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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In recent years important progress has been made in the study of linear
operators by extending to closed operators many fundamental theorems
which were known for bounded operators. The applications of the
theory permit unification of a series of diverse questions in differential
equations, which leads to significant results with substantial simplification.
The aim of this book is to present a systematic treatment of the
theory of unbounded linear operators in normed linear (not necessarily
Hilbert) spaces with applications to differential equations. Except for
the last chapter, the material is quite self-contained. The reader is
assumed to be familiar with metric spaces and real variable theory.
The book could be introduced in a course in functional analysis,
particularly when linear operators are studied. By considering differential
or integro-differential equations from the point of view of operator
theory, the material may also be useful to those concerned with the more
applied aspects of analysis.
Chapter I gives the elementary theory of normed linear spaces and,
in particular, Hilbert space, which is used throughout the book. In
Chapters II and IV, the basic theory of unbounded linear operators is
developed with the normed linear spaces assumed complete and the operators
assumed closed only when needed. The surprising thing is that the
proofs are as uncomplicated as the proofs for bounded operators. Thus,
the reader who is meeting the theory of linear operators for the first time
is not distracted by any of the additional complications which one expects
when the operator is not required to be bounded. Chapter III introduces
the class of strictly singular operators which includes the class of compact
operators. The main reason for considering such a class is to generalize
the Riesz-Schauder theory for compact operators. In Chapter V some
of the main theorems concerning perturbations of unbounded operators
are given and are later applied to ordinary differential operators. In
Chapter VI a number of the powerful theorems proved in the earlier
chapters are used to examine existence and uniqueness of solutions of
certain differential equations. The reader is not required to have any
previous knowledge of differential equations. Chapter VII sketches the
interplay between functional analysis and ((hard” classical analysis in
the Atudy of elliptic partial differential equation:;;.
For thoRO unncllllltinLcd with the Rubjcet matter, examples and motivu.
t.ion~ for <:crLuin deliniLiollH lind proofH n.ro mentioned in ordor to give
some feeling for what is going on. Simple notation is used so that it is not
necessary to refer continually to a collection of symbols in the rear of
the book.
In the spring of 1964 I had the great privilege of visiting a number of
mathematicians in various parts of the Soviet Union in order to discuss
the contents of this book. To Professors 1. M. Gelfand and O. Ladyzenskaya
goes my gratitude for inviting me to speak about my work at
their respective seminars at Moscow State University and the University
of Leningrad. I wish to thank Professors M. S. Birman, 1. C. Gokhberg,
M. G. Krein, A. S. Markus, and M. A. Naimark for their valuable suggestions
and comments. I am especially indebted to Professors Gokhberg
and Markus for their advice during the three days we spent together in
Kishinev.
My gratitude goes to Professors G. Stampacchia and H. G. Tillman
for arranging my stay at the Universities of Pisa and Mainz, respectively,
where I benefited from their knowledge and experience. Most of the
manuscript was written while I was on leave at the University of Pisa.
I wish to express my profound thanks to Professor T. Kato, who was
kind enough to show me portions of his manuscript concerned with perturbation
theory. Our conversation at Berkeley and our correspondence
have been of great help to me.
My appreciation is extended to Professors T. W. Gamelin, G. C. Rota,
and R. J. Whitley, who read portions of the manuscript and gave suggestions.
Professor Rota was a constant source of encouragement to me in
the preparation of this manuscript.
I am especially indebted to Professor R. S. Freeman and my class of
1964-1965 for going through the entire manuscript. Professor Freeman
was also kind enough to discuss partial differential equations with me
these many months and to offer valuable suggestions.
By writing this book, I have come to realize fully why authors express
their gratitude to their typists. It was indeed my good fortune to have
Mrs. Ouida Taylor type the manuscript for me. Her accuracy, speed, and
artistic layout of each page saved me many months of tedious work.
Finally, my appreciation and thanks go to the Mathematics Division
of the Air Force Office of Scientific Research for supporting the major
portion of this book under grant number AF OSR 495-64.

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h

The goal of physics is to provide an understanding of the physical world by developing theories
based on experiments. A physical theory is essentially a guess, usually expressed mathematically,
about how a given physical system works. The theory makes certain predictions
about the physical system which can then be checked by observations and experiments. If the
predictions turn out to correspond closely to what is actually observed, then the theory
stands, although it remains provisional. No theory to date has given a complete description of
all physical phenomena, even within a given subdiscipline of physics. Every theory is a work in
progress.
The basic laws of physics involve such physical quantities as force, velocity, volume, and
acceleration, all of which can be described in terms of more fundamental quantities. In mechanics,
the three most fundamental quantities are length (L), mass (M), and time (T); all
other physical quantities can be constructed from these three.

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Everyone has observed chemical reactions involving pyrotechnic mixtures.
Beautiful 4th of July fireworks, highway distress signals,
solid fuel boosters for the Space Shuttle, and the black powder used
by muzzle-loading rifle enthusiasts all have a common technical background.
The chemical principles underlying these high-energy materials
have been somewhat neglected in the twentieth century by academic
and industrial researchers. Most of the recent work has been goaloriented
rather than fundamental in nature (e.g. , produce a deeper
green flame). Many of the significant results are found in military
reports, and chemical fundamentals must be gleaned from many pages
of test results.
Much of today’s knowledge is carried in the heads of experienced
personnel. Many of these workers acquired their initial training during
World War II, and they are presently fast approaching (if not
already past) retirement age. This is most unfortunate for future
researchers. Newcomers have a difficult time acquiring the skills and
knowledge needed to begin productive experiments. A background
in chemistry is helpful, but much of today’s modern chemistry curriculum
will never be used by someone working in pyrotechnics and
explosives. Further, the critical education in how to safely mix,
handle, and store high-energy materials is not covered at all in today’s
schools and must be acquired in “on-the-job” training.
This book is an attempt to provide an introduction to the basic
principles of high-energy chemistry to newcomers and to serve as a
review for experienced personnel. It can by no means substitute
for the essential “hands on” experience and training necessary to
safely work in the field, but I hope that it will be a helpful companion.
An attempt has been made to keep chemical theory simple and
directly applicable to pyrotechnics and explosives. The level approaches
that of an introductory college course, and study of this
text may prepare persons to attend professional meetings and seminars
dealing with high-energy materials and enable them to intelligently
follow the material being presented. In particular, the International
Pyrotechnic Seminars, hosted biannually by the Illinois
Institute of Technology Research Institute in conjunction with the
International Pyrotechnics Society, have played a major role in
bringing researchers together to discuss current work. The Proceedings
of the nine seminars held to date contain a wealth of information
that can be read and contemplated by persons with adequate
introduction to the field of high-energy chemistry.
I would like to express my appreciation to Mr. Richard Seltzer of
the American Chemical Society and to Dr. Maurits Dekker of Marcel
Dekker, Inc. for their encouragement and their willingness to recognize
pyrotechnics as a legitimate branch of modern chemistry. I
am grateful to Washington College for a sabbatical leave in 1983 that
enabled me to finalize the manuscript. I would also like to express
my thanks to many colleagues in the field of pyrotechnics who have
provided me with data as well as encouragement, and to my 1983 and
1984 Summer Chemistry Seminar groups at Washington College for
their review of draft versions of this book. I also appreciate the
support and encouragement given to me by my wife and children as
I concentrated on this effort.
Finally, I must acknowledge the many years of friendship and
collaboration that I enjoyed with Dr. Joseph H. McLain, former
Chemistry Department Chairman and subsequently President of
Washington College. It was his enthusiasm and encouragement that
dragged me away from the norbornyl cation and physical organic
chemistry into the fascinating realm of pyrotechnics and explosives.
The field of high-energy chemistry lost an important leader when
Dr. McLain passed away in 1981.
John A. Conkling

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For high school students. Keyed to school syllabuses, it defines every mathematical
term and concept most people will ever need to know, in language that is easy to
understand.

Clear, concise, and informative
Newly revised and expanded, The Facts On File Dictionary of Mathematics includes
terms of general interest to the high school and general reader. It contains
more than 2,500 entries that explain, clearly and concisely, the most important
and commonly used terms in mathematics. Over 90 line drawings illustrate complex
mathematical concepts, and extensive cross references ensure the Dictionary’s
accessibility and ease of use. Over 200 new terms have been included with
emphasis on those in applied mathematics and computer science.

This dictionary is one of a series designed for use in schools. It is intended for students
of mathematics, but we hope that it will also be helpful to other science students
and to anyone interested in science. Facts On File also publishes dictionaries in
a variety of disciplines, including biology, chemistry, forensic science, marine science,
physics, space and astronomy, and weather and climate.
The Facts On File Dictionary of Mathematics was first published in 1980 and the
third edition was published in 1999. This fourth edition of the dictionary has been
extensively revised and extended. The dictionary now contains over 2,000 headwords
covering the terminology of modern mathematics. A totally new feature of
this edition is the inclusion of over 800 pronunciations for terms that are not in
everyday use. A number of appendixes have been included at the end of the book
containing useful information, including symbols and notation, symbols for physical
quantities, areas and volumes, expansions, derivatives, integrals, trigonometric formulae,
a table of powers and roots, and a Greek alphabet. There is also a list of Web
sites and a bibliography. A guide to using the dictionary has also been added to this
latest version of the book.
We would like to thank all the people who have cooperated in producing this book.
A list of contributors is given on the acknowledgments page. We are also grateful to
the many people who have given additional help and advice.

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