. Sah* M. Kontsevich G. Cherlin A. Radul
J.L. Dupont A.L. Rosenberg I.M. Gelfand N. White The Gelfand
Mathematical Seminar volumes stimulate the birth of significant
ideas in contemporary mathematics and remain invaluable reference
material. * indicates deceased contributor (Production: please
ensure that appropriate symbol be incorporated onto the final back
cover design)
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. (See below for details of other books in this series.) 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. Unabridged republication of edition published by The M.I.T. Press, Cambridge, Massachusetts, 1969. Foreword. Introduction. Problems for Independent Solution.
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 unifying theme of Volume 6 is the study
of representations of the general linear group of order two over
various fields and rings of number-theoretic nature, most
importantly over local fields ($p$-adic fields and fields of power
series over finite fields) and over the ring of adeles.
Representation theory of the latter group naturally leads to the
study of automorphic functions and related number-theoretic
problems. The book contains a wealth of information about discrete
subgroups and automorphic representations, and can be used both as
a very good introduction to the subject and as a valuable
reference.
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 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.
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 unifying theme of this collection of papers by the very
creative Russian mathematician I. M. Gelfand and his co-workers is
the representation theory of groups and lattices. Two of the papers
were inspired by application to theoretical physics; the others are
pure mathematics though all the papers will interest mathematicians
at quite opposite ends of the subject. Dr. G. Segal and Professor
C-M. Ringel have written introductions to the papers which explain
the background, put them in perspective and make them accessible to
those with no specialist knowledge in the area.
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