pref.tex

Concrete_Mathematics_2nd_Ed_TeX_Source_Code

The margins also include
\g This was the most enjoyable course I've ever had. But it might be nice
to summarize the material as you go~along.\g
direct quotations from famous mathematicians of past generations,
giving the actual words in which they announced some of their
fundamental discoveries. Somehow it seems appropriate to
mix the words of "Leibniz", "Euler", "Gauss", and others with those of
the people who will be continuing the work. Mathematics is an
ongoing endeavor for people everywhere; many strands are being woven
into one rich fabric.

This book contains more than 500 "exercises", divided into six
\g I see:\par Concrete mathematics means drilling.\g
"!exercises, levels of"
categories:

\def\\#1{\smallskip\item{$\bullet$}{\subsubtitle#1\enspace}\ignorespaces}
\\{\kern-.05em Warmups} are exercises that {\sc every reader} should try to
	do when first reading the material.
\\{Basics} are exercises to develop facts that are best learned by trying
\g The homework was tough but I learned a lot. It was worth every hour.\g
	one's own derivation rather than by reading somebody else's.
\\{Homework exercises} are problems intended to deepen an understanding
	of material in the current chapter.
\\{Exam problems} typically involve ideas from two or more chapters
	simultaneously; they are generally intended for use in take-home
\g Take-home exams are vital\dash---keep them.\g
	exams (not for in-class exams under time pressure).
\\{Bonus problems} go beyond what an average student of concrete
	mathematics is expected to handle while taking a course based
\g Exams were harder than the homework led me to expect.\g
	on this book; they extend the text in interesting ways.
\\{Research problems} may or may not be humanly solvable, but the ones
	presented here seem to be worth a try (without time pressure).
\smallskip\noindent
"Answers" to all the exercises appear in Appendix A, often with
additional information about related results. (Of course, the ``answers''
to research problems are incomplete; but even in these cases,
partial results or hints
are given that might prove to be helpful.) Readers are encouraged to
look at the answers, especially the answers to the warmup problems,
but only {\sc after} making a serious attempt to solve the problem
\g Cheaters may pass this course by just copying the answers, but they're
only "cheating" themselves.\g
without peeking.

We have tried in Appendix C to give proper credit to the sources of
each exercise, since a great deal of creativity and/or luck often goes into the
design of an instructive problem. Mathematicians have unfortunately
developed a tradition of borrowing exercises without any acknowledgment;
we believe that the opposite tradition, practiced for example by books
and magazines about chess (where names, dates, and locations of original chess
problems are routinely specified) is far superior. However, we have not
\g Difficult exams don't take into account students who have
other classes to~prepare~for.\g
been able to pin down the sources of many problems that have become
part of the folklore. If any reader knows the origin of an exercise for
which our citation is missing or inaccurate, we would be glad to learn
the details so that we can correct the omission in subsequent
editions of this book.

The "typeface" used for mathematics throughout this book is a new
design by Hermann "Zapf"~[|knuth-zapf|],
commissioned by the "American Mathematical
Society" and developed with the help of a committee that included
B.~"Beeton", R.\thinspace P. "Boas", L.\thinspace K. "Durst",
D.\thinspace E. "Knuth",
P.~"Murdock", R.\thinspace S. "Palais", P.~"Renz",
 E.~"Swanson", S.\thinspace B. "Whidden",
and W.\thinspace B. "Woolf". The underlying philosophy of Zapf's design is
to capture the flavor of mathematics as it might be written by a
mathematician with excellent handwriting. A handwritten rather than
mechanical style is appropriate because people generally create mathematics
with pen, pencil, or chalk. (For example, one of the trademarks of
the new design is the symbol for zero, `$0$', which is slightly pointed
at the top because a handwritten zero rarely closes together smoothly
\g I'm unaccustomed to this face.\g
when the curve returns to its starting point.) The letters are upright,
not italic, so that subscripts, superscripts, and accents are more easily
fitted with ordinary symbols. This new type family has been named
{\sl "AMS~Euler"}, after the great Swiss mathematician Leonhard "Euler"
(1707--1783) who discovered so much of mathematics as we know it today.
The alphabets include Euler Text ($Aa\,Bb\,Cc$ through $Xx\,Yy\,Zz$),
Euler Fraktur ($\frak Aa\,Bb\,Cc$ through $\frak Xx\,Yy\,Zz$), and
Euler Script Capitals ($\scr A\,B\,C$ through $\scr X\,Y\,Z$), as
well as Euler Greek ($A\alpha\,B\beta\,\Gamma\gamma$ through
$X\chi\,\Psi\psi\,\Omega\omega$) and special symbols such as
$\wp$ and~$\aleph$. We are especially pleased to be able to inaugurate the
Euler family of typefaces in this book, because Leonhard Euler's spirit
truly lives on every page: Concrete mathematics is Eulerian mathematics.

The authors are extremely grateful to
\g Dear prof: Thanks for (1)~the "pun"s, (2)~the subject matter.\g
Andrei "Broder", Ernst~"Mayr",
Andrew "Yao", and Frances "Yao", who contributed
greatly to this book during the years that they taught Concrete
Mathematics at Stanford.
 Furthermore we offer 1024 thanks to the
teaching assistants who creatively transcribed what took place in class
each year and who helped to design the examination questions; their
names are listed in Appendix~C\null. This book, which is essentially a
compendium of sixteen years' worth of lecture notes, would have
been impossible without their first-rate work.

\looseness=-1
Many other people have helped to make this book a reality. For example,
\g I don't see how what I've learned will ever help me.\g
we wish to commend the students at "Brown", "Columbia", "CUNY",
"Princeton", "Rice", and "Stanford"
who contributed the choice graffiti and helped to debug our first drafts.
Our contacts at "Addison-Wesley" were especially efficient and helpful;
in particular, we wish to thank our publisher (Peter "Gordon"), production
supervisor (Bette "Aaronson"), designer (Roy "Brown"), and copy editor
(Lyn "Dupr\'e"). The "National Science Foundation" and the "Office of Naval
Research" have given invaluable support.
"Cheryl Graham" was tremendously helpful as we prepared the index.
 And above all, we wish to thank our wives (Fan, Jill, and Amy)
"!Knuth, Jill" "!Chung, Fan" "!Patashnik, Amy"
\g I had a lot of trouble in this class, but I know it sharpened my
math skills and my thinking skills.\g
for their patience, support, encouragement, and ideas.

This second edition features a new Section 5.8, which describes some
important ideas that Doron "Zeilberger" discovered shortly after the first
edition went to press. Additional improvements to the first printing can
also be found on almost every page.

We have tried to produce a perfect book, but we are imperfect authors.
Therefore we solicit help in correcting any mistakes that we've made.
A~"reward" of \$2.56 will gratefully be paid to the first finder of
any error, whether it is mathematical, historical, or typographical.
\g I would advise the casual student to stay away from this course.\g

\smallskip
{\advance\baselineskip-1pt
\halign to\hsize{\sl#\hfil\tabskip=0pt plus 1fil&\hfil#\tabskip=0pt\cr
Murray Hill, New Jersey&\dash---RLG\cr
"!Knuth, Don" "!Graham, Ron" "!Patashnik, Oren"
and Stanford, California&DEK\cr
May 1988 and October 1993&OP\cr}
}
\vfill\eject
\tracingpages=1
\beginchapter {} A Note on Notation

SOME OF THE SYMBOLISM in this book has not (yet?) become standard.
Here is a list of "notation"s that might be unfamiliar to readers
who have learned similar material from other books,
together with the page numbers where these notations are explained.
(See the general index, at the end of the book, for references to more
standard notations.)

\smallskip
\begingroup \advance\baselineskip 7pt plus 1pt \advance\lineskip 7pt plus 1pt
\advance\lineskiplimit7pt
\halign to\hsize{\displaymath#$\hfill\tabskip10pt plus20pt&
 #\hfil&\hfil#\tabskip0pt\cr
\openup5pt
\hbox{\sl Notation}&\sl Name&\sl Page\cr
\noalign{\vskip1pt}
\ln x&natural logarithm: $\log_e x$&|nn:ln|\cr
\lg x&binary logarithm: $\log_2 x$&|nn:lg|\cr
\log x&common logarithm: $\log_{10}x$&|nn:log|\cr
\lfloor x\rfloor&floor: $\max@@
 \{\,n\mid n\le x,\hbox{ integer $n\,\}$}$&|nn:floor|\cr
\lceil x\rceil&ceiling: $\min@@
 \{\,n\mid n\ge x,\hbox{ integer $n\,\}$}$&|nn:ceil|\cr
x\bmod y&remainder: $x-y\lfloor x/y\rfloor$&|nn:mod|\cr
\{x\}&fractional part: $x\bmod 1$&|nn:fracpt|\cr
\noalign{\vskip2pt}
\sum f(x)\,\delta x&indefinite summation&|nn:indef-sum|\cr
\sum\nolimits_a^b f(x)\,\delta x&definite summation&|nn:def-sum|\cr
\noalign{\vskip2pt}
x\_n&falling factorial power: $x!/(x-n)!$&|nn:fall|,\thinspace|nn:gfall|\cr
x\_^n&rising factorial power: $\Gamma(x+n)/\Gamma(x)$&|nn:rise|,\thinspace
                                                      |nn:grise|\cr
n\?&subfactorial: $n!/0!-n!/1!+\cdots+(-1)^nn!/n!$&|nn:subfact|\cr
\Re z&real part: $x$, if $z=x+iy$&|nn:realpart|\cr
\noalign{\g \vskip-15.5pt If you don't under\-stand what the x denotes at the
 bottom of this page, try asking your Latin professor
 instead of your math professor.\g}
\Im z&imaginary part: $y$, if $z=x+iy$&|nn:realpart|\cr
H_n&harmonic number: $1/1+\cdots+1/n$&|nn:hn|\cr
H_n^{(x)}&generalized harmonic number: $1/1^x+\cdots+1/n^x$&|nn:hn-gen|\cr
f^{(m)}(z)&$m$th derivative of $f$ at $z$&|nn:mth-deriv|\cr
{n\brack@m@}&Stirling cycle number (the ``first kind'')&|nn:brack|\cr
{n\brace m}&Stirling subset number (the ``second kind'')&|nn:brace|\cr
{n\euler m}&Eulerian number&|nn:euler|\cr
{\Euler nm}&Second-order Eulerian number&|nn:euler2|\cr
\noalign{\g Prestressed \hbox{concrete} mathematics is
concrete mathe\-matics that's preceded by
 a bewildering list of~notations.\g\vskip6pt}
(a_m\ldots a_0)_b&radix notation for $\sum_{k=0}^m a_kb^k$&|nn:radix|\cr
K(a_1,\ldots,a_n)&continuant polynomial&|nn:continuant|\cr
\hyp{a,b}cz&hypergeometric function&|nn:hyp|\cr
\#A&cardinality: number of elements in the set $A$&|nn:card|\cr
[z^n]\,f(z)&coefficient of $z^n$ in $f(z)$&|nn:coeff-brack|\cr
[\alpha\dts\beta]&closed interval: the set
 $\{x\mid \alpha\le x\le\beta\}$&|nn:interval|\cr
\[m=n]&$1\,$ if $m=n$, otherwise $0\,$*&|nn:iverson|\cr
\[m\divides n]&$1\,$ if $m$ divides $n$, otherwise $0\,$*&|nn:div|\cr
\[m\edivides n]&$1\,$ if $m$ exactly divides $n$, otherwise $0\,$*&|nn:ediv|\cr
\[m\rp n]&$1\,$ if $m$ is relatively prime to $n$, otherwise $0\,$*&|nn:rp|\cr}
\endgroup\bigskip\noindent
\llap{*}In general, if $S$ is any statement that can be true or false, the
bracketed notation
$\[S]$ stands for $1$~if $S$~is true, $0$ otherwise.

Throughout this text, we use single-quote marks (`\dots') to delimit
text as it is {\it written},
double-quote marks (``\dots'')
"!quotation marks"
 for a phrase as it is {\it spoken}. Thus, the string of letters `string'
\g Also `nonstring' is a~string.\g
is sometimes called a ``string.\qback''

An expression of the form `$a/bc$' means the same as `$a/(bc)$'. Moreover,
"!parenthesis conventions"
$\log x/\!@\log y=(\log x)/(\log y)$ and $2@n!=2(n!)$.

\bye