Multidimensional analysis and discrete models

253 Pages·1995·92.741 MB·English

by  Dezin, Aleksej Alekseevič, Aleksanova, Irene

Multidimensional
Analvsis
and
~ i s c i e tMe odels
Aleksei A. Dezin
Steklov Mathematical Institute
Russian Academy of Sciences
Moscow
Translated from the Russian by Irene Aleksanova
Boca Raton London New York
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Library of Congress Cataloging-in-Publication Data
Dezin, A. A. (Alekse i Alekseevich)
[Mnogomerny i analiz i diskretnye modeli. English]
Multidimensional analysis and discrete models / A.A. Dezin ;
translated from the Russian by Irene Aleksanova.
p. cm.
Includes bibliographical references and index.
ISBN 0-8493-9425-2 (alk. paper)
1. Multivariate analysis. 2. Mathematical models. I. Title.
QA278.D4413 1995
530.1’5-dc20
95-16807
A Library of Congress record exists under LC control number: 95016807
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the original copies may be apparent.
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been unable to contact.
ISBN 13: 978-1-315-89576-5 (hbk)
ISBN 13: 978-1-351-07486-5 (ebk)
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Preface
During the last decades, much research effort was devoted to the
study of topological, geometrical, and algebraic structures underly-
ing the basic ideas of classical analysis. The results obtained made
it possible, in a number of cases, to construct certain intrinsically
defined discrete models (finite or "countable") for the problems of
analysis and mathematical physics. It is to be understood, however,
that when speaking about these models we do not mean just ap-
proximation (in one way or another) of the given continual object.
What we do mean is the construction of discrete counterparts of
these objects.
It should also be mentioned that, recently, similar ideas have
started to penetrate computational practice. 'freatment of discrete
models is of some methodological interest as well since it allows us to
acquire a better understanding of the nature of this or that relation
or statement.
Formally, the main subject matter of the book is the description
of some regular methods of constructing intrinsically defined discrete
models for special classes of continual objects.
In fact, I had a more extensive task in mind. I wanted (1) to
demonstrate (and do it at as "low" a level as possible) the interac-
tion of ideas and methods of such parts of mathematics as, for ex-
ample, classical and functional analysis, Riemannian geometry, and
algebraic topology; and (2) to demonstrate this interaction not when
studying such a special and difficult problem as the index of the gen-
eral elliptical operator on a smooth manifold, but when investigating
the connection between the Laplace operator, the multidimensional
analogs of Cauchy-Riemann equations (or operations of vector anal-
ysis), and the corresponding difference equations.
The book is addressed, first of all, to those who have chosen
mathematical physics as their main field of interest. It is assumed
that the potential readers would like to acquire a more intimate
knowledge of the nature of some foreign objects they have to deal
with in the course of their research, without resorting to the study
of fundamental works devoted to Riemannian geometry, geometrical
theory of integration, algebraic topology and the like, which they
have no wish, time, or energy to pursue systematically. It is also my
belief that, in a number of cases, the study of special finite models
can be very useful and can simplify the acquaintance with some
related divisions of mathematics.
This approach has considerably influenced the treatment of the
material in the book. The reader is supposed to be more or less
familiar with partial differential equations, to have some notion of
differential geometry, and to have never been concerned with prob-
lems or definitions of homology theory.
I have few words about the arrangement of the material. The
introductory chapter should give a fairly clear idea of what is meant
by a finite model and Chapter 1 of how the multidimensional analysis
should be understood. In Chapter 2, the reader will see (perhaps not
without a certain surprise or even disappointment) that analysis on
the Riemannian manifold is treated as the theory of special class
of the first-order partial differential equations closely related to the
operators of classical vector analysis, the Laplace operator, and wave
operator. Such concepts as curvature, connectedness, G-structures,
and the like are not mentioned at all.
Chapters 3-5 are devoted to models proper. Chapter 3 considers
the objects of classical mathematical physics, Chapter 4 deals with
quantum mechanics and the field theory, and Chapter 5 considers
some general aspects of the theory of discrete equations which do
not claim to be connected to physics in any way.
For a sufficiently advanced reader, all chapters are practically
self-contained (modulo necessary notations and definitions).
A sufficiently complete formal outline of the subject matter is
given in the table of contents. More detailed and less formal infor-
mation can be found in the introductions and preliminary notes to
each chapter and section.
Contents
To the Reader 1
Chapter 0. One-Dimensional Models 3
0. Introductory Remarks . . . 3
1. Models on the Real Line . . 4
1.1. Combinatorial real line 4
1.2. Multiplications ..... 6
1.3. ·Equations and problems . 8
1.4. Norms, step functions, and approximations 11
2. Models on a Circle . . . . . . . . . . . . . . . . 13
Chapter 1. Formal Structures 21
0. Introductory Remarks 21
1. Topology and Metrics 22
1.0. Preliminary notes 22
1.1. Topological space 22
1.2. Metric space . . . 24
2. Groups and Complexes 25
2.0. Preliminary remarks 25
2.1. Definitions . . .. . . 26
2.2. The simplest examples of complexes 28
2.3. Cohomology . . . . 32
2.4. Tensor products . . . . . . . . . 33
2.5. Concluding remarks . . . . . . . 35
3. Linear Space and Related Structures 36
3.0. Preliminary remarks . 36
3.1. Initial definitions . . . . . . . . . 36
3.2. Norm, metric, topology . . . . . . . . . . . . . . . . 37
3.3. Linear maps . . . . . . . . . . . . . . . . . . . . . . 38
3.4. Functionals, the inner product, the Euclidean space 43
3.5. Subspaces and the exterior multiplication 46
3.6. Tensor algebra . . . . . . . . . . . . . . . . 50
3.7. Linear representations of groups . . . . . . 52
4. Infinitesimal Operations and Smooth Manifolds 54
4.0. Preliminary remarks 54
4.1. Differentiation . . 55
4.2. Smooth manifolds . 56
4.3. Integration . . . . . 60
4.4. The Stokes formula and the exterior differentiation . 64
5. Hilbert Space and Differential Operators . 68
5.0. Preliminary remarks ..... 68
5.1. An abstract Hilbert space .. 69
5.2. Functional spaces !HI and W . 72
5.3. Differential operators 75
5.4. Mollifiers . . . . . . . . . . . 79
Chapter 2. Analysis on Riemannian Manifolds 85
0. Introductory Remarks . . . 85
1. The Riemannian Structure . . . . . 86
1.1. The metric tensor . . . . . . . . 86
1.2. Polar and spherical coordinates . 89
2. The Orthogonal Decompositions of the Spaces IHih
and the Poisson Equation . . . . . . . . . . . . 90
2.0. Preliminary remarks . . . . . . . . . . . . . . . 90
2.1. Riemannian formalism and vector analysis . . 90
2.2. Orthogonal decompositions and the Poisson equation 93
2.3. Theorems of de Rham, Hodge, and Kodaira . 97
3. First-Order Invariant Systems . 98
3.0. Preliminary remarks . . . . . . . . . . . . . . 98
3.1. Classical invariant systems . . . . . . . . . . 98
3.2. The Cauchy-Riemann multidimensional equations
and the index . . . . . . . . . . . . . . . . . 101
3.3. Regular invariant systems; the spectrum . . . . 105
3.4. Splitting and the Lorentz metric . . . . . . . . 106
4. Boundary Value Problems for Invariant Systems . 109
4.0. Preliminary remarks . . . . . . . . . . . . 109
4.1. Glueing of the double . . . . . . . . . . . 110
4.2. Invariant systems in bounded domains . . 113
4.3. A disk, a ball, and a cube . . . . . . . . . 117
4.4. Time-dependent systems . . . . . . . . . 121
5. Special Constructions on Manifolds with Boundaries . 124
5.1. Index in boundary value problems . 124
5.2. Orthogonal decompositions . . . . . . . . 127
5.3. The Cauchy integral . . . . . . . . . . . . 128
6. Certain Equations of Mathematical Physics . 130
6.0. Preliminary remarks . . 130
6.1. Euler equations . . . . . . . . . . . . . . . 130
6.2. Certain special flows . . . . . . . . . . . . 133
6.3. The Navier-Stokes equations and linearization . 137
6.4. Maxwell equations . . . . . . . . . . . . . . . . . 139
Chapter 3. The Model of Euclidean Space
and Difference Operators 143
0. Introductory Remarks . . . . . . . . . . . . . . . . . 143
1. The Combinatorial Model of the Euclidean Space . . 144
1.0. Preliminary remarks . . . . . . . . . .. ... . . 144
1.1. The model of the real line . . . . . . . . . . . . . 144
1.2. A model of the n-dimensional Euclidean space . 146
1.3. Subdivisions and the limit space . . . . . . . . 149
2. Difference Operators and the Principal Problems . 151
2.0. Preliminary remarks . .. . ... ... . .. . . 151
2.1. Difference operators d, 8 . . . . . . . . . . . . . 151
2.2. Orthogonal decompositions and cohomology . 154
2.3. Natural equations . . . . . . . . . . . . . . . . 155
3. A Two-Dimensional Case: Boundary Value Problems
and Approximations . . . . 158
3.0. Preliminary remarks . . . . 158
3.1. Continual objects . . . . . 159
3.2. Combinatorial structures . 160
3.3. Equations and problems . . 162
3.4. Orthogonal decompositions . 166
3.5. Step functions . . . . . . . . 169
3.6. Approximation and the limiting process . 172
4. Discrete Analogs of Certain Hydrodynamics Relations . 177
4.0. Preliminary remarks ...... . . 177
4.1. The minimal natural pattern .. . 179
4.2. A tripled pattern and solvability . 181
4.3. Nonstationary equations . . 183
5. A Double Complex . . . 184
6. Some Open Problems . . . . . 186
Chapter 4. Models in Quantum Physics 189
0. Introductory Remarks . . . . . . 189
1. Models in Quantum Mechanics . 190
1.1. Classical single mass point . 190
1.2. Model of a mass point . . . . 193
1.3. Model of a quantified object . 195
1.4. The correspondence principle . 201
2. Models in Quantum Field Theory. . 205
2.0. Preliminary remarks . . . . . . . 205
2.1. The main features of transition to the field theory . 206
2.2. The Fock space . . . . . . . . . . . . . . 207
2.3. The Minkowski space and axiomatics . 212
2.4. Fields operators . . . . . . . . . . . . 215
2.5. Scattering and perturbation theory . . 218
2.6. Additional remarks . . . . . . . . . . 221
Chapter 5. Structural Analysis of Discrete Equations 223
0. Introductory Remarks . . . . . . . 223
1. Perturbation of Model Equations . 224
1.0. Preliminary remarks . . 224
1.1. One-dimensional case . . . . . 224
1.2. Two-dimensional case .... . 227
2. The Formal Theory of Solvability . . 229
References 235
Index 241
To the Reader
The book is divided into chapters, chapters into sections, and
sections into subsections. The numbering of formulas, theorems,
and statements is its own within every section. When a relation or
theorem is cited in the framework of a given section, only its number
is indicated, and when a relation or theorem from some other section
is cited, that section or subsection is indicated in addition to the
number. When necessary, a chapter is also indicated.
A number in brackets means a reference to the corresponding
number in the list of references. The reference does not mean that
the indicated book or article is the only source (or the initial source)
of the revelant information.
The use of Halmos' sign • meaning the end of a proof (maybe
only outlined) or emphasizing the absence of a proof is not com-
pletely formalized. In certain cases it is omitted.
As a rule, definitions are not separated as a special paragraph but
are usually entered into the main text. Concepts being defined are
given in italics. Elements of the text which are specially emphasized
are given in bold face.
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