COSMIC STRINGS IN THE WIRE APPROXIMATION

This free book is a comprehensive survey of the current state of knowledge about the
dynamics and gravitational properties of cosmic strings treated in the idealized
classical approximation as line singularities described by the Nambu-Goto
action. The author’s purpose is to provide a standard reference to all work that
has been published since the mid-1970s and to link this work together in a
single conceptual framework and a single notational formalism. A working
knowledge of basic general relativity is assumed. The ebook will be essential
reading for researchers and postgraduate students in mathematics, theoretical
physics, and astronomy interested in cosmic strings.

One of the most striking successes of modern science has been to reduce the
complex panoply of dynamical phenomena we observe in the world around usfrom
the build-up of rust on a car bumper to the destructive effects of cyclonic
winds-to the action of only four fundamental forces: gravity, electromagnetism,
and the strong and weak nuclear forces. This simple picture of four fundamental
forces, which became evident only after the isolation of the strong and weak
nuclear forces in the 1930s, was simplified even further when Steven Weinberg in
1967 and Abdus Salam in 1968 independently predicted that the electromagnetic
and weak forces would merge at high temperatures to form a single electroweak
force.
The Weinberg-Salam model of electroweak unification was the first practical
realization of the Higgs mechanism, a theoretical device whereby a system of
initially massless particles and fields can be given a spectrum of masses by
coupling it to a massive scalar field. The model has been extremely successful
not only in describing the known weak reactions to high accuracy, but also in
predicting the masses of the carriers of the weak force, the W± and ZO bosons,
which were experimentally confirmed on their discovery in 1982-83.

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This free ebook is about one big idea: You can synthesize a variety of complicated
functions from pure sinusoids in much the same way that you produce a major chord
by striking nearby C, E, G keys on a piano. A geometric version of this idea forms
the basis for the ancient Hipparchus-Ptolemy model of planetary motion (Almagest,
2nd century see Fig. 1.2). It was Joseph Fourier (Analytical Theory of Heat, 1815),
however, who developed modern methods for using trigonometric series and integrals
as he studied the flow of heat in solids. Today, Fourier analysis is a highly
evolved branch of mathematics with an incomparable range of applications and with
an impact that is second to none (see Appendix 1). If you are a student in one of
the mathematical, physical, or engineering sciences, you will almost certainly find
it necessary to learn the elements of this subject. My goal in writing this book is
to help you acquire a working knowledge of Fourier analysis early in your career.
If you have mastered the usual core courses in calculus and linear algebra, you
have the maturity to follow the presentation without undue difficulty. A few of the
proofs and more theoretical exercises require concepts (uniform continuity, uniform
convergence, . . . ) from an analysis or advanced calculus course. You may choose to
skip over the difficult steps in such arguments and simply accept the stated results.
The text has been designed so that you can do this without severely impacting
your ability to learn the important ideas in the subsequent chapters. In addition, I
will use a potpourri of notions from undergraduate courses in differential equations
[solve y
(x) + ?y(x) = 0, y
(x) = xy(x), y

(x) + ?2y(x) = 0, . . . ], complex analysis
(Euler’s formula: ei? = cos ?+i sin ?, arithmetic for complex numbers, . . . ), number
theory (integer addition and multiplication modulo N, Euclid’s gcd algorithm, . . . ),
probability (random variable, mean, variance, . . . ), physics (F = ma, conservation
of energy, Huygens’ principle, . . . ), signals and systems (LTI systems, low-pass
filters, the Nyquist rate, . . . ), etc. You will have no trouble picking up these
concepts as they are introduced in the text and exercises.
If you wish, you can find additional information about almost any topic in
this book by consulting the annotated references at the end of the corresponding
chapter. You will often discover that I have abandoned a traditional presentation
in favor of one that is in keeping with my goal of making these ideas accessible
to undergraduates. For example, the usual presentation of the Schwartz theory
of distributions assumes some familiarity with the Lebesgue integral and with
a graduate-level functional analysis course. In contrast, my development of ?,
X, . . . in Chapter 7 uses only notions from elementary calculus. Once you master
this theory, you can use generalized functions to study sampling, PDEs, wavelets,
probability, diffraction, . . . .
The exercises (541 of them) are my greatest gift to you! Read each chapter
carefully to acquire the basic concepts, and then solve as many problems as you
can. You may find it beneficial to organize an interdisciplinary study group, e.g.,
mathematician + physicist + electrical engineer. Some of the exercises provide
routine drill: You must learn to find convolution products, to use the FT calculus,
to do routine computations with generalized functions, etc. Some supply historical
perspective: You can play Gauss and discover the FFT, analyze Michelson and
Stratton’s analog supercomputer for summing Fourier series, etc. Some ask for
mathematical details: Give a sufficient condition for . . . , given an example of . . . ,
show that, . . . . Some involve your personal harmonic analyzers: Experimentally
determine the bandwidth of your eye, describe what would you hear if you replace
notes with frequencies f1, f2, . . . by notes with frequencies C/f1, C/f2, . . . . Some
prepare you for computer projects: Compute ? to 1000 digits, prepare a movie for
a vibrating string, generate the sound file for Risset’s endless glissando, etc. Some
will set you up to discover a pattern, formulate a conjecture, and prove a theorem.
(It’s quite a thrill when you get the hang of it!) I expect you to spend a lot of time
working exercises, but I want to help you work efficiently. Complicated results are
broken into simple steps so you can do (a), then (b), then (c), . . . until you reach
the goal. I frequently supply hints that will lead you to a productive line of inquiry.
You will sharpen your problem-solving skills as you take this course.

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This free ebook volume is a self-contained, exhaustive exposition of the extrapolation
methods theory, and of the various algorithms and procedures for accelerating
the convergence of scalar and vector sequences. Many subroutines (written in
FORTRAN 77) with instructions for their use are provided on a floppy disk in
order to demonstrate to those working with sequences the advantages of the use
of extrapolation methods. Many numerical examples showing the effectiveness of
the procedures and a consequent chapter on applications are also provided –
including some never before published results and applications. Although
intended for researchers in the field, and for those using extrapolation methods
for solving particular problems, this volume also provides a valuable resource
for graduate courses on the subject.

The aim of this free ebook is twofold. First it is a self-contained and, as
much as possible, exhaustive exposition of the theory of extrapolation
methods and of the various algorithms and procedures for accelerating
the convergence of scalar and vector sequences. Our second aim is to
convince people working with sequences to use extrapolation methods
and to help them in this respect. This is the reason why we provide many
subroutines (written in FORTRAN 77) with their directions for use. We
also include many numerical examples showing the effectiveness of the
procedures and a quite consequent chapter on applications. In order to
reduce the size of the free book the proofs of the theoretical results have been
omitted and replaced by references to the existing literature. However,
on the other hand, some results and applications are given here for the
first time. We have also included suggestions for further research.

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Handbook of Integral Equations, Second Edition, a unique reference
for engineers and scientists, contains over 2,500 integral equationswith
solutions, aswell as analytical and numerical methods for solving linear
and nonlinear equations. It considers Volterra, Fredholm,Wiener–Hopf,Hammerstein,
Urysohn, and other equations,which arise inmathematics, physics,
engineering sciences, economics, etc. In total, the number of equations
described is an order of magnitude greater than in any other book available.
The second edition has been substantially updated, revised, and extended.
It includes new chapters on mixed multidimensional equations, methods
of integral equations for ODEs and PDEs, and about 400 new equations with
exact solutions. It presents a considerable amount of new material on Volterra,
Fredholm, singular, hypersingular, dual, and nonlinear integral equations,
integral transforms, and special functions. Many examples were added
for illustrative purposes. The new edition has been increased by a total
of over 300 pages. Note that the first part of the book can be used as a
database of test problems for numerical and approximate methods for solving
linear and nonlinear integral equations. We would like to express our deep
gratitude to Alexei Zhurov and Vasilii Silvestrov for fruitful discussions.
We also appreciate the help of Grigory Yosifian in translating new sections
of this book and valuable remarks. The authors hope that the handbookwill prove
helpful for a wide audience of researchers, college and university teachers,
engineers, and students in various fields of appliedmathematics, mechanics,
physics, chemistry, biology, economics, and engineering sciences.
A. D. Polyanin
A. V. Manzhirov

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This book contains a collection of general mathematical results, formulas,
and integrals that occur throughout applications of mathematics. Many of
the entries are based on the updated fifth edition of Gradshteyn and Ryzhik’s
”Tables of Integrals, Series, and Products,” though during the preparation of
the book, results were also taken from various other reference works.
The material has been arranged in a straightforward manner, and for the
convenience of the user a quick reference list of the simplest and most
frequently used results is to be found in Chapter 0 at the front of the book.
Tab marks have been added to pages to identify the twelve main subject areas
into which the entries have been divided and also to indicate the main
interconnections that exist between them. Keys to the tab marks are to
be found inside the front and back covers.
The Table of Contents at the front of the book is sufficiently detailed to
enable rapid location of the section in which a specific entry is to be found,
and this information is supplemented by a detailed index at the end of the book.
In the chapters listing integrals, instead of displaying them in their canonical
form, as is customary in reference works, in order to make the tables more
convenient to use, the integrands are presented in the more general form in
which they are likely to arise. It is hoped that this will save the user the
necessity of reducing a result to a canonical form before consulting the tables.
Wherever it might be helpful, material has been added explaining the idea underlying
a section or describing simple techniques that are often useful in the application
of its results. Standard notations have been used for functions, and a list of these
together with their names and a reference to the section in which they occur or are
defined is to be found at the front of the book. As is customary with tables of
indefinite integrals, the additive arbitrary constant of integration has always
been omitted. The result of an integration may take more than one form, often
depending on the method used for its evaluation, so only the most common forms
are listed. A user requiring more extensive tables, or results involving the less
familiar special functions, is referred to the short classified reference list at
the end of the book. The list contains works the author found to be most useful
and which a user is likely to find readily accessible in a library, but it is in
no sense a comprehensive bibliography. Further specialist references are to
be found in the bibliographies contained in these reference works. Every effort
has been made to ensure the accuracy of these tables and, whenever possible,results
have been checked by means of computer symbolic algebra and integration programs,
but the final responsibility for errors must rest with the author.

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