This free ebook differs from others in the field in that it has been prepared very
much with students and their needs in mind, having been classroom tested over
many years. It is a true “learner’s book” made for students who require a
deeper understanding of probability and statistics. It presents the fundamentals
of the subject along with concepts of probabilistic modelling, and the process
of model selection, verification and analysis. Furthermore, the inclusion of
more than 100 examples and 200 exercises (carefully selected from a wide range
of topics), along with a solutions manual for instructors, means that this text
is of real value to students and lecturers across a range of engineering
disciplines.
This free ebook was written for an introductory one-semester or two-quarter course
in probability and statistics for students in engineering and applied sciences. No
previous knowledge of probability or statistics is presumed but a good understanding
of calculus is a prerequisite for the material.
The development of this book was guided by a number of considerations
observed over many years of teaching courses in this subject area, including the
following:
*As an introductory course, a sound and rigorous treatment of the basic
principles is imperative for a proper understanding of the subject matter
and for confidence in applying these principles to practical problem solving.
A student, depending upon his or her major field of study, will no doubt
pursue advanced work in this area in one or more of the many possible
directions. How well is he or she prepared to do this strongly depends on
his or her mastery of the fundamentals.
*It is important that the student develop an early appreciation for applications.
Demonstrations of the utility of this material in nonsuperficial applications
not only sustain student interest but also provide the student with
stimulation to delve more deeply into the fundamentals.
*Most of the students in engineering and applied sciences can only devote one
semester or two quarters to a course of this nature in their programs.
Recognizing that the coverage is time limited, it is important that the material
be self-contained, representing a reasonably complete and applicable body of
knowledge.
The choice of the contents for this free ebook is in line with the foregoing
observations. The major objective is to give a careful presentation of the
fundamentals in probability and statistics, the concept of probabilistic modeling,
and the process of model selection, verification, and analysis. In this text,
definitions and theorems are carefully stated and topics rigorously treated
but care is taken not to become entangled in excessive mathematical details.
Practical examples are emphasized; they are purposely selected from many
different fields and not slanted toward any particular applied area. The same
objective is observed in making up the exercises at the back of each chapter.
A Transition to Abstract Mathematics was written under the assumption that
students do not yet know how to read upper level mathematics texts. Since the
primary purpose of the book is to teach students to write with formal rigor, and
since I naturally presume they do not yet appreciate exposition written in that
form, two overriding features of style defined the first edition: a loose and informal
expository style of writing, and an airtight composition and organization of
the logic, so that no student could ever say that any necessary detail had been
overlooked or omitted. Consequently, the scope of the first edition was rather
narrow and forward focused, where every exercise had an important role in the
story and there were no characters too peripheral to the plot.
I believe the second edition maintains the benefits of the first edition’s features
but is improved in several ways. First, the exposition is still written to the
student, but it is tighter and more efficient than before. Second, there are many
more exercises than in the first edition. Many of these are essential in that they
are the logical basis of later results. The Instructor’s Guide and Solutions Manual
points out which exercises simply must be either assigned or at least discussed
because they undergird later results. Others may be assigned, discussed casually,
or omitted altogether.
A third and major change to the second edition is that exercises are now integrated
into the flow of the material instead of being placed at the end of each
section. I believe this arrangement has several advantages. It better facilitates
the students’ understanding of how the mathematics is built, one step at a time,
because it requires their continual participation in that process at every step. In the
second edition, the text speaks clearly to the students and then presents them with
exercises right on the heels of every new concept. It also should make daily course
organization easier for the instructor, in that it is always clear which exercises may
be assigned after a particular day’s class meeting.
Other changes to the second edition include a reorganization of the material
that comprised Chapter 2 in the first edition. Introductory proof-writing material
on set and real number properties has now been divided into two chapters, and the
order of the material basically reversed from the first edition. Thus the students
first theorems involve basic algebraic properties of numbers, which might be a
simpler place for them to begin to write proofs than set properties. Chapter 1
now includes a section that enumerates different techniques of proof writing, with
plenty of examples but no expectation that a student yet knows how or in what
circumstances to employ these techniques. Finally, with exercises integrated into
the exposition, certain sections that were quite long in the first edition have now
been divided into more sections of more manageable length.
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.
About Eric Weisstein, Creator of MathWorld
Eric Weisstein
Eric W. Weisstein began compiling scientific encyclopedias as a high school student nearly twenty years ago. Born in Bloomington, Indiana in 1969, Weisstein studied physics and astronomy at Cornell University and Caltech and received his Ph.D. from Caltech in 1996. In 1995, Weisstein took the vast collection of mathematical facts that he had been accumulating since his teenage years and began to deploy them on the early internet. These pioneering efforts at organizing and presenting online content helped define a paradigm that has subsequently been followed by other large-scale informational projects on the web.
Weisstein is responsible for the award-winning MathWorld site–the world’s most widely accessed online mathematics resource. He is also a prolific communicator of mathematics, having authored most of MathWorld’s nearly 13,000 articles.
Weisstein joined Wolfram Research in 1999 and unveiled the MathWorld website at mathworld.wolfram.com later that year. As a member of the Scientific Information Group at Wolfram Research, Weisstein has led the development of MathWorld, continuing to expand its scope and depth and fulfilling his vision for bringing accessible mathematical and scientific knowledge to the widest possible audience.
An expert Mathematica user since the 1990s, Weisstein has not only used Mathematica to develop content for MathWorld, but also continues to work closely with the main development teams at Wolfram Research, providing input and advice for future features of Mathematica. To date, he has authored more than 4000 Mathematica notebooks containing useful algorithms and visualizations that are freely downloadable from MathWorld.
Weisstein is a sought-after speaker on mathematics communication, scientific computing, and knowledge management on the internet. He has participated in a number of important standards initiatives and led Wolfram Research’s contribution on a National Science Digital Library project.
Weisstein is an advocate for author’s rights, especially in the area of electronic publication, as well as a consultant for the CBS television crime drama NUMB3RS. As a committed encyclopedist, he continues to assemble informational sites on other scientific and scholarly topics.
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.
Free Ebooks Download | Most Popular Ebooks
Free Ebooks Download! Most Popular Ebooks!
Entries (RSS) and Comments (RSS).
Free Website Directory