This textbook presents the foundations of
tensor calculus and
the elements of tensor analysis. In addition,
the authors
consider numerous applications of tensors
to geometry, mechanics
and physics.
While developing tensor calculus, the authors
emphasize its
relationship with linear algebra. Necessary
notions and theorems
of linear algebra are introduced and proved
in connection with
the construction of the apparatus of tensor
calculus; prior
knowledge is not assumed. For simplicity
and to enable the reader
to visualize concepts more clearly, all exposition
is conducted
in three-dimensional space. The principal
feature of the book is
that the authors use mainly orthogonal tensors,
since such
tensors are important in applications to
physics and engineering.
With regard to applications, the authors
construct the general
theory of second-degree surfaces, study the
inertia tensor as
well as the stress and strain tensors, and
consider some problems
of crystallophysics. The last chapter introduces
the elements of
tensor analysis.
All notions introduced in the book, and also
the obtained
results, are illustrated with numerous examples
discussed in the
text. Each section of the book presents problems
(a total over
300 problems are given). Examples and problems
are intended to
illustrate, reinforce and deepen the presented
material. There
are answers to most of the problems, as well
as hints and
solutions to selected problems at the end
of the book.
Contents:
Vector Spaces
Multilinear Forms and Tensors
Linear Transformations and Second-Order Tensors
Reduction of the Matrix of Linear Transformation
to the Simplest
Form
The General Theory of Second-Degree Surfaces
Applications of Tensor Calculus to Some Problems
of Mechanics and
Physics
Foundations of Tensor Analysis
Readership: Upper level undergraduates, graduate
students and
researchers in physics and engineering.
340pp (approx.) Pub. date: Scheduled Winter
2003
ISBN 981-238-505-3
ISBN 981-238-506-1(pbk)
Although the Fields Medal does not have the
same public
recognition as the Nobel Prizes, they share
a similar
intellectual standing. It is restricted to
one field ・that of
mathematics ・and an age limit of 40 has
become an accepted
tradition. Mathematics has in the main been
interpreted as pure
mathematics, and this is not so unreasonable
since major
contributions in some applied areas can be
(and have been)
recognized with Nobel Prizes.
A list of Fields Medallists and their contributions
provides a
bird's-eye view of mathematics over the past
60 years. It
highlights the areas in which, at various
times, greatest
progress has been made. This volume does
not pretend to be
comprehensive, nor is it a historical document.
On the other
hand, it presents contributions from Fields
Medallists and so
provides a highly interesting and varied
picture.
The second edition of Fields Medallists'
Lectures features
additional contributions from the following
Medallists: Kunihiko
Kodaira (1954), Richard E Borcherds (1998),
William T Gowers (1998),
Maxim Kontsevich (1998), Curtis T McMullen
(1998) and Vladimir
Voevodsky (2002).
Readership: Mathematicians and mathematical
physicists.
Review of the 1st Editon
"The essays have an intrinsic interest
as the writers have
participated in mathematics at the highest
level. They reinforce
the impression of an essential unity in the
subject in spite of a
seeming diversity in research activities."
SIAM Review, 1999
900pp (approx.) Pub. date: Scheduled Winter
2003
ISBN 981-238-256-9
ISBN 981-238-259-3(pbk)
This book is an alternative and highly engaging
introduction
to the highlights of a typical undergraduate
mathematics course.
Building on very simple principles, it develops
these
mathematical highlights, known to every well-rounded
mathematician, in an intuitive and entertaining
way. The aim of
the book is to motivate and inspire the reader
to discover and
understand some of these truly amazing mathematical
structures
and ideas which are frequently not fully
grasped, pass unnoticed
or simply swamped in an undergraduate mathematics
course. For the
experienced mathematician the book offers
refreshing, often
enlightening, hindsight. For the novice it
is an exciting
intellectual journey.
Contents:
Numbers
Analysis
Algebra
Calculus and Differential Equations
Probability
Theoretical Physics
Appendices:
Exercises for the Reader
Further Reading
Basic Mathematical Background
Dictionary of Symbols
Readership: Mathematics students and teachers.
550pp (approx.) Pub. date: Scheduled Winter
2003
ISBN 981-238-554-1
ISBN 981-238-555-X(pbk)
August 2003 | Hardback | 592 pages 170
line diagrams 548 exercises | ISBN: 0-521-82332-3
Uniting dozens of seemingly disparate results
from different
fields, this book combines concepts from
mathematics and computer
science to present the first integrated treatment
of sequences
generated by ‘finite automata’. The authors
apply the theory
to the study of automatic sequences and their
generalizations,
such as Sturmian words and k-regular sequences.
And further, they
provide applications to number theory (particularly
to formal
power series and transcendence in finite
characteristic),
physics, computer graphics, and music. Starting
from first
principles wherever feasible, basic results
from combinatorics on
words, numeration systems, and models of
computation are
discussed. Thus this book is suitable for
graduate students or
advanced undergraduates, as well as for mature
researchers
wishing to know more about this fascinating
subject. Results are
presented from first principles wherever
feasible, and the book
is supplemented by a collection of 460 exercises,
85 open
problems, and over 1600 citations to the
literature.
Contents
Preface; 1. Stringology; 2. Number theory
and algebra; 3.
Numeration systems; 4. Finite automata and
other models of
computation; 5. Automatic sequences; 6. Uniform
morphisms and
automatic sequences; 7. Morphic sequences;
8. Frequency of
letters; 9. Characteristic words; 10. Subwords;
11. Cobham’s
theorem; 12. Formal power series; 13. Automatic
real numbers; 14.
Multidimensional automatic sequences; 15.
Automaticity; 16. k-regular
sequences; 17. Physics; A. Hints … for selected
exercises.
August 2003 | Paperback | 488 pages
| ISBN: 0-521-53931-5
(London Mathematical Society Lecture Note
Series, vol. 309)
Corings and comodules are fundamental algebraic
structures, which
can be thought of as both dualisations and
generalisations of
rings and modules. Introduced by Sweedler
in 1975, only recently
they have been shown to have far reaching
applications ranging
from the category theory including differential
graded categories
through classical and Hopf-type module theory
to non-commutative
geometry and mathematical physics. This is
the first extensive
treatment of the theory of corings and their
comodules. In the
first part, the module-theoretic aspects
of coalgebras over
commutative rings are described. Corings
are then defined as
coalgebras over non-commutative rings. Topics
covered include
module-theoretic aspects of corings, such
as the relation of
comodules to special subcategories of the
category of modules (sigma-type
categories), connections between corings
and extensions of rings,
properties of new examples of corings associated
to entwining
structures, generalisations of bialgebras
such as bialgebroids
and weak bialgebras, and the appearance of
corings in non-commutative
geometry.
Contents
Preface; Notations; 1. Coalgebras and comodules;
2. Bialgebras
and hopf algebras; 3. Corings and comodules;
4. Corings and
extensions of rings; 5. Corings and entwining
structures; 6. Weak
corings and entwining; Appendix.