footnode.html

wqeqwvrw rkjqhwrjwq jkhrjqwhrwq jkhrwq

.
.
.
.
</PRE><DT><A NAME="11300">...prime</em></A><DD>
Two numbers <I>x</I> and <I>y</I> are
<em>relatively prime</em><A NAME=11239>&#160;</A><A NAME=11240>&#160;</A>
if there is no number other than one
that divides both <I>x</I> and <I>y</I> evenly.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="11715">...type.</A><DD>
Typically, on 32-bit computer 925#925.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="13873">...<I>i</I>.</A><DD>What else would it be?
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="15293">...NAME=15292>&#160;</A>.</A><DD>
Isomorphic is a fancy word that means
being of identical or similar form or shape or structure.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="16465">...<tt>div</tt>.</A><DD>
Of course, in C++ we can overload<A NAME=16093>&#160;</A>
the built-in operators.
However, having done so we would write an infix expression
rather than the prefix one.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="20181">...Landis</A><DD>
Russian mathematicians G. M. Adel'son-Vel'ski&#305; and E. M. Landis
published this result in 1962.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="21731">...trees.</A><DD>
Obviously since B-Trees are <I>M</I>-way trees,
the ``B'' in <em>B-Tree</em> does not stand for <em>binary</em>.
B-Trees were invented by R. Bayer and E. McCright in 1972,
so the ``B'' either stands for <em>balanced</em>
or <em>Bayer</em>-take your pick.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="27283">...HREF="page385.html#exercisepqueuesbinom"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>).</A><DD>
Isaac Newton<A NAME=27133>&#160;</A> discovered the binomial theorem in 1676
but did not publish a proof.
Leonhard Euler<A NAME=27134>&#160;</A> attempted a proof in 1774.
Karl Friedrich Gauss<A NAME=27135>&#160;</A>
produced the first correct proof in 1812.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="28783">...subsets.</A><DD>
<em>Stirling numbers of the second kind</em><A NAME=28651>&#160;</A>
are given by the formula
<P>1655#1655<P>
where <I>n</I><I>&gt;</I>0 and 1656#1656.
<P>
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="30487">...<tt>p</tt>.</A><DD>
Actually, the programmer must be careful when doing this kind of thing.
Because C++ cannot guarantee the order in which globals declared in
separate files are constructed,
in some circumstances it may be possible for <tt>operator new</tt>
and <tt>operator delete</tt> to be called
before the storage pool <tt>p</tt> has been initialized
by its constructor.
The solution to this is to use a technique
such as <em>double construction</em><A NAME=30389>&#160;</A>[<A HREF="page619.html#plauger" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page619.html#plauger">33</A>].
Such trickery is beyond the scope of this text.
<em>Caveat emptor</em>.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="32227">...<tt>Log2Ceil</tt>.</A><DD>
We assume that there exists the function
<tt>unsigned int Log2Ceil (unsigned int n);</tt>
which computes 1782#1782.
It is possible to compute this function in constant time
using simple bit manipulations.
Most computers have instructions that make the implementation trivial.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="33614">...space.</A><DD>
The reader may find it instructive to compare
Program&nbsp;<A HREF="page450.html#progsolution2c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page450.html#progsolution2c"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> with Program&nbsp;<A HREF="page268.html#progtree1c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page268.html#progtree1c"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> and Program&nbsp;<A HREF="page552.html#proggraph1c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page552.html#proggraph1c"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="33620">...space.</A><DD>
The reader may find it instructive to compare
Program&nbsp;<A HREF="page451.html#progsolution3c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page451.html#progsolution3c"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> with Program&nbsp;<A HREF="page270.html#progtree2c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page270.html#progtree2c"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> and Program&nbsp;<A HREF="page555.html#proggraph2c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page555.html#proggraph2c"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="33623">...class.</A><DD>
The declaration of the class itself has been omitted since
it follows directly from Programs&nbsp;<A HREF="page449.html#progsolution2h" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page449.html#progsolution2h"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> and&nbsp;<A HREF="page450.html#progsolution3h" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page450.html#progsolution3h"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="33649">...NAME=33394>&#160;</A>.</A><DD>
The table is named in honor of <em>Blaise Pascal</em><A NAME=33396>&#160;</A>
who published a treatise on the subject in&nbsp;1653.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="34152">...number!</A><DD>
Prime numbers of the form 2015#2015 are known as
<em>Mersenne primes</em><A NAME=33976>&#160;</A>.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="33980">...2018#2018.</A><DD>
For convenience, we use the notation 2019#2019
to denote 2020#2020.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="34600">...time.</A><DD>
In this chapter we assume that 601#601
and that 1296#1296.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="35376">...NAME=35375>&#160;</A>.</A><DD>
Unfortunately, the fame of bubble sort exceeds by far its practical value.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE><DT><A NAME="37868">...zero.</A><DD>
There is also the symmetrical case in which <I>i</I> is always <I>n</I>-1.
<PRE>.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
</PRE> </DL>
<P><ADDRESS>
<img src="bruno.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/bruno.gif" alt="Bruno" align=right>
<a href="javascript:if(confirm('http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/copyright.html  \n\nThis file was not retrieved by Teleport Pro, because it is addressed on a domain or path outside the boundaries set for its Starting Address.  \n\nDo you want to open it from the server?'))window.location='http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/copyright.html'" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/copyright.html">Copyright &#169; 1997</a> by <a href="javascript:if(confirm('http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/signature.html  \n\nThis file was not retrieved by Teleport Pro, because it is addressed on a domain or path outside the boundaries set for its Starting Address.  \n\nDo you want to open it from the server?'))window.location='http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/signature.html'" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/signature.html">Bruno R. Preiss, P.Eng.</a>  All rights reserved.
<P>
</ADDRESS>
</BODY>
</HTML>