...102#102.
The notation 103#103 denotes the floor function , which is defined as follows: For any real number x, 104#104 is the greatest integer less than or equal to x. While we are on the subject, there is a related function, the ceiling function , written 105#105. For any real number x, 106#106 is the smallest integer greater than or equal to x. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...119#119.
In fact, we would normally write 120#120, but we have not yet seen the 1#1 notation which is introduced in Chapter gif. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...rule
Guillaume François Antoine de L'Hôpital, marquis de Sainte-Mesme, is known for his rule for computing limits which states that if 357#357 and 358#358, then

359#359

where f'(n) and g'(n) are the first derivatives with respect to n of f(n) and g(n), respectively. The rule is also effective if 360#360 and 361#361. 

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...commensurate.
Functions which are commensurate  are functions which can be compared one with the other. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...436#436.
This notion of the looseness (tightness ) of an asymptotic bound is related to but not exactly the same as that given in Definition gif. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...numbers.
Fibonacci numbers are named in honor of Leonardo Pisano (Leonardo of Pisa), the son of Bonaccio (in Latin, Filius Bonaccii), who discovered the series in 1202. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...numbers.
These running times were measured on an Intel Pentium III, which has a 1 GHz clock and 256MB RAM under the WindowsME operating system. The programs were compiled using the C# compiler provided with the Microsoft .NET beta SDK (csc) and run under the Microsoft common language runtime. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...represents.
The address attribute is sometimes called its l-value  and the value attribute is sometimes called is r-value . This terminology arises from considering the semantics of an assignment statement such as y = x. The meaning of such as statement is ``take the value of variable x and store it in memory at the address of variable y.'' So, when a variable appears on the right-hand-side of an assignment, we use its r-value; and when it appears on the left-hand-size, we use its l-value. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...class
For a complete list of the methods defined in the System.Object class, you should consult The C# Language Specifications[22]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...NAME=7497> .
The word deque is usually pronounced like ``deck'' and sometimes like ``deek.'' 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...order
A total order is a relation, say <, defined on a set of elements, say 798#798, with the following properties:
  1. For all pairs of elements 799#799, such that 800#800, exactly one of either i<j or j<i holds. (All elements are commensurate ).
  2. For all triples 801#801, 802#802. (The relation 394#394 is transitive ).

(See also Definition gif). 

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...

825#825

This is the Swedish word for the number two. The symbol å in the Unicode character set  can be represented in a C# program using the Unicode escape  ``u00E5''. 

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...

825#825

I have been advised that a book with out sex will never be a best seller. ``Sex'' is the Swedish word for the number six. 

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...prime
Two numbers x and y are relatively prime   if there is no number other than one that divides both x and y evenly. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...i.
What else would it be? 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...NAME=14631> .
Isomorphic is a fancy word that means being of identical or similar form or shape or structure. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...Landis
Russian mathematicians G. M. Adel'son-Vel'skiı and E. M. Landis published this result in 1962. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...trees.
Obviously since B-Trees are M-way trees, the ``B'' in B-Tree does not stand for binary. B-Trees were invented by R. Bayer and E. McCright in 1972, so the ``B'' either stands for balanced or Bayer-take your pick. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...HREF="page386.html#exercisepqueuesbinom">gif).
Isaac Newton  discovered the binomial theorem in 1676 but did not publish a proof. Leonhard Euler  attempted a proof in 1774. Karl Friedrich Gauss  produced the first correct proof in 1812. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...subsets.
Stirling numbers of the second kind  are given by the formula

1619#1619

where n>0 and 1620#1620. 

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...structures.
Mark-and-sweep garbage collection is described by John McCarthy in a paper on the LISP language published in 1960. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...space.
The reader may find it instructive to compare Program gif with Program gif and Program gif. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...space.
The reader may find it instructive to compare Program gif with Program gif and Program gif. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...NAME=33188> .
The table is named in honor of Blaise Pascal  who published a treatise on the subject in 1653. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...number!
Prime numbers of the form 1917#1917 are known as Mersenne primes . 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...1920#1920.
For convenience, we use the notation 1921#1921 to denote 1922#1922. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...NAME=35116> .
Unfortunately, the fame of bubble sort exceeds by far its practical value. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...zero.
There is also the symmetrical case in which i is always n-1. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

Bruno Copyright © 2001 by Bruno R. Preiss, P.Eng. All rights reserved.